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| | ==Text== | | ==Text== |
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| − | * D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole
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| | | | |
| | * 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 | | * 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) == | | ==Topics List C (Proofs and Fundamentals) == |
Foundations of Mathematics (3-0) 3 Credit Hours
Course Catalog
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
Topics List C (Proofs and Fundamentals)
| 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
|
|
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
|
|
5.1-5.2
|
Prerequisites
|
Outcomes
|
Examples
|
| 10
|
Relations II
|
|
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
|
Topics List D (Proofs and Fundamentals) Wiki Format
| Date |
Sections |
Topics |
Prerequisite Skills |
Student Learning Outcomes
|
| 1.
|
|
|
|
- Motivation for rigorous mathematics from a historical perspective
- An understanding of where and why this course is going
|
| 1.
|
|
|
|
- Motivation for rigorous mathematics from a historical perspective
- An understanding of where and why this course is going
|
| 1.
|
|
|
|
- Motivation for rigorous mathematics from a historical perspective
- An understanding of where and why this course is going
|
| 1.
|
|
|
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- Motivation for rigorous mathematics from a historical perspective
- An understanding of where and why this course is going
|
| 2.
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| 2.
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See also
