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Introduction to the mathematics of discrete structures with emphasis on structures for computer science.
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<strong>UTSA Department of Mathematics</strong>
  
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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]
  
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'''Catalog entry'''
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== Undergraduate Studies ==
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===STEM Core===
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* [[MAT1073]] College Algebra for Scientists and Engineers
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* [[MAT1093]] Precalculus
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* [[MAT1193]] Calculus for Biosciences
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* [[MAT1214]] Calculus I (4 credit hours)
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* [[MAT1213]] Calculus I (3 credit hours)
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* [[MAT1224]] Calculus II (4 credit hours)
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* [[MAT1223]] Calculus II (3 credit hours)
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* [[MAT2214]] Calculus III (4 credit hours)
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* [[MAT2213]] Calculus III (3 credit hours)
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* [[MAT2233]] Linear Algebra
  
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''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
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===Minor in Mathematics===
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To receive a minor, students must complete at least 18 semester credit hours, including 6 hours at the upper-division level at UTSA, and must achieve a grade point average of at least 2.0 (on a 4.0 scale) on all work used to satisfy the requirements of a minor. See [https://catalog.utsa.edu/undergraduate/bachelorsdegreeregulations/minors/  UTSA's Undergraduate Catalog]
  
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''Contents'':
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The Minor in Mathematics requires the Calculus series plus linear algebra, and upper division courses in either the Math Major or the Data & Applied Science Core
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(1) Propositional logic: Axioms and Rules of Inference. Limitations of propositional logic: Informal introduction to quantifiers and syllogisms.
 
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(2) Predicate Logic: Existential and universal quantification, free variables and substitutions. Discussion of the various axiomatic systems for first-order logic (including axioms and rules of inference). The power and the limitations of axiomatic systems for logic: Informal discussion of the completeness and incompleteness theorems.
 
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(3) Sets and boolean algebras: Operations on sets. Correspondence between finitary set operations and propositional logic. Correspondence between infinitary operations and quantifiers. The power and limitations of the language of set theory: Informal discussion of the set-theoretic paradoxes and the need for axiomatic systems for set theory.
 
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(4) Relations: Special relations: Equivalence relations, partially ordered sets, maximum/minimum, maximal/minimal elements, least upper bounds and greatest lower bounds, totally ordered sets.
 
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(5) Functions: Operations of functions, direct image and inverse image.
 
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(6) Well-ordered sets: Correspondence between well-ordering relations and induction. Correspondence between well-ordering relations and choice functions.
 
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(7) Introduction to computability. Classical models of computation (recursive functions, and Turing models). Limitations of computation (the Halting Problem, fast-growing functions). Contemporary models of computation.
 
  
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===Data & Applied Science Core===
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==== Lower Division ====
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* [[MDC1213]] Sociocultural Foundations of Mathematics, Data Science, and Computing
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* [[MAT2253]] Applied Linear Algebra (3 credit hours)
  
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'''Sample textbooks''':
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==== Upper Division ====
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* [[MAT4133]]/[[MAT5133]] Mathematical Biology
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* [[MAT4813]]/[[MAT5163]] Mathematical Physics 1
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* [[MAT4XXX]]/[[MAT5XXX]] Mathematical Physics 2
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* [[MAT4XXX]]/[[MAT5XXX]] Introduction to Quantum Information Science and Engineering
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* [[MAT4373]]/[[MAT5373]] Mathematical Foundations of Statistics I (discrete & continuous PDFs)
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* [[MAT4383]]/[[MAT5383]] Mathematical Foundations of Statistics II (statistical inference)
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* [[MDC4153]]/[[MDC5153]] Mathematical Foundations of Data Analytics
  
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[1] Gordon Pace, ''Mathematics of Discrete Structures for Computer Science'', Springer, 2012
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===Math Major===
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==== Lower Division ====
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* [[MAT1313]] Algebra and Number Systems
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* [[MAT2313]] Combinatorics and Probability
  
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[2] Vladlen Koltun, ''Discrete Structures Lecture Notes, Stanford University'', 2008. Freely available [https://web.stanford.edu/class/cs103x/cs103x-notes.pdf here.]
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==== Upper Division ====
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* [[MAT3003]] Discrete Mathematics
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* <del>[[MAT3013]] Foundations of Mathematics</del> Course transitioning to be replaced by [[MAT3003]] Discrete Mathematics (below).
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* <del>[[MAT3213]] Foundations of Analysis</del> Course transitioning to be replaced by [[MAT3333]] Fundamentals of Analysis and Topology (below).
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* [[MAT3333]] Fundamentals of Analysis and Topology
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* [[MAT3233]] Modern Algebra
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* [[MAT3333]] Fundamentals of Analysis and Topology
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* [[MAT3313]] Logic and Computability
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* [[MAT3613]] Differential Equations I
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* [[MAT3623]] Differential Equations II
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* [[MAT3633]] Numerical Analysis
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* [[MAT3223]] Complex Variables
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* [[MAT4033]] Linear Algebra II
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* [[MAT4213]] Real Analysis I
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* [[MAT4223]] Real Analysis II
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* [[MAT4233]] Modern Abstract Algebra
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* [[MAT4273]] Topology
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* [[MAT4283]] Computing for Mathematics
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* [[MAT4323]] Applied Graph Theory
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* [[MAT4373]] Mathematical Statistics I
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* [[MAT43XX]]/[[MAT53XX]] Mathematics of Quantum Computing
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* [[MAT4YYY]]/[[MAT5YYY]] Mathematics of AI
  
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===Business===
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* [[MAT1053]] Algebra for Business
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* [[MAT1133]] Calculus for Business
  
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===Math for Liberal Arts===
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* [[MAT1043]] Introduction to Mathematics
  
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=== Elementary Education ===
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* [[MAT1023]] College Algebra
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* [[MAT1153]] Essential Elements in Mathematics I
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* [[MAT1163]] Essential Elements in Mathematics II
  
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=== General Math Studies===
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* [[MAT3233]] Modern Algebra
  
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== Graduate Studies ==
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=== Core Courses ===
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Core courses, common across all M.Sc. tracks, must be at least 50% of the credit 30 hours needed to obtain a M.Sc. degree. The following courses add up to 15 credit hours.
  
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* [[MAT5283]] Linear Algebra
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* [[MAT5203]] Real Analysis I
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* [[MAT5603]] Numerical Analysis
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* [[MAT5603]] Differential Equations I
  
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==Topics List==
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{| class="wikitable sortable"
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=== Qualifying Exam Track: Algebra-Cryptography  ===
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! Week !! Topic !! Sections from Pace's book !! !! Sections from Pace's book !! Prerequisites.
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|-
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* [[MAT 5173]] Abstract Algebra I
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|  1 
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* [[MAT 5183]] Abstract Algebra II
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|| [[Propositional logic]]
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* [[MAT 5123]] Introduction to Cryptography
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|| 2.1-2.4
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* [[MAT 5143]] Cryptography II (Post-Quantum Cryptography)
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||
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* Proofs
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=== Qualifying Track: Mathematics of Information, Data and Communications  ===
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* boolean models
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* connections between boolean models and proofs
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* [[MAT5123]] Introduction to Cryptography
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|| MAT1313 or CS2233/2231, or equivalent.
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* [[MAT5143]] Cryptography II
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|-
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* [[MAT5433]] Discrete Mathematics II (Graph Theory)
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|  2 
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* [[MAT5153]] Data Analytics
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|| [[Completeness and soundness]]
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* [[MAT5XXX]] Quantum Information
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|| 2.5
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* [[MAT5ZZZ]] Mathematics of AI
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* Completeness and soundness of propositional logic
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=== Qualifying Exam Track: Scientific Computing ===
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||
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|-
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* [[MAT5603]] Numerical Analysis
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|  5-6 
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* [[MAT5293]] Numerical Linear Algebra
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|| [[Predicate calculus]]
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* [[MAT5XXX]] Quantum Information
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|| 3.1-3.5
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* [[MAT5163]] Probability and Computing
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* Limits of propositional logic
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=== Qualifying Exam Track: Analysis ===
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* free variables and substitution.
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* [[MAT5203]] Real Analysis I
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|-
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* [[MAT5213]] Real Analysis II
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|  7 
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* [[MAT5223]] Complex Analysis I
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|| [[Sets and boolean algebras]]
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* [[MAT5233]] Complex Analysis II
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|| 4.1-4.5
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* [[MAT5553]] Harmonic Analysis
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* [[MAT5553]] Functional Analysis
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* Set comprehension.
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* Finitary and general operations on sets.
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=== Qualifying Exam Track: Differential Equations ===
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|-
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* [[MAT5653]] Differential Equations I
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|  8 
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* [[MAT5673]] Partial Differential Equations I
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|| [[Sets and boolean algebras]]
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* [[MAT5683]] Partial Differential Equations II
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|| 4.6
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=== Qualifying Exam Track: Discrete Mathematics and Logic ===
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* Boolean algebras and boolean rings.
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* [[MAT5423]] Discrete Mathematics
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|-
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* [[MAT5433]] Discrete Mathematics II (Graph Theory)
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|  9 
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* [[MAT5443]] Logic and Computability
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|| [[Relations]]  
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* [[MAT5453]] Quantum Logic
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|| 5.1-5.7
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=== Qualifying Exam Track: Topology and Geometry ===
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* Relations and sets
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* Inverse of a relation and composition of relations
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* [[MAT5243]] Topology I
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* Beyond binary relations
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* [[MAT5253]] Topology II
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* Differential Geometry
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|-
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|  10 
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=== Qualifying Exam Track: Mathematical Physics  ===
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|| [[Classifying Relations]]  
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|| 6.1-6.3
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* [MAT5863]] Mathematical Physics I
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* Mathematical Physics II*
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* Totality
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* Quantum Information*
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* Surjectivity
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* Differential Geometry*
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* Injectivity
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* [[MAT5453]] Quantum Logic
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* Functionality
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=== Qualifying Exam Track: Mathematics of Life Sciences  ===
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|-
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|  11-12 
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* [[MAT5133]] Mathematical Biology
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|| [[Discrete structures]]  
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|| 7.1-8.4
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Plus one of the following three courses:
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* Graphs
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* [[MAT5323]] Mathematical Modeling
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* Semigroups
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* Systems Biology*
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* groups
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* Optimal Control and Calculus of Variations*.
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|-
 
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|  13-14 
 
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|| [[Reasoning about programs]]  
 
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|| 10.1-10.4
 
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* Algorithms
 
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* Program semantics
 
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* Uncomputability
 
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|}
 

Latest revision as of 11:59, 21 August 2025

UTSA Department of Mathematics

To edit tables in each course below, you can use MediaWiki-to-Excel converter and/or the Excel-to-MediaWiki converter

Undergraduate Studies

STEM Core

  • MAT1073 College Algebra for Scientists and Engineers
  • MAT1093 Precalculus
  • MAT1193 Calculus for Biosciences
  • MAT1214 Calculus I (4 credit hours)
  • MAT1213 Calculus I (3 credit hours)
  • MAT1224 Calculus II (4 credit hours)
  • MAT1223 Calculus II (3 credit hours)
  • MAT2214 Calculus III (4 credit hours)
  • MAT2213 Calculus III (3 credit hours)
  • MAT2233 Linear Algebra

Minor in Mathematics

To receive a minor, students must complete at least 18 semester credit hours, including 6 hours at the upper-division level at UTSA, and must achieve a grade point average of at least 2.0 (on a 4.0 scale) on all work used to satisfy the requirements of a minor. See UTSA's Undergraduate Catalog

The Minor in Mathematics requires the Calculus series plus linear algebra, and upper division courses in either the Math Major or the Data & Applied Science Core

Data & Applied Science Core

Lower Division

  • MDC1213 Sociocultural Foundations of Mathematics, Data Science, and Computing
  • MAT2253 Applied Linear Algebra (3 credit hours)

Upper Division

Math Major

Lower Division

  • MAT1313 Algebra and Number Systems
  • MAT2313 Combinatorics and Probability

Upper Division

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

Graduate Studies

Core Courses

Core courses, common across all M.Sc. tracks, must be at least 50% of the credit 30 hours needed to obtain a M.Sc. degree. The following courses add up to 15 credit hours.


Qualifying Exam Track: Algebra-Cryptography

Qualifying Track: Mathematics of Information, Data and Communications

Qualifying Exam Track: Scientific Computing

Qualifying Exam Track: Analysis

Qualifying Exam Track: Differential Equations

  • MAT5653 Differential Equations I
  • MAT5673 Partial Differential Equations I
  • MAT5683 Partial Differential Equations II

Qualifying Exam Track: Discrete Mathematics and Logic

Qualifying Exam Track: Topology and Geometry

Qualifying Exam Track: Mathematical Physics

  • [MAT5863]] Mathematical Physics I
  • Mathematical Physics II*
  • Quantum Information*
  • Differential Geometry*
  • MAT5453 Quantum Logic

Qualifying Exam Track: Mathematics of Life Sciences

Plus one of the following three courses:

  • MAT5323 Mathematical Modeling
  • Systems Biology*
  • Optimal Control and Calculus of Variations*.
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