Department of Mathematics at UTSA - User contributions [en]

MAT5343

2025-01-24T14:15:49Z

Duynguyenvu.hoang:

MAT 5343 Differential Geometry

'''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022)

'''Catalog entry''':

''Prerequisite'': A course in Linear Algebra and Calculus sequence

''Contents'':
This course offers an in-depth study of differential geometry, focusing on the geometric properties of differentiable manifolds. Topics include smooth manifolds, vector fields, differential forms, tensors, connections, Riemannian metrics, and curvature. Emphasis will be placed on understanding the interplay between geometry and topology, as well as applications in physics, particularly in general relativity.

==Topics List==
{| class="wikitable sortable"
! Week !! Topic !! Chapter from textbook !! SLO
|-
| 1
|| [[Curves in space and on the plane]]
||
|| Understand how to compute the curvature of a space/planar curve
|-
| 2
|| [[Moving frames, Serret-Frenet formulas]]
|| 2.1-2.5
|| Understand how apply the Serret-Frenet formulas, determine curve from given curvature and torsion
|-
| 3
|| [[Surfaces in space, introduction to differential forms]]
|| 3.1-3.4
|| Parameterize surfaces in space, compute and interpret first and second fundamental form
|-
| 4
|| [[Gaussian curvature]]
|| 3.5-3.7
|| Compute and interpret Gaussian curvature, understand Theorema Egregium.
|-
| 5
|| [[Covariant differentiation on surfaces and parallel transport]]
|| 3.8-3.11
|| Become fluent with tensor calculus and covariant differentiation
|-
| 6
|| [[Gauss-Codazzi equations and Riemann curvature tensor]]
|| 3.13-3.14
|| Be able to compute Riemann curvature tensor, become familiar with symmetry properties of the Riemann tensor
|-
| 7
|| [[Riemannian manifolds I]]
|| 5.1-5.5
|| Understand definition of Riemannian manifolds
|-
| 8
|| [[Riemannian manifolds II]]
|| 5.6-5.14
|| Understand properties of Riemannian manifolds
|-
|-
| 9
|| [[Introduction into Lorentzian manifolds and relativity]]
|| instructor provided material
|| Understand the difference between Riemannian and Lorentzian manifolds
|-
| 10
|| [[Einstein manifolds]]
|| instructor provided material
|| Understand where Einstein equations come from
|-
| 11
|| [[Global theorems in differential geometry]]
|| 3.7, 5.4
|| Hopf-Rinow and Gauss-Bonnet theorem
|-
| 12
|| [[Differential forms and integral theorems]]
|| instructor provided material
|| Be able to apply Stokes' theorem
|-
| 13
|| [[Introduction to Lie groups]]
|| 6.1-6.5
|| Be able to compute generators of a Lie group
|-
| 14
|| [[Auxillary topics]]
||
||
|}

MAT5343

2025-01-24T13:30:33Z

Duynguyenvu.hoang:

MAT 5343 Differential Geometry

'''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022)

'''Catalog entry''':

''Prerequisite'': A course in Linear Algebra and Calculus sequence

''Contents'':
This course offers an in-depth study of differential geometry, focusing on the geometric properties of differentiable manifolds. Topics include smooth manifolds, vector fields, differential forms, tensors, connections, Riemannian metrics, and curvature. Emphasis will be placed on understanding the interplay between geometry and topology, as well as applications in physics, particularly in general relativity.

==Topics List==
{| class="wikitable sortable"
! Week !! Topic !! Chapter from textbook !! SLO
|-
| 1
|| [[Curves in space and on the plane]]
||
|| Understand how to compute the curvature of a space/planar curve
|-
| 2
|| [[Moving frames, Serret-Frenet formulas]]
|| 2.1-2.5
|| Understand how apply the Serret-Frenet formulas, determine curve from given curvature and torsion
|-
| 3
|| [[Surfaces in space, introduction to differential forms]]
|| 3.1-3.4
|| Parameterize surfaces in space, compute and interpret first and second fundamental form
|-
| 4
|| [[Gaussian curvature]]
|| 3.5-3.7
|| Compute and interpret Gaussian curvature
|-
| 5
|| [[Covariant differentiation on surfaces and parallel transport]]
|| 3.8-3.11
||
|-
| 6
|| [[Gauss-Codazzi equations and Riemann curvature tensor]]
|| 3.13-3.14
||
|-
| 7
|| [[Riemannian manifolds I]]
|| 5.1-5.5
||
|-
| 8
|| [[Riemannian manifolds II]]
|| 5.6-5.14
||
|-
| 9
|| [[Introduction into Lorentzian manifolds and relativity]]
|| instructor provided material
||
|-
| 10
|| [[Einstein manifolds]]
|| instructor provided material
||
|-
| 11
|| [[Global theorems in differential geometry]]
|| 3.7, 5.4
|| Hopf-Rinow and Gauss-Bonnet theorem
|-
| 12
|| [[Differential forms and integral theorems]]
|| instructor provided material
||
|-
| 13
|| [[Introduction to Lie groups]]
|| 6.1-6.5
||
|-
| 14
|| [[Auxillary topics]]
||
||
|}

MAT5343

2025-01-23T21:48:49Z

Duynguyenvu.hoang:

MAT5343

2025-01-23T21:34:47Z

Duynguyenvu.hoang:

MAT 5343 Differential Geometry

'''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022)

'''Catalog entry'''

''Prerequisite'': A course in Linear Algebra and Calculus sequence

''Contents'':
(1)

==Topics List==
{| class="wikitable sortable"
! Week !! Topic !! Chapter from textbook !! SLO
|-
| 1
|| [[Curves in space and on the plane]]
|| 1
|| Understand how to compute the curvature of a space/planar curve
|-
| 2-4
|| [[Quantum computing]]
|| 2
||
|-
| 5-6
|| [[Hash-based Digital Signature Schemes]]
|| 3
||
|-
| 7-8
|| [[Code-based cryptography]]
|| 4
||
|-
| 9-10
|| [[Lattice-based Cryptography]]
|| 5
||
|-
| 11-12
|| [[Multivariate Public Key Cryptography]]
|| 6
||
|-
| 13-end
|| [[Homomorphic encryption]]
||
||
|}

MAT5343

2025-01-23T21:32:49Z

Duynguyenvu.hoang:

MAT 5343 Differential Geometry

'''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022)

'''Catalog entry'''

''Prerequisite'': A course in Linear Algebra and Calculus sequence

''Contents'':
(1)

==Topics List==
{| class="wikitable sortable"
! Week !! Topic !! Chapter 1 from the Bernstein-Johannes Buchmann-Dahmen book !! MAT1313, or CS2233/2231, or instructor consent.
|-
| 1-3
|| [[Introduction to post-quantum cryptography]]
|| 1
|| MAT1313 or CS2233/2231, or equivalent.
|-
| 2-4
|| [[Quantum computing]]
|| 2
||
|-
| 5-6
|| [[Hash-based Digital Signature Schemes]]
|| 3
||
|-
| 7-8
|| [[Code-based cryptography]]
|| 4
||
|-
| 9-10
|| [[Lattice-based Cryptography]]
|| 5
||
|-
| 11-12
|| [[Multivariate Public Key Cryptography]]
|| 6
||
|-
| 13-end
|| [[Homomorphic encryption]]
||
||
|}

MAT5343

2025-01-23T21:30:19Z

Duynguyenvu.hoang: Created page with "MAT 5343 Differential Geometry '''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022) '''Catalog entry''' ''Prerequisite'': Linear Algebra and Ca..."

MAT 5343 Differential Geometry

'''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022)

'''Catalog entry'''

''Prerequisite'': Linear Algebra and Calculus sequence

''Contents'':
(1) Introduction to post-quantum cryptography.
(2) Quantum computing.
(3) Hash-based Digital Signature Schemes.
(4) Code-based cryptography.
(5) Lattice-based Cryptography
(6) Multivariate Public Key Cryptography.
(7) Homomorphic encryption.

==Topics List==
{| class="wikitable sortable"
! Week !! Topic !! Chapter 1 from the Bernstein-Johannes Buchmann-Dahmen book !! MAT1313, or CS2233/2231, or instructor consent.
|-
| 1-3
|| [[Introduction to post-quantum cryptography]]
|| 1
|| MAT1313 or CS2233/2231, or equivalent.
|-
| 2-4
|| [[Quantum computing]]
|| 2
||
|-
| 5-6
|| [[Hash-based Digital Signature Schemes]]
|| 3
||
|-
| 7-8
|| [[Code-based cryptography]]
|| 4
||
|-
| 9-10
|| [[Lattice-based Cryptography]]
|| 5
||
|-
| 11-12
|| [[Multivariate Public Key Cryptography]]
|| 6
||
|-
| 13-end
|| [[Homomorphic encryption]]
||
||
|}

MAT4143

2023-03-24T14:17:25Z

Duynguyenvu.hoang: /* Course description */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics. This course is also suitable for Mathematics majors, who can deepen their understanding of applications and proofs of major theorems in Functional Analysis.

'''Textbooks:'''

* Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press.
* Grimshaw, R. (1993). Nonlinear Ordinary Differential Equations (1st ed.). Routledge.
* R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Wiley Classics Library, John Wiley & Sons Inc., New York, 1989.
* Bender, C. and Orszag, S. Advanced Mathematical Methods for Scientists and Engineers. Mc-Graw Hill.
* Kreyszig, E. (1989). Introductory Functional Analysis with Applications. Wiley.
* Methods of Applied Mathematics. Todd Arbogast and Jerry L. Bona. Department of Mathematics, and Institute for Computational Engineering and Sciences, University of Texas at Austin, 2008.
* P. J. Olver , Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993.
* Rutherford Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover Publications

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
*Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||

||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* Systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||

||
* Phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* Standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
* Soliton solutions, infinite hierarchy of conservation laws
|}

MAT4143

2023-03-24T14:00:34Z

Duynguyenvu.hoang: /* Course description */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics. This course is also suitable for Mathematics majors, who can deepen their understanding of applications and proofs of major theorems in Functional Analysis.

'''Textbooks:'''

* Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press.
* Grimshaw, R. (1993). Nonlinear Ordinary Differential Equations (1st ed.). Routledge.
* R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Wiley Classics Library, John Wiley & Sons Inc., New York, 1989.
* Bender, C. and Orszag, S. Advanced Mathematical Methods for Scientists and Engineers. Mc-Graw Hill.
* Kreyszig, E. (1989). Introductory Functional Analysis with Applications. Wiley.
* Methods of Applied Mathematics. Todd Arbogast and Jerry L. Bona. Department of Mathematics, and Institute for Computational Engineering and Sciences, University of Texas at Austin, 2008.
* P. J. Olver , Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993.
* Rutherford Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover
Publications

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
*Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||

||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* Systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||

||
* Phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* Standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
* Soliton solutions, infinite hierarchy of conservation laws
|}

MAT4143

2023-03-24T14:00:06Z

Duynguyenvu.hoang: /* Course description */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics. This course is also suitable for Mathematics majors, who can deepen their understanding of applications and proofs of major theorems in Functional Analysis.

'''Textbooks:'''

* Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press.
* Grimshaw, R. (1993). Nonlinear Ordinary Differential Equations (1st ed.). Routledge.
* R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differ�ential Equations, Wiley Classics Library, John Wiley & Sons Inc., New York, 1989.
* Bender, C. and Orszag, S. Advanced Mathematical Methods for Scientists and Engi�neers. Mc-Graw Hill.
* Kreyszig, E. (1989). Introductory Functional Analysis with Applications. Wiley.
* Methods of Applied Mathematics. Todd Arbogast and Jerry L. Bona. Department of
Mathematics, and Institute for Computational Engineering and Sciences, University of
Texas at Austin, 2008.
* P. J. Olver , Applications of Lie Groups to Differential Equations, Second Edition,
Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993.
* Rutherford Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover
Publications

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
*Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||

||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* Systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||

||
* Phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* Standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
* Soliton solutions, infinite hierarchy of conservation laws
|}

MAT 5673

2023-03-24T13:57:48Z

Duynguyenvu.hoang: /* Course description */

==Course description==
Partial differential equations arise in many different areas as one tries
to describe the behavior of a system ruled by some law. Typically, this has to do with
some physical process such as heat diffusion in a material, vibrations of a bridge, circulation
of fluids, the behavior of microscopic particles or the evolution of the universe as a whole.
Modeling by means of partial differential equations has been successful in other disciplines
as well, like in the case of the Black-Scholes equation for stock options pricing and the
Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are
an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.

'''Textbooks:
'''
* P. Olver: Introduction to Partial Differential Equations (Undergraduate Texts in Mathematics) 1st ed. 2014, Corr. 3rd printing 2016
* L.C. Evans: Partial Differential Equations: Second Edition (Graduate Studies in Mathematics) 2nd Edition

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Introduction and classification of PDE, Calculus review
||
Multivariable Calculus, Chain Rule
||
* Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
|-
|Week 2
||
*
||
Applied examples of PDE
||
Multivariable Calculus, Chain Rule
||
* Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
|-
|Week 3
||
*
||
The method of characteristics for first-order quasilinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving quasilinear first-order equations using the method of characteristics
|-
|Week 4
||
*
||
The method of characteristics for first-order fully nonlinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
|-
|Week 5
||
*
||
Heat and wave equation on the whole real line
||
Differentiation of integrals with respect to a parameter, integration by parts
||
* Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
|-
|Week 6
||
*
||
Initial-boundary value problem for heat and wave equation I
||
Partial derivatives, chain rule
||
* Separation of variables method for heat and wave equation
|-
|Week 7
||
*
||
Initial-boundary value problem for heat and wave equation II, introduction to Fourier series
||
Partial derivatives, chain rule
||
* Forming more general solutions out of infinite superposition of basic solutions
|-
|Week 8
||
*
||
Introduction to Fourier series
||
Infinite series
||
* Orthonormal systems of functions, spectral method for the wave and heat equation
|-
|Week 9
||
*
||
Schroedinger equation
||
Complex numbers
||
* Basic properties of Schroedinger equation, particle in a potential well
|-
|Week 10
||

||
Qualitative properties of PDE's
||
Differentiation of integrals with respect to parameter
||
* Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
|-
|Week 11
||
*
||
Introduction to numerical methods for PDE (optional)
||
Derivatives, Calculus, Matrices, Linear Algebra
||
* Basic finite difference schemes for first-order quasilinear equations, CFL condition
|-
|Week 12
||

||
Introduction to the Laplace and Poisson equation
||
*
||
* Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
|-
|Week 13
||
*
||
Introduction to the Calculus of Variations
||
Differentiation of an integral with respect to a parameter, parametric surfaces
||
* Compute the variational derivative of a functional
|-
|Week 14
||
*
||
Review, advanced topics
||
*
||
*
|}

AIM 5113

2023-03-23T14:17:10Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
This course introduces students to mathematical techniques useful in an industrial setting.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Interpolation, grid spacing, intro to MATLAB. Basis functions. Numerical integration. Latex and MATLAB basics.
|-
|Week 2
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Regression, Least Squares, Inner Products
|-
|Week 3
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Hilbert Space, Trigonometric polynomials
|-
|Week 4
||
*
||
Statistical Reasoning, Distributions
||

||
* Understanding probabilistic and statistical reasoning, Excel, data visualization
|-
|Week 5
||
*
||
Monte Carlo Method
||

||
* Numerical integration, implementing the Monte-Carlo method
|-
|Week 6
||
*
||
Introduction to ODEs
||

||
* Statement of basic theorems, higher-order equations
|-
|Week 7
||
*
||
Boundary conditions, phase portraits, Finite difference methods.
||

||
* Learn to implement finite difference methods
|-
|Week 8
||
*
||
Frequency domain methods, Laplace transforms
||
*
||
* Applying transform methods to solve ODEs
|-
|Week 9
||
*
||
Control theory
||
*
||
* Designing and implementing control for systems of ODEs
|-
|Week 10
||
*
||
Introduction to PDEs
||
Derivatives
||
* Basic PDEs such as Laplace and Wave equation
|-
|Week 11
||
*
||
Signal processing and data acquisition
||
*
||
* z-transform, Discrete Fourier Transform
|-
|Week 12
||
*
||
Fourier and Laplace transform revisited
||
*
||
* Using transform methods for signal processing
|-
|Week 13
||
*
||
Optimization I
||

||
* Linear vs Nonlinear programming
|-
|Week 14
||
*
||
Optimization II
||
*
||
* Optimal control, Neural Networks
|}

AIM 5113

2023-03-23T14:16:25Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
This course introduces students to mathematical techniques useful in an industrial setting.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Interpolation, grid spacing, intro to MATLAB. Basis functions. Numerical integration. Latex and MATLAB basics.
|-
|Week 2
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Regression, Least Squares, Inner Products
|-
|Week 3
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Hilbert Space, Trigonometric polynomials
|-
|Week 4
||
*
||
Statistical Reasoning, Distributions
||

||
* Understanding probabilistic and statistical reasoning, Excel, data visualization
|-
|Week 5
||
*
||
Monte Carlo Method
||

||
* Numerical integration, implementing the Monte-Carlo method
|-
|Week 6
||
*
||
Introduction to ODEs
||

||
* Statement of basic theorems, higher-order equations
|-
|Week 7
||
*
||
Boundary conditions, phase portraits, Finite difference methods.
||

||
* Learn to implement finite difference methods
|-
|Week 8
||
*
||
Frequency domain methods, Laplace transforms
||
*
||
* Applying transform methods to solve ODEs
|-
|Week 9
||
*
||
Control theory
||
*
||
* Designing and implementing control for systems of ODEs
|-
|Week 10
||
*
||
Introduction to PDEs
||
Derivatives
||
* Basic PDEs such as Laplace and Wave equation
|-
|Week 11
||
*
||
Signal processing and data acquisition
||
*
||
* z-transform, Discrete Fourier Transform
|-
|Week 12
||
*
||
Fourier and Laplace transform revisited
||
*
||
*
|-
|Week 13
||
*
||
Optimization I
||

||
* Linear vs Nonlinear programming
|-
|Week 14
||
*
||
Optimization II
||
*
||
* Optimal control, Neural Networks
|}

AIM 5113

2023-03-23T14:07:11Z

Duynguyenvu.hoang:

==Course description==
This course introduces students to mathematical techniques useful in an industrial setting.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Interpolation, grid spacing, intro to MATLAB. Basis functions. Numerical integration. Latex and MATLAB basics.
|-
|Week 2
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Regression, Least Squares, Inner Products
|-
|Week 3
||
*
||
Model Fitting, Basis Functions and Applications of the Inner Product
||

||
* Hilbert Space, Trigonometric polynomials
|-
|Week 4
||
*
||
Monte-Carlo methods (continued)
||

||
* Learn to implement Monte-Carlo methods in Python/MATLAB
|-
|Week 5
||
*
||
Introduction to Data Acquisition and Manipulation
||

||
* z-transform, filters, stability, error analysis, aliasing
|-
|Week 6
||
*
||
Discrete Fourier transform
||

||
* Properties of the Fourier transform
|-
|Week 7
||
*
||
Linear Regression and Optimization Problems
||

||
*
|-
|Week 8
||
*
||
Cost-Benefit Analysis
||
*
||
*
|-
|Week 9
||
*
||
Basic of Economic Analysis
||
*
||
*
|-
|Week 10
||
*
||
Introduction to Ordinary Differential Equations
||
Derivatives
||
*
|-
|Week 11
||
*
||
Introduction to Partial Differential Equations
||
*
||
*
|-
|Week 12
||
*
||
Finite Differences
||
*
||
*
|-
|Week 13
||
*
||
Writing Technical Reports
||

||
*
|-
|Week 14
||
*
||
Review, advanced topics
||
*
||
*
|}

MAT 5673

2023-03-23T13:50:33Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Partial differential equations arise in many different areas as one tries
to describe the behavior of a system ruled by some law. Typically, this has to do with
some physical process such as heat diffusion in a material, vibrations of a bridge, circulation
of fluids, the behavior of microscopic particles or the evolution of the universe as a whole.
Modeling by means of partial differential equations has been successful in other disciplines
as well, like in the case of the Black-Scholes equation for stock options pricing and the
Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are
an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Introduction and classification of PDE, Calculus review
||
Multivariable Calculus, Chain Rule
||
* Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
|-
|Week 2
||
*
||
Applied examples of PDE
||
Multivariable Calculus, Chain Rule
||
* Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
|-
|Week 3
||
*
||
The method of characteristics for first-order quasilinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving quasilinear first-order equations using the method of characteristics
|-
|Week 4
||
*
||
The method of characteristics for first-order fully nonlinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
|-
|Week 5
||
*
||
Heat and wave equation on the whole real line
||
Differentiation of integrals with respect to a parameter, integration by parts
||
* Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
|-
|Week 6
||
*
||
Initial-boundary value problem for heat and wave equation I
||
Partial derivatives, chain rule
||
* Separation of variables method for heat and wave equation
|-
|Week 7
||
*
||
Initial-boundary value problem for heat and wave equation II, introduction to Fourier series
||
Partial derivatives, chain rule
||
* Forming more general solutions out of infinite superposition of basic solutions
|-
|Week 8
||
*
||
Introduction to Fourier series
||
Infinite series
||
* Orthonormal systems of functions, spectral method for the wave and heat equation
|-
|Week 9
||
*
||
Schroedinger equation
||
Complex numbers
||
* Basic properties of Schroedinger equation, particle in a potential well
|-
|Week 10
||

||
Qualitative properties of PDE's
||
Differentiation of integrals with respect to parameter
||
* Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
|-
|Week 11
||
*
||
Introduction to numerical methods for PDE (optional)
||
Derivatives, Calculus, Matrices, Linear Algebra
||
* Basic finite difference schemes for first-order quasilinear equations, CFL condition
|-
|Week 12
||

||
Introduction to the Laplace and Poisson equation
||
*
||
* Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
|-
|Week 13
||
*
||
Introduction to the Calculus of Variations
||
Differentiation of an integral with respect to a parameter, parametric surfaces
||
* Compute the variational derivative of a functional
|-
|Week 14
||
*
||
Review, advanced topics
||
*
||
*
|}

MAT4143

2023-03-23T13:49:28Z

Duynguyenvu.hoang:

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics. This course is also suitable for Mathematics majors, who can deepen their understanding of applications and proofs of major theorems in Functional Analysis.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
*Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||

||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* Systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||

||
* Phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* Standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
* Soliton solutions, infinite hierarchy of conservation laws
|}

MAT4143

2023-03-23T13:42:43Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
*Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||

||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* Systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||

||
* Phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* Standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
* Soliton solutions, infinite hierarchy of conservation laws
|}

MAT4143

2023-03-23T13:41:25Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
*Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||

||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* Systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||

||
* Phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* Standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
*
|}

MAT4143

2023-03-23T13:39:03Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
*Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||

||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||

||
* phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
*
|}

MAT4143

2023-03-23T13:36:20Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||

||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||
Differentiation of integrals with respect to parameter
||
* phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
*
|}

MAT4143

2023-03-23T13:36:02Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||
Multivariable Calculus, Chain Rule
||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||
Differentiation of integrals with respect to parameter
||
* phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* standard examples, qualitative properties, conservation laws

|-
|Week 12
||

||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
*
|}

MAT4143

2023-03-23T13:35:10Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Complex Analysis Part I
||

||
* Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|-
|Week 2
||
*
||
Complex Analysis Part II
||

||
Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|-
|Week 3
||
*
||
Complex Analysis Part III
||
Multivariable Calculus, Chain Rule
||
* Harmonic functions and Poisson's formula
|-
|Week 4
||
*
||
Tensor Calculus Basics I
||

||
* Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|-
|Week 5
||
*
||
Tensor Caluclus Basics II
||

||
* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|-
|Week 6
||
*
||
Applied Functional Analysis Part I
||

||
* Hilbert spaces and inner products, orthogonality and completeness.
|-
|Week 7
||
*
||
Applied Functional Analysis Part II
||

||
* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|-
|Week 8
||
*
||
Applied Functional Analysis Part III
||

||
* Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|-
|Week 9
||
*
||
Overview about ordinary differential equations I
||

||
* systems of nonlinear/linear equations, basic existence and uniqueness theorems
|-
|Week 10
||

||
Overview about ordinary differential equations II
||
Differentiation of integrals with respect to parameter
||
* phase-plane, linearization, stability, chaos
|-
|Week 11
||
*
||
PDE's of Mathematical Physics
||

||
* standard examples, qualitative properties, conservation laws

|-
|Week 12
||
Matrices, Linear Algebra
||
Introduction to Lie Groups and Symmetries I
||
*
||
* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|-
|Week 13
||
*
||
Introduction to Lie Groups and Symmetries II
||

||
* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|-
|Week 14
||
*
||
KdV equation, completely integrable systems
||
*
||
*
|}

MAT4143

2023-03-23T13:27:47Z

Duynguyenvu.hoang: Created page with "==Course description== Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used i..."

==Course description==
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Introduction and classification of PDE, Calculus review
||
Multivariable Calculus, Chain Rule
||
* Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
|-
|Week 2
||
*
||
Applied examples of PDE
||
Multivariable Calculus, Chain Rule
||
* Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
|-
|Week 3
||
*
||
The method of characteristics for first-order quasilinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving quasilinear first-order equations using the method of characteristics
|-
|Week 4
||
*
||
The method of characteristics for first-order fully nonlinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
|-
|Week 5
||
*
||
Heat and wave equation on the whole real line
||
Differentiation of integrals with respect to a parameter, integration by parts
||
* Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
|-
|Week 6
||
*
||
Initial-boundary value problem for heat and wave equation I
||
Partial derivatives, chain rule
||
* Separation of variables method for heat and wave equation
|-
|Week 7
||
*
||
Initial-boundary value problem for heat and wave equation II, introduction to Fourier series
||
Partial derivatives, chain rule
||
* Forming more general solutions out of infinite superposition of basic solutions
|-
|Week 8
||
*
||
Introduction to Fourier series
||
Infinite series
||
* Orthonormal systems of functions, spectral method for the wave and heat equation
|-
|Week 9
||
*
||
Schroedinger equation
||
Complex numbers
||
* Basic properties of Schroedinger equation, particle in a potential well
|-
|Week 10
||
Complex numbers
||
Qualitative properties of PDE's
||
Differentiation of integrals with respect to parameter
||
* Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
|-
|Week 11
||
*
||
Introduction to numerical methods for PDE (optional)
||
Derivatives, Calculus
||
* Basic finite difference schemes for first-order quasilinear equations, CFL condition
|-
|Week 12
||
Matrices, Linear Algebra
||
Introduction to the Laplace and Poisson equation
||
*
||
* Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
|-
|Week 13
||
*
||
Introduction to the Calculus of Variations
||
Differentiation of an integral with respect to a parameter, parametric surfaces
||
* Compute the variational derivative of a functional
|-
|Week 14
||
*
||
Review, advanced topics
||
*
||
*
|}

MAT 5673

2023-03-23T13:21:15Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Partial differential equations arise in many different areas as one tries
to describe the behavior of a system ruled by some law. Typically, this has to do with
some physical process such as heat diffusion in a material, vibrations of a bridge, circulation
of fluids, the behavior of microscopic particles or the evolution of the universe as a whole.
Modeling by means of partial differential equations has been successful in other disciplines
as well, like in the case of the Black-Scholes equation for stock options pricing and the
Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are
an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Introduction and classification of PDE, Calculus review
||
Multivariable Calculus, Chain Rule
||
* Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
|-
|Week 2
||
*
||
Applied examples of PDE
||
Multivariable Calculus, Chain Rule
||
* Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
|-
|Week 3
||
*
||
The method of characteristics for first-order quasilinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving quasilinear first-order equations using the method of characteristics
|-
|Week 4
||
*
||
The method of characteristics for first-order fully nonlinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
|-
|Week 5
||
*
||
Heat and wave equation on the whole real line
||
Differentiation of integrals with respect to a parameter, integration by parts
||
* Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
|-
|Week 6
||
*
||
Initial-boundary value problem for heat and wave equation I
||
Partial derivatives, chain rule
||
* Separation of variables method for heat and wave equation
|-
|Week 7
||
*
||
Initial-boundary value problem for heat and wave equation II, introduction to Fourier series
||
Partial derivatives, chain rule
||
* Forming more general solutions out of infinite superposition of basic solutions
|-
|Week 8
||
*
||
Introduction to Fourier series
||
Infinite series
||
* Orthonormal systems of functions, spectral method for the wave and heat equation
|-
|Week 9
||
*
||
Schroedinger equation
||
Complex numbers
||
* Basic properties of Schroedinger equation, particle in a potential well
|-
|Week 10
||
Complex numbers
||
Qualitative properties of PDE's
||
Differentiation of integrals with respect to parameter
||
* Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
|-
|Week 11
||
*
||
Introduction to numerical methods for PDE (optional)
||
Derivatives, Calculus
||
* Basic finite difference schemes for first-order quasilinear equations, CFL condition
|-
|Week 12
||
Matrices, Linear Algebra
||
Introduction to the Laplace and Poisson equation
||
*
||
* Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
|-
|Week 13
||
*
||
Introduction to the Calculus of Variations
||
Differentiation of an integral with respect to a parameter, parametric surfaces
||
* Compute the variational derivative of a functional
|-
|Week 14
||
*
||
Review, advanced topics
||
*
||
*
|}

MAT 5653

2023-03-14T14:05:00Z

Duynguyenvu.hoang: Created page with "==Topics List== {| class="wikitable sortable" ! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes |- |Week 1 || * || Basic Concepts of Ordinary D..."

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Basic Concepts of Ordinary Differential Equations
||

||
*
|-
|Week 2
||
*
||
Equilibrium Points, Phase Plane
||

||
*
|-
|Week 3
||
*
||
ODE Models in Science and Engineering
||

||

|-
|Week 4
||
*
||
Picard-Lindeloef Theorem and Applications
||

||
* Picard Iteration, Contraction
|-
|Week 5
||
*
||
Peano Theorem
||

||
*
|-
|Week 6
||
*
||
Dependence of Solutions on Initial Values, Maximal Interval of Existence
||

||
*
|-
|Week 7
||
*
||
Conservative Systems
||

||
*
|-
|Week 8
||
*
||
Linear Systems and Exponentials
||
*
||
*
|-
|Week 9
||
*
||
Stable Manifold Theorem
||
*
||
*
|-
|Week 10
||
*
||
Hartman-Grobman Theorem
||

||
*
|-
|Week 11
||
*
||
Stability and Lyapunov functions
||
*
||
*
|-
|Week 12
||
*
||
Introduction to Bifurcation Theory/Hopf Bifurcation
||
*
||
*
|-
|Week 13
||
*
||
Epidemic and Population Models
||
*
||
*
|-
|Week 14
||
*
||
Review, advanced topics
||
*
||
*
|}

AIM 5113

2023-03-13T14:20:52Z

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Introduction to statistical reasoning and random variables
||

||
* Uniform, Gaussian and Poisson distributions
|-
|Week 2
||
*
||
Introduction to statistical reasoning and random variables (continued)
||

||
* Taguchi quality control
|-
|Week 3
||
*
||
Monte-Carlo methods
||

||
* Computing integrals using the Monte-Carlo method
|-
|Week 4
||
*
||
Monte-Carlo methods (continued)
||

||
* Learn to implement Monte-Carlo methods in Python/MATLAB
|-
|Week 5
||
*
||
Introduction to Data Acquisition and Manipulation
||

||
* z-transform, filters, stability, error analysis, aliasing
|-
|Week 6
||
*
||
Discrete Fourier transform
||

||
* Properties of the Fourier transform
|-
|Week 7
||
*
||
Linear Regression and Optimization Problems
||

||
*
|-
|Week 8
||
*
||
Cost-Benefit Analysis
||
*
||
*
|-
|Week 9
||
*
||
Basic of Economic Analysis
||
*
||
*
|-
|Week 10
||
*
||
Introduction to Ordinary Differential Equations
||
Derivatives
||
*
|-
|Week 11
||
*
||
Introduction to Partial Differential Equations
||
*
||
*
|-
|Week 12
||
*
||
Finite Differences
||
*
||
*
|-
|Week 13
||
*
||
Writing Technical Reports
||

||
*
|-
|Week 14
||
*
||
Review, advanced topics
||
*
||
*
|}

MAT 5673

2023-03-12T15:29:51Z

Duynguyenvu.hoang: /* Topics List */

==Course description==
Partial differential equations arise in many different areas as one tries
to describe the behavior of a system ruled by some law. Typically, this has to do with
some physical process such as heat diffusion in a material, vibrations of a bridge, circulation
of fluids, the behavior of microscopic particles or the evolution of the universe as a whole.
Modeling by means of partial differential equations has been successful in other disciplines
as well, like in the case of the Black-Scholes equation for stock options pricing and the
Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are
an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.
==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
Introduction and classification of PDE, Calculus review
||
Multivariable Calculus, Chain Rule
||
* Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
|-
|Week 2
||
*
||
Applied examples of PDE
||
Multivariable Calculus, Chain Rule
||
* Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
|-
|Week 3
||
*
||
The method of characteristics for first-order quasilinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving quasilinear first-order equations using the method of characteristics
|-
|Week 4
||
*
||
The method of characteristics for first-order fully nonlinear equations
||
Multivariable Calculus, Chain Rule
||
* Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
|-
|Week 5
||
*
||
Heat and wave equation on the whole real line
||
Differentiation of integrals with respect to a parameter, integration by parts
||
* Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
|-
|Week 6
||
*
||
Initial-boundary value problem for heat and wave equation I
||
Partial derivatives, chain rule
||
* Separation of variables method for heat and wave equation
|-
|Week 7
||
*
||
Initial-boundary value problem for heat and wave equation II, introduction to Fourier series
||
Partial derivatives, chain rule
||
* Forming more general solutions out of infinite superposition of basic solutions
|-
|Week 8
||
*
||
Introduction to Fourier series
||
*
||
* Orthonormal systems of functions, spectral method for the wave and heat equation
|-
|Week 9
||
*
||
Schroedinger equation
||
*
||
* Basic properties of Schroedinger equation, particle in a potential well
|-
|Week 10
||
*
||
Qualitative properties of PDE's
||
Differentiation of integrals with respect to parameter
||
* Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
|-
|Week 11
||
*
||
Introduction to numerical methods for PDE (optional)
||
*
||
* Basic finite difference schemes for first-order quasilinear equations, CFL condition
|-
|Week 12
||
*
||
Introduction to the Laplace and Poisson equation
||
*
||
* Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
|-
|Week 13
||
*
||
Introduction to the Calculus of Variations
||
Differentiation of an integral with respect to a parameter
||
* Compute the variational derivative of a functional
|-
|Week 14
||
*
||
Review, advanced topics
||
*
||
*
|}

MAT 5673

2023-03-12T14:54:03Z

Duynguyenvu.hoang:

MAT5143

2023-03-10T23:07:54Z

Duynguyenvu.hoang: /* Topics List */

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
*
||
*
||
*
|-
}

MAT5143

2023-03-10T23:06:54Z

Duynguyenvu.hoang: /* Topics List */

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week I
||
*
||
*
||
*
||
*
|-
}

MAT5143

2023-03-10T23:06:16Z

Duynguyenvu.hoang: Created Math Phys page

MAT 5673

2023-03-10T20:25:58Z

Duynguyenvu.hoang: /* Topics List */

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
*
||
*
||
*
||
* Learning outcome
|}

MAT 5673

2023-03-10T20:23:28Z

Duynguyenvu.hoang: Created page for PDE1.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Order of Differential Equations]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Explain the basic notion of the order of a differential equation.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Solutions of Differential Equations]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Explain the basic notion of solutions of differential equations.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Initial Value Problem|Initial Value Problem (IVP)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Explain the basic notion of the initial values problem.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Cauchy Problem]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Explain the Cauchy Problem
* Explain the basic notion of existence and uniqueness of a solution to the Cauchy Problem.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Separation of Variables (1st Order)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Determine separable differential equations of the first order.
* Apply direct methods to evaluate exact solutions of separable differential equations of the first order.
|-
|Week II
||
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
||
* [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Determine homogeneous differential equations of the first order.
* Apply direct methods to evaluate exact solutions of homogeneous differential equations of the first order (substitutions).
* Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
|-
|Week II
||
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
||
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Determine linear differential equations of the first order.
* Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
|-
|Week II
||
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
||
* [[Integrating Factor]]
||
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
||
* Apply integrating factor to solve linear differential equations of the first order.
* Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
|-
|Week III
||
* Ahmad and Ambrosetti 2014, Ch. 3
||
* [[Bernoulli Equations (1st Order)]]
||
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
||
* Determine Bernoulli of the first order.
* Apply direct methods to evaluate exact solutions of Bernoulli of the first order.
|-
|Week III
||
* Ahmad and Ambrosetti 2014, Ch. 3
||
* [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
||
* [[Integrating Factor]] for exact equations.
* [[Partial Derivatives]]
||
* Determine Exact Differential Equations of the first order.
* Apply direct methods to evaluate exact solutions of Exact Differential Equations of the first order.
* Use the integrating factor technique for exact equations.
|-
|Week IV
||
* Ahmad and Ambrosetti 2014, Chaps. 1-3
||
* Overview of the solutions methods discussed so far (Chapters 1-3).
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
* [[Partial Derivatives]]
* First-order differential equations:
:- [[Separation of Variables (1st Order)]]
:- [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
:- [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
:- [[Bernoulli Equations (1st Order)]]
:- [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
||
* Determine the type of different classes of differential equations of the first order: separable, linear, homogeneous, Bernoulli, exact.
* Use direct methods to solve first order differential equations solved and not solved for the first derivative.
|-
|Week V
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Linear Independence of Functions]].
||
* [[Linear Independence of Vectors]].
||
* Understanding of Linear Independence of Functions.
|-
|Week V
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Linear Independence of Functions|Linear Dependence of Functions]].
||
* [[Linear Dependence of Vectors]].
||
* Understanding of Linear Dependence of Functions.
|-
|Week V
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Wronskian]]
||
* [[Linear Independence of Functions]].
* [[Linear Independence of Functions|Linear Dependence of Functions]].
* [[Determinant]].
||
* Showing linear independence of two functions using the Wronskian.
* Showing linear independence of two solutions of Linear Second-Order ODEs using the Wronskian.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Reduction of the Order]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Apply of the reduction of the order technique for second-order ODEs with a given solution.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Homogeneous Differential Equations|Linear Homogeneous Equations]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Determine homogeneous classes of differential equations of the second and higher order.
* Determine linear and non-linear classes of differential equations of the second and higher order.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Abel’s Theorem]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Determine Wronskian for a second-order ODE with 2 given solutions.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Fundamental Solutions]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Determine fundamental solutions.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Linear Differential Equations|Linear Non-homogeneous Equations]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Determine non-homogeneous classes of differential equations of the second and higher order.
* Determine linear and non-linear classes of differential equations of the second and higher order
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Variation Of Parameters (2nd Order)|Variation of Parameters (2nd Order)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
* [[Quadratic Equations]].
* [[Solutions of Linear Systems]].
||
* Apply of the variation of parameters technique for second-order ODEs.
|-
|Week VII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Variation Of Parameters (2nd Order)|Variation of Parameters (2nd Order)]] (continued)
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
* [[Quadratic Equations]].
* [[Solutions of Linear Systems]].
||
* Apply variation of parameters technique for second-order ODEs.
|-
|Week VII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Method of Undetermined Coefficients (2nd Order)]]
||
* [[Quadratic Equations]].
* [[Systems of Linear Equations]].
||
* Apply method of undetermined coefficients technique for second-order ODEs.
|-
|Week VIII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Non-linear 2nd Order ODEs]]
||
* [[Algebraic Equations]]
* [[Reduction of the Order]]
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Methods for nonlinear second-order ODEs.
* Apply reduction of the order method to some nonlinear second-order ODEs.
|-
|Week VIII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
||
* [[Variation Of Parameters|Variation of Parameters (2nd Order)]]
||
* Apply variation of parameters technique for higher-order ODEs
|-
|Week VIII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
||
* [[Method of Undetermined Coefficients (2nd Order)]]
||
* Apply method of undetermined coefficients technique for higher-order ODEs
|-
|Week IX
||
* Ahmad and Ambrosetti 2014, Ch. 6
||
* [[Linear Differential Equations|Linear Differential Equations (Higher Order)]]
||
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
* [[Variation Of Parameters|Variation of Parameters (Higher Order)]].
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]].
||
* Methods for linear higher-order ODEs
|-
|Week X
||
* Ahmad and Ambrosetti 2014, Chaps. 5, 6
||
* Overview of the solutions methods for second and higher order differential equations.
||
* [[Algebraic Equations]]
* Direct methods for second and higher-order ODEs:
:- [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
:- [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
||
* Evaluate the exact solutions of important classes of differential equations such as second order differential equations as well as some higher order differential equations.
|-
|Week X
||
* Ahmad and Ambrosetti 2014, Chaps. 10
||
* [[Power Series Solutions]]
||
* [[Power Series Induction]]
||
Apply power series method to evaluate solutions of first-order and second-order ODEs.
|-
|Week XI
||
* Ahmad and Ambrosetti 2014, Chaps. 10
||
* [[Power Series Solutions]] (continued)
||
* [[Power Series Induction]]
||
Apply power series method to evaluate solutions of first-order and second-order ODEs.
|-
|Week XII
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[Laplace Transform]]
||
* [[Functions]] of Single Variable.
* [[Continuity]] of functions of single variables.
* [[Derivatives]] of functions of single variables.
* [[Improper Integrals]] of functions of single variables with infinite limits.
||
* Definition and main properties of the L-transform.
|-
|Week XIII
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[Inverse Laplace Transform]]
||
* [[Laplace Transform]]
* [[Complex Derivatives]]
||
* Apply the theorem(s) for inverse L-transform.
|-
|Week XIV
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[Laplace Transform to ODEs]]
||
* [[Linear Differential Equations|Linear Equations]]
* [[Laplace Transform]]
* [[Inverse Laplace Transform]]
||
* Apply the Laplace transform as solution technique.
|-
|Week XIV
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[Laplace Transform to ODEs|Laplace Transform to Systems of ODEs]]
||
* [[Solutions of Linear Systems]].
* [[Laplace Transform]].
* [[Inverse Laplace Transform]].
||
* Apply the Laplace transform as solution technique.
|-
|Week XV
||
* Ahmad and Ambrosetti 2014
||
* Overview of the solutions methods discussed.
||
* [[Separation of Variables (1st Order)]]
* [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
* [[Integrating Factor]]
* [[Bernoulli Equations (1st Order)]]
* [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
* [[Reduction of the Order]]
* [[Method of Undetermined Coefficients (2nd Order)]]
* [[Non-linear 2nd Order ODEs]]
* [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
* [[Linear Differential Equations|Linear Differential Equations (Higher Order)]]
* [[Power Series Solutions]]
* [[Laplace Transform to ODEs]]
* [[Laplace Transform to ODEs|Laplace Transform to Systems of ODEs]]
||
* Apply all solutions methods discussed.
|}

Main Page

2023-03-10T16:13:35Z

Duynguyenvu.hoang: /* M.Sc. Track in Applied & Industrial Mathematics */ Changed "Numerical Linear Algebra" to "Computing for Mathematics", updated course number

<strong>UTSA Department of Mathematics</strong>

To edit tables in each course below, you can use [https://tableconvert.com/mediawiki-to-excel MediaWiki-to-Excel converter] and/or the [https://tableconvert.com/excel-to-mediawiki Excel-to-MediaWiki converter]

==STEM Core==
* [[MAT1073]] College Algebra for Scientists and Engineers
* [[MAT1093]] Precalculus
* [[MAT1193]] Calculus for Biosciences
* [[MAT1214]] Calculus I (4 credit hours)
* [[MAT1224]] Calculus II (4 credit hours)
* [[MAT2214]] Calculus III (4 credit hours)
* [[MAT2233]] Linear Algebra

==Data & Applied Science Core==
* [[MDC1213]] AI in the Modern World
* [[MAT1213]] Calculus I (3 credit hours)
* [[MAT1223]] Calculus II (3 credit hours)
* [[MAT2213]] Calculus III (3 credit hours)
* [[MAT2243]] Applied Linear Algebra (3 credit hours)
* [[MATxxxx]] Mathematical Biology
* [[MATxxxx]] Mathematical Physics
* [[MATxxxx]] Mathematical Statistics I (discrete & continuous PDFs)
* [[MATxxxx]] Mathematical Statistics II (statistical inference)
* [[MDC4413]] Data Analytics

==Math Major==
* [[MAT1313]] Algebra and Number Systems
* [[MAT2313]] Combinatorics and Probability
* [[MAT3013]] Foundations of Mathematics
* [[MAT3213]] Foundations of Analysis
* [[MAT3613]] Differential Equations I
* [[MAT3623]] Differential Equations II
* [[MAT3633]] Numerical Analysis
* [[MAT3223]] Complex Variables
* [[MAT4213]] Real Analysis I
* [[MAT4223]] Real Analysis II
* [[MAT4233]] Modern Abstract Algebra
* [[MAT4273]] Topology

==Business==
* [[MAT1053]] Algebra for Business
* [[MAT1133]] Calculus for Business

==Math for Liberal Arts==
* [[MAT1043]] Introduction to Mathematics

== Elementary Education ==
* [[MAT1023]] College Algebra
* [[MAT1153]] Essential Elements in Mathematics I
* [[MAT1163]] Essential Elements in Mathematics II

== General Math Studies==
* [[MAT3233]] Modern Algebra

== Core M.Sc. Studies ==
* [[MAT5173]] Algebra (fall odd years)
* [[MAT5203]] Analysis I (fall odd years)
* [[MAT5283]] Linear Algebra (fall odd years)
* [[MAT5001]] Discrete Mathematics I (fall even years)
* [[MAT4373]]/[[MAT5373]] Mathematical Statistics I (fall even years)

== Qualifying Examination Tracks ==
* [[MAT5223]] Cryptography (spring even years) (Pure, Applied Tracks)
* [[MAT5213]] Analysis II (spring even years) (Pure Track)
* [[MAT5113]] Computing for Mathematics (spring even years) (Pure, Applied Tracks)
* [[MAT5002]] Discrete Mathematics II (spring odd years) (Pure, Applied Tracks)
* [[MAT4383]]/[[MAT5383]] Mathematical Statistics II (fall even years) (Applied Tracks)

== M.Sc. Track in Pure Mathematics ==

* [[MAT3313/5003]] Logic and Computability
* [[MAT5243]] General Topology
* [[MAT5253]] General Topology II
* [[MAT5183]] Algebra II
* [[MAT5223]] Theory of Functions of a Complex Variable
* [[MAT5343]] Differential Geometry

== M.Sc. Track in Applied & Industrial Mathematics ==

* [[MAT4153]] [[MAT5153]] Data Analytics
* [[AIM 5113]] Introduction to Industrial Mathematics
* [[MAT 5113]] Computing for Mathematics
* [[MAT 5653]] Differential Equations I
* [[MAT 5673]] Partial Differential Equations

== M.Sc. Track in College Mathematics Education ==

== M.Sc. Track in K-12 Mathematics Education ==

Main Page

2023-03-10T12:53:52Z

Duynguyenvu.hoang: /* M.Sc. Track in Applied & Industrial Mathematics */

<strong>UTSA Department of Mathematics</strong>

To edit tables in each course below, you can use [https://tableconvert.com/mediawiki-to-excel MediaWiki-to-Excel converter] and/or the [https://tableconvert.com/excel-to-mediawiki Excel-to-MediaWiki converter]

==STEM Core==
* [[MAT1073]] College Algebra for Scientists and Engineers
* [[MAT1093]] Precalculus
* [[MAT1193]] Calculus for Biosciences
* [[MAT1214]] Calculus I (4 credit hours)
* [[MAT1224]] Calculus II (4 credit hours)
* [[MAT2214]] Calculus III (4 credit hours)
* [[MAT2233]] Linear Algebra

==Data & Applied Science Core==
* [[MDC1213]] AI in the Modern World
* [[MAT1213]] Calculus I (3 credit hours)
* [[MAT1223]] Calculus II (3 credit hours)
* [[MAT2213]] Calculus III (3 credit hours)
* [[MAT2243]] Applied Linear Algebra (3 credit hours)
* [[MATxxxx]] Mathematical Biology
* [[MATxxxx]] Mathematical Physics
* [[MATxxxx]] Mathematical Statistics I (discrete & continuous PDFs)
* [[MATxxxx]] Mathematical Statistics II (statistical inference)
* [[MDC4413]] Data Analytics

==Math Major==
* [[MAT1313]] Algebra and Number Systems
* [[MAT2313]] Combinatorics and Probability
* [[MAT3013]] Foundations of Mathematics
* [[MAT3213]] Foundations of Analysis
* [[MAT3613]] Differential Equations I
* [[MAT3623]] Differential Equations II
* [[MAT3633]] Numerical Analysis
* [[MAT3223]] Complex Variables
* [[MAT4213]] Real Analysis I
* [[MAT4223]] Real Analysis II
* [[MAT4233]] Modern Abstract Algebra
* [[MAT4273]] Topology

==Business==
* [[MAT1053]] Algebra for Business
* [[MAT1133]] Calculus for Business

==Math for Liberal Arts==
* [[MAT1043]] Introduction to Mathematics

== Elementary Education ==
* [[MAT1023]] College Algebra
* [[MAT1153]] Essential Elements in Mathematics I
* [[MAT1163]] Essential Elements in Mathematics II

== General Math Studies==
* [[MAT3233]] Modern Algebra

== Core M.Sc. Studies ==
* [[MAT5173]] Algebra (fall odd years)
* [[MAT5203]] Analysis I (fall odd years)
* [[MAT5283]] Linear Algebra (fall odd years)
* [[MAT5001]] Discrete Mathematics I (fall even years)
* [[MATxxx]] Mathematical Statistics I (fall even years)

== Qualifying Examination Tracks ==
* [[MAT5223]] Cryptography (spring even years) (Pure, Applied Tracks)
* [[MAT5213]] Analysis II (spring even years) (Pure Track)
* [[MATxxx]] Computational Mathematics (spring even years) (Pure, Applied Tracks)
* [[MAT5002]] Discrete Mathematics II (spring odd years) (Pure, Applied Tracks)
* [[MATxxx]] Mathematics Statistics II (fall even years) (Applied Tracks)

== M.Sc. Track in Pure Mathematics ==

* [[MAT3313/5003]] Logic and Computability
* [[MAT5243]] General Topology
* [[MAT5253]] General Topology II
* [[MAT5183]] Algebra II
* [[MAT5223]] Theory of Functions of a Complex Variable
* [[MAT5343]] Differential Geometry

== M.Sc. Track in Applied & Industrial Mathematics ==

* [[MAT4XXX/5XXX]] Data Analytics
* [[AIM 5113]] Introduction to Industrial Mathematics
* [[MAT 5283]] Linear Algebra and Matrix Theory
* [[MAT 5293]] Numerical Linear Algebra
* [[MAT 5653]] Differential Equations I
* [[MAT 5673]] Partial Differential Equations

== M.Sc. Track in College Mathematics Education ==

== M.Sc. Track in K-12 Mathematics Education ==

MAT3013

2020-07-27T13:37:58Z

Duynguyenvu.hoang:

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

* D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

* Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', 2nd ed, Springer (2011). https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List B ==

{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==Topics List C (Proofs and Fundamentals) ==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction
||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Logical statements
|| 1.1-1.2
||
||
* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going

|-
|2
|| Informal logic
||
* Statements
* Relation between statements
* Valid Arguments
* Quantifiers
|| 1.1-1.5
|| Prerequisites
|| Outcomes
|| Examples

|-
|3
|| Strategies for proofs
||
* Why we need proofs
* Direct proofs
* Proofs by contrapositive and contradiction
* Cases and If and Only If
|| 2.2-2.4
|| Prerequisites
|| Outcomes
|| Examples

|-
|4
|| Writing Mathematics/Set theory I
||
* Basic concepts
* Operations and constructions with sets
|| 2.6, 3.1-3.3
|| Prerequisites
|| Outcomes
|| Examples

|-
|5
|| Set theory II
||
* Family of sets
* Axioms of set theory
||3.4-3.5
|| Prerequisites
|| Outcomes
|| Examples

|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Functions I
||
* Definition of functions
* Image and inverse image
* Composition and inverse functions
||4.1-4.3
|| Prerequisites
|| Outcomes
|| Examples

|-
|8
||Functions II
||
* Injectivity, surjectivity and bijectivity
* Sets of functions
|| 4.4-4.5
|| Prerequisites
|| Outcomes
|| Examples

|-
|9
||Relations I
||
* Relations
* Congruence
||5.1-5.2
|| Prerequisites
|| Outcomes
|| Examples

|-
|10
||Relations II
||
* Equivalence relations
||4.3-4
|| Prerequisites
|| Outcomes
|| Examples

|-
|11
||Finite and infinite sets II
||
* Introduction
* Properties of natural numbers
||6.1-6.2
|| Prerequisites
|| Outcomes
|| Examples

|-
|12
||
* Catch up and review
* Midterm 2

|-
|13
|| Finite and infinite sets II
||
* Mathematical induction
* Recursion
||6.2-6.3
|| Prerequisites
|| Outcomes
|| Examples

|-
|14
|| Finite and infinite sets III
||
*Cardinality of sets
* Finite sets and countable sets
*Cardinality of number systems
|| 6.4 - 6.7
|| Prerequisites
|| Outcomes
|| Examples

|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-27T13:28:30Z

Duynguyenvu.hoang: /* Topics List B */

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

* D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

* Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', 2nd ed, Springer (2011). https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List C (Proofs and Fundamentals) ==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Logical statements

|| 1.1-1.2

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Informal logic
||
* Statements
* Relation between statements
* Valid Arguments
* Quantifiers

|| 1.1-1.5
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Strategies for proofs
||
* Why we need proofs
* Direct proofs
* Proofs by contrapositive and contradiction
* Cases and If and Only If
|| 2.2-2.4
|| informal logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Writing Mathematics/Set theory I
||
* Basic concepts
* Operations and constructions with sets
|| 2.6, 3.1-3.3
||
*Writing math
*Basic concepts of set theory
*Set operations
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Set theory II
||
* Family of sets
* Axioms of set theory
||3.4-3.5
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic axiom of set theory
||
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Functions I
||
* Definition of functions
* Image and inverse image
* Composition and inverse functions
||4.1-4.3
||Set theory
||Gain basic concepts about functions.
||
*
*

|-
|8
||Functions II
||
* Injectivity, surjectivity and bijectivity
* Sets of functions
||4.4-4.5
||
||
*
*

|| Examples
|-
|9
||Relations I
||
* Relations
* Congruence
||5.1-5.2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of relations.
||

|-
|10
||Relations II
||
* Equivalence relations
||4.3-4
||Functions 1
||
*
||
*

|-
|11
||Finite and infinite sets II
||
* Introduction
* Properties of natural numbers
||6.1-6.2
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2

|-
|13
|| Finite and infinite sets II
||
* Mathematical induction
* Recursion
||6.2-6.3
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
|| Finite and infinite sets III
||
*Cardinality of sets
* Finite sets and countable sets
*Cardinality of number systems
|| 6.4 - 6.7
|| Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==Topics List C (Proofs and Fundamentals)==

{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-27T13:27:10Z

Duynguyenvu.hoang: /* Topics List B*/

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

* D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

* Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', 2nd ed, Springer (2011). https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List B==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Logical statements

|| 1.1-1.2

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Informal logic
||
* Statements
* Relation between statements
* Valid Arguments
* Quantifiers

|| 1.1-1.5
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Strategies for proofs
||
* Why we need proofs
* Direct proofs
* Proofs by contrapositive and contradiction
* Cases and If and Only If
|| 2.2-2.4
|| informal logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Writing Mathematics/Set theory I
||
* Basic concepts
* Operations and constructions with sets
|| 2.6, 3.1-3.3
||
*Writing math
*Basic concepts of set theory
*Set operations
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Set theory II
||
* Family of sets
* Axioms of set theory
||3.4-3.5
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic axiom of set theory
||
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Functions I
||
* Definition of functions
* Image and inverse image
* Composition and inverse functions
||4.1-4.3
||Set theory
||Gain basic concepts about functions.
||
*
*

|-
|8
||Functions II
||
* Injectivity, surjectivity and bijectivity
* Sets of functions
||4.4-4.5
||
||
*
*

|| Examples
|-
|9
||Relations I
||
* Relations
* Congruence
||5.1-5.2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of relations.
||

|-
|10
||Relations II
||
* Equivalence relations
||4.3-4
||Functions 1
||
*
||
*

|-
|11
||Finite and infinite sets II
||
* Introduction
* Properties of natural numbers
||6.1-6.2
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2

|-
|13
|| Finite and infinite sets II
||
* Mathematical induction
* Recursion
||6.2-6.3
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
|| Finite and infinite sets III
||
*Cardinality of sets
* Finite sets and countable sets
*Cardinality of number systems
|| 6.4 - 6.7
|| Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==Topics List C (Proofs and Fundamentals)==

{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-27T13:23:26Z

Duynguyenvu.hoang: /* Topics List B */

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

* D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

* Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', 2nd ed, Springer (2011). https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List B==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Logical statements

|| 1.1-1.2

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Informal logic
||
* Statements
* Relation between statements
* Valid Arguments
* Quantifiers

|| 1.1-1.5
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Strategies for proofs
||
* Why we need proofs
* Direct proofs
* Proofs by contrapositive and contradiction
* Cases and If and Only If
|| 2.2-2.4
|| informal logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Writing Mathematics/Set theory I
||
* Basic concepts
* Operations and constructions with sets
|| 2.6, 3.1-3.3
||
*Writing math
*Basic concepts of set theory
*Set operations
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Set theory II
||
* Family of sets
* Axioms of set theory
||3.4-3.5
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic axiom of set theory
||
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Functions I
||
* Definition of functions
* Image and inverse image
* Composition and inverse functions
||4.1-4.3
||Set theory
||Gain basic concepts about functions.
||
*
*

|-
|8
||Functions II
||
* Injectivity, surjectivity and bijectivity
* Sets of functions
||4.4-4.5
||
||
*
*

|| Examples
|-
|9
||Relations I
||
* Relations
* Congruence
||5.1-5.2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of relations.
||

|-
|10
||Relations II
||
* Equivalence relations
||4.3-4
||Functions 1
||
*
||
*

|-
|11
||Finite and infinite sets II
||
* Introduction
* Properties of natural numbers
||6.1-6.2
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2

|-
|13
|| Finite and infinite sets II
||
* Mathematical induction
* Recursion
||6.2-6.3
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
|| Finite and infinite sets III
||
*Cardinality of sets
* Finite sets and countable sets
*Cardinality of number systems
|| 6.4 - 6.7
|| Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==Topics List C (Proofs and Fundamentals)==

{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-27T12:57:01Z

Duynguyenvu.hoang:

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

* D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

* Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', 2nd ed, Springer (2011). https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List B==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==Topics List C (Proofs and Fundamentals)==

{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-27T12:54:38Z

Duynguyenvu.hoang: /* Text */

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

* D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

* Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', 2nd ed, Springer (2011). https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List B==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-27T12:52:17Z

Duynguyenvu.hoang: /* Text */

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

* D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

* Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', Springer

https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List B==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-27T12:51:34Z

Duynguyenvu.hoang: /* Text */

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', Springer

https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List B==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-27T12:51:07Z

Duynguyenvu.hoang: /* Text */

Foundations of Mathematics (3-0) 3 Credit Hours
==Course Catalog==

[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3013. Foundations of Mathematics]. (3-0) 3 Credit Hours.

Prerequisite: [[MAT1214]]. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly [[MAT2243]]. Credit cannot be earned for [[MAT3013]] and [[MAT2243]].) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

Ethan D. Bloch, ''Proofs and Fundamentals: A First Course in Abstract Mathematics'', Springer
https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

==Topics List A==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|1.0
||
* 1.1
||
* Historical remarks
* Overview of the course and its goals
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Proofs]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Logic]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Axioms]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|1.0
||
* 1.1
||
* [[Propositions]]
||

||
* Motivation for rigorous mathematics from a historical perspective
* An understanding of where and why this course is going
|-
|2.0
||
* 1.2-3
||
* [[Logical Operators]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Values]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Truth Tables]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|2.0
||
* 1.2-3
||
* [[Quantifiers]]
||

||
* Gain the prerequisites for writing and evaluating proofs.
|-
|3.0
||
* 1.4-6
||
* [[Methods for Proofs]]
||
* [[Propositions]]
* [[Logical Operators]]
||
* Start proving elementary results.
|-
|4.0
||
* 2.1-3
||
* [[Basic Concepts of Set Theory]]
||

||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Operations with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|4.0
||
* 2.1-3
||
* [[Constructions with sets]]
||
* [[Basic Concepts of Set Theory]]
||
* How to start working with sets
|-
|5.0
||
* 2.4-6
||
* [[Mathematical Induction]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|5.0
||
* 2.4-6
||
* [[Counting Principles]]
||
* [[Natural Numbers]]
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
|-
|6.0
||

||
* Catch up and review
* Midterm 1
||

||

|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Cartesian Products Subsets]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|7.0
||
* 3.1-3
||
* [[Equivalence Relations]]
||
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Gain basic concepts about relations.
|-
|8.0
||
* 3.4-5
||
* [[Partial Orders]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|8.0
||
* 3.4-5
||
* [[Graphs]]
||
* [[Equivalence Relations]]
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|-
|9.0
||
* 4.1-2
||
* [[Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|9.0
||
* 4.1-2
||
* [[Constructions With Functions]]
||
* [[Equivalence Relations]]
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
||
* Gain basic rigorous knowledge of functions.
|-
|10.0
||
* 4.3-4
||
* [[One-to-One]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function is one-to-one with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Onto]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Determine whether a function onto with proofs.
|-
|10.0
||
* 4.3-4
||
* [[Compositional Inverse]]
||
* [[Functions]]
* [[Constructions With Functions]]
||
* Finding inverses
|-
|11.0
||
* 4.5-6
||
* [[Images of Subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find images of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Preimages of subsets]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||
* Find preimages of subsets under functions, with proofs.
|-
|11.0
||
* 4.5-6
||
* [[Sequences]]
||
* [[One-to-One]]
* [[Onto]]
* [[Compositional Inverse]]
||

|-
|12.0
||

||
* Catch up and review
* Midterm 2
||

||

|-
|13.0
||
* 5.1-2
||
* [[Finite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
|-
|13.0
||
* 5.1-2
||
* [[Infinite Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
|-
|13.0
||
* 5.1-2
||
* [[Equivalent Sets]]
||
* [[Functions]]
* [[Operations with sets]]
* [[Constructions with sets]]
||

|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of countable sets.
|-
|14.0
||
* 5.3-5
||
* [[Uncountable Sets]]
||
* [[Finite Sets]]
* [[Infinite Sets]]
* [[Equivalent Sets]]
||
* Learn properties of uncountable sets.
|-
|15.0
||

||
* Catch up and review for Final
* Study Days
||

||

|}

==Topics List B==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]