Difference between revisions of "MAT3013"

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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 C (Proofs and Fundamentals) ==
+
==Topics List==
−
{| 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
 
−
|}
 
−
==Topics List D (Proofs and Fundamentals) Wiki Format ==
 
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
 
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
 
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
 
|-
 
|-
−
|1.
+
| <!-- * Week -->
 +
1.
 
||
 
||
 +
<!-- * Sections -->
 
* 1.1-1.2
 
* 1.1-1.2
 
||
 
||
−
* [[Historical remarks]]
+
<!-- * Topics -->
−
||
 
−
* Prerequisites
 
−
||
 
−
* Motivation for rigorous mathematics from a historical perspective
 
−
* An understanding of where and why this course is going
 
−
|-
 
−
|1.
 
−
||
 
−
* 1.1-1.2
 
−
||
 
−
* [[Overview of the course and its goals]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Motivation for rigorous mathematics from a historical perspective
 
−
* An understanding of where and why this course is going
 
−
|-
 
−
|1.
 
−
||
 
−
* 1.1-1.2
 
−
||
 
−
* [[Ideas of proofs and logic]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Motivation for rigorous mathematics from a historical perspective
 
−
* An understanding of where and why this course is going
 
−
|-
 
−
|1.
 
−
||
 
−
* 1.1-1.2
 
−
||
 
−
* [[Logical statements]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Motivation for rigorous mathematics from a historical perspective
 
−
* An understanding of where and why this course is going
 
−
|-
 
−
|2.
 
−
||
 
−
* 1.1-1.5
 
−
||
 
 
* [[Statements]]
 
* [[Statements]]
 +
* [[Sentential Logic]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
−
||
 
−
* Outcomes
 
−
|-
 
−
|2.
 
−
||
 
−
* 1.1-1.5
 
−
||
 
−
* [[Relation between statements]]
 
 
||
 
||
−
* Prerequisites
+
<!-- * Outcomes -->
−
||
+
* Identify syntactically correct formulas in sentential logic.
−
* Outcomes
+
* Translate compound statements in informal language to formal propositional sentences.
 +
* Find the interpretation of a sentential formula given interpretations of the propositional symbols therein.
 
|-
 
|-
−
|2.
+
| <!-- * Week -->
 +
2.
 
||
 
||
−
* 1.1-1.5
+
<!-- * Sections -->
 +
* 1.3-1.4
 
||
 
||
−
* [[Valid Arguments]]
+
<!-- * Topics -->
 +
* [[Logical Implication]]
 +
* [[Logical Equivalence]]
 +
* [[Deductive Rules]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sentential Logic]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 +
* Express informally stated relations between sentences in terms of semantic implication and equivalence.
 +
* State and recognize basic rules of deductive reasoning and their correct application.
 +
* Use the rules of deduction to prove basic semantic relations (implication or equivalence) between formal interpretations of propositional formulas.
 +
* Distinguish between correct and incorrect applications of deductive rules.
 
|-
 
|-
−
|2.
+
| <!-- * Week -->
 +
3.
 
||
 
||
−
* 1.1-1.5
+
<!-- * Sections -->
 +
* 1.5-2.2
 
||
 
||
 +
<!-- * Topics -->
 
* [[Quantifiers]]
 
* [[Quantifiers]]
 +
* [[Mathematical Proofs]]
 +
* [[Proofs:Direct]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sentential Logic]]
 +
* [[Deductive Rules]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|3.
+
| <!-- * Week -->
 +
4.
 
||
 
||
−
* 2.2-2.4
+
<!-- * Sections -->
 +
* 2.3-2.4
 
||
 
||
−
* [[Why we need proofs]]
+
<!-- * Topics -->
 +
* [[Proofs:Contraposition]]
 +
* [[Proofs:Contradiction]]
 +
* [[Proofs:Cases]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Mathematical Proofs]]
 +
* [[Proofs:Direct]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|3.
+
| <!-- * Week -->
 +
5.
 
||
 
||
−
* 2.2-2.4
+
<!-- * Sections -->
 +
* 2.4-2.6
 
||
 
||
−
* [[Direct proofs]]
+
<!-- * Topics -->
 +
* [[Proofs:Biconditionals]]
 +
* [[Proofs:Quantifiers]]
 +
* [[Writing Mathematics]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
−
||
+
* [[Quantifiers]]
−
* Outcomes
+
* [[Mathematical Proofs]]
−
|-
 
−
|3.
 
−
||
 
−
* 2.2-2.4
 
−
||
 
−
* [[Proofs by contrapositive and contradiction]]
 
−
||
 
−
* Prerequisites
 
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|3.
+
| <!-- * Week -->
 +
6.
 
||
 
||
−
* 2.2-2.4
+
<!-- * Sections -->
 +
* 3.1-3.3
 
||
 
||
−
* [[Cases and If and Only If]]
+
<!-- * Topics -->
 +
* [[Sets:Definitions]]
 +
* [[Sets:Operations]]
 +
* [[Sets:Families]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|4.
+
| <!-- * Week -->
 +
7.
 
||
 
||
−
* 2.6, 3.1-3.3
+
<!-- * Sections -->
 
||
 
||
−
* [[Basic concepts]]
+
<!-- * Topics -->
 +
* Review of Chapters 1-3.
 +
* Midterm exam.
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|4.
+
| <!-- * Week -->
−
||
+
8.
−
* 2.6, 3.1-3.3
 
−
||
 
−
* [[Operations and constructions with sets]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|5.
 
−
||
 
−
* 3.4-3.5
 
−
||
 
−
* [[Family of sets]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|5.
 
−
||
 
−
* 3.4-3.5
 
−
||
 
−
* [[Axioms of set theory]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|6.
 
−
||
 
−
 
 
−
||
 
−
 
 
−
||
 
−
 
 
−
||
 
−
 
 
−
|-
 
−
|7.
 
 
||
 
||
 +
<!-- * Sections -->
 
* 4.1-4.3
 
* 4.1-4.3
 
||
 
||
−
* [[Definition of functions]]
+
<!-- * Topics -->
 +
* [[Functions:Definition]]
 +
* [[Functions:Forward Image]]
 +
* [[Functions:Forward Image|Functions:Inverse Image]]
 +
* [[Functions:Composition]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sets:Definitions]]
 +
* [[Sets:Operations]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|7.
+
| <!-- * Week -->
 +
9.
 
||
 
||
−
* 4.1-4.3
+
<!-- * Sections -->
 +
* 4.3-4.4
 
||
 
||
−
* [[Image and inverse image]]
+
<!-- * Topics -->
 +
* [[Functions:Inverses]]
 +
* [[Functions:Injective]]
 +
* [[Functions:Surjective]]
 +
* [[Functions:Bijective]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Functions:Definition]]
 +
* [[Functions:Composition]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|7.
+
| <!-- * Week -->
 +
10.
 
||
 
||
−
* 4.1-4.3
+
<!-- * Sections -->
 +
* 5.1 & 5.3
 
||
 
||
−
* [[Composition and inverse functions]]
+
<!-- * Topics -->
−
||
+
* [[Relations]]
−
* Prerequisites
+
* [[Functions as Relations]]
−
||
+
* [[Equivalence Relations]]
−
* Outcomes
 
−
|-
 
−
|8.
 
 
||
 
||
−
* 4.4-4.5
+
<!-- * Prerequisites -->
 +
* [[Sets:Definitions]]
 +
* [[Sets:Operations]]
 
||
 
||
−
* [[Injectivity, surjectivity and bijectivity]]
+
<!-- * Outcomes -->
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
 
|-
 
|-
−
|8.
+
| <!-- * Week -->
−
||
+
11.
−
* 4.4-4.5
 
−
||
 
−
* [[Sets of functions]]
 
 
||
 
||
−
* Prerequisites
+
<!-- * Sections -->
 +
* 6.1-6.3
 
||
 
||
−
* Outcomes
+
<!-- * Topics -->
−
|-
+
* [[Natural Numbers:Postulates]]
−
|9.
+
* [[Natural Numbers:Well-Ordering]]
−
||
+
* [[Proofs:Induction]]
−
* 5.1-5.2
 
 
||
 
||
 +
<!-- * Prerequisites -->
 +
* [[Sets:Definitions]]
 +
* [[Functions:Definition]]
 
* [[Relations]]
 
* [[Relations]]
 
||
 
||
−
* Prerequisites
+
<!-- * Outcomes -->
−
||
 
−
* Outcomes
 
−
|-
 
−
|9.
 
−
||
 
−
* 5.1-5.2
 
−
||
 
−
* [[Congruence]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
 
|-
 
|-
−
|10.
+
| <!-- * Week -->
 +
12.
 
||
 
||
−
* 4.3-4
+
<!-- * Sections -->
−
||
+
* 6.3-6.4
−
* [[Equivalence relations]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|11.
 
−
||
 
−
* 6.1-6.2
 
−
||
 
−
* [[Introduction]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|11.
 
−
||
 
−
* 6.1-6.2
 
−
||
 
−
* [[Properties of natural numbers]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|12.
 
−
||
 
−
 
 
−
||
 
−
 
 
−
||
 
−
 
 
−
||
 
−
 
 
−
|-
 
−
|13.
 
−
||
 
−
* 6.2-6.3
 
−
||
 
−
* [[Mathematical induction]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|13.
 
−
||
 
−
* 6.2-6.3
 
 
||
 
||
 +
<!-- * Topics -->
 +
* [[Proofs:Induction|Induction:Variants]]
 
* [[Recursion]]
 
* [[Recursion]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Proofs:Induction]]
 +
* [[Functions:Definition]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|14.
+
| <!-- * Week -->
 +
13.
 
||
 
||
−
* 6.4 - 6.7
+
<!-- * Sections -->
 +
* 6.5
 
||
 
||
−
* [[Cardinality of sets]]
+
<!-- * Topics -->
 +
* [[Sets:Cardinality]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sets:Definitions]]
 +
* [[Equivalence Relations]]
 +
* [[Functions:Injective]]
 +
* [[Functions:Bijective]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|14.
+
| <!-- * Week -->
 +
14.
 
||
 
||
−
* 6.4 - 6.7
+
<!-- * Sections -->
 +
* 6.6-6.7
 
||
 
||
−
* [[Finite sets and countable sets]]
+
<!-- * Topics -->
 +
* [[Sets:Finite]]
 +
* [[Sets:Countable]]
 +
* [[Sets:Uncountable]]
 +
* [[Cardinality of important sets|Cardinality:𝐍]]
 +
* [[Cardinality of important sets|Cardinality:𝐙]]
 +
* [[Cardinality of important sets|Cardinality:𝐐]]
 +
* [[Cardinality of important sets|Cardinality:𝐑]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sets:Cardinality]]
 +
* [[Natural Numbers:Postulates]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|14.
+
| <!-- * Week -->
 +
15.
 
||
 
||
−
* 6.4 - 6.7
+
<!-- * Sections -->
 
||
 
||
−
* [[Cardinality of number systems]]
+
<!-- * Topics -->
 +
* Catch-up and review for final exam.
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
−
||
 
−
* Outcomes
 
−
|-
 
−
|15.0
 
−
||
 
−
 
 
 
||
 
||
 +
<!-- * Outcomes -->
 +
|}
  
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|}
 
 
==See also==
 
==See also==
  
 
* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]
 
* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

Latest revision as of 14:15, 14 October 2021

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

Date Sections Topics Prerequisite Skills Student Learning Outcomes

1.

  • 1.1-1.2
  • Identify syntactically correct formulas in sentential logic.
  • Translate compound statements in informal language to formal propositional sentences.
  • Find the interpretation of a sentential formula given interpretations of the propositional symbols therein.

2.

  • 1.3-1.4
  • Express informally stated relations between sentences in terms of semantic implication and equivalence.
  • State and recognize basic rules of deductive reasoning and their correct application.
  • Use the rules of deduction to prove basic semantic relations (implication or equivalence) between formal interpretations of propositional formulas.
  • Distinguish between correct and incorrect applications of deductive rules.

3.

  • 1.5-2.2

4.

  • 2.3-2.4

5.

  • 2.4-2.6

6.

  • 3.1-3.3

7.

  • Review of Chapters 1-3.
  • Midterm exam.

8.

  • 4.1-4.3

9.

  • 4.3-4.4

10.

  • 5.1 & 5.3

11.

  • 6.1-6.3

12.

  • 6.3-6.4

13.

  • 6.5

14.

  • 6.6-6.7

15.

  • Catch-up and review for final exam.

See also

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=MAT3013&oldid=2281"