Difference between revisions of "MAT3013"

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Foundations of Mathematics (3-0) 3 Credit Hours
 
Foundations of Mathematics (3-0) 3 Credit Hours
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==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==
 
==Description==
Line 20: Line 25:
 
==Text==
 
==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==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
−
! Week !! Session !! Topics !! Section !! Prerequisite Skills !! Learning Outcomes !! Examples !
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! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
−
 
 
 
|-
 
|-
−
|1
+
| <!-- * Week -->
−
 
+
1.
−
|| Introduction
+
||
−
 
+
<!-- * Sections -->
 +
* 1.1-1.2
 
||
 
||
−
* Historical remarks
+
<!-- * Topics -->
−
* Overview of the course and its goals
+
* [[Statements]]
−
* Ideas of proofs and logic
+
* [[Sentential Logic]]
−
* Axioms and propositions
 
−
 
 
−
|| 1.1
 
−
 
 
 
||
 
||
−
 
+
<!-- * Prerequisites -->
 
||
 
||
−
 
+
<!-- * Outcomes -->
−
* Motivation for rigorous
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* Identify syntactically correct formulas in sentential logic.
−
mathematics from a
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* Translate compound statements in informal language to formal propositional sentences.
−
historical perspective
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* Find the interpretation of a sentential formula given interpretations of the propositional symbols therein.
−
* An understanding of where
 
−
and why this course is
 
−
going
 
 
|-
 
|-
−
|2
+
| <!-- * Week -->
−
|| Propositional logic
+
2.
 
||
 
||
−
* Logical operators
+
<!-- * Sections -->
−
* Truth values
+
* 1.3-1.4
−
* Truth tables
+
||
−
* Quantifiers
+
<!-- * Topics -->
−
 
+
* [[Logical Implication]]
−
|| 1.2-3
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* [[Logical Equivalence]]
 +
* [[Deductive Rules]]
 
||
 
||
−
|| Gain the prerequisites for
+
<!-- * Prerequisites -->
−
writing and evaluating
+
* [[Sentential Logic]]
−
proofs.
 
 
||
 
||
−
* connectives
+
<!-- * Outcomes -->
−
* conditionals
+
* Express informally stated relations between sentences in terms of semantic implication and equivalence.
−
* biconditionals
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* 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
+
| <!-- * Week -->
−
|| Proof methods
+
3.
−
|| Methods for proofs
+
||
−
|| 1.4-6
+
<!-- * Sections -->
−
|| Propositional logic
+
* 1.5-2.2
−
|| Start proving elementary results.
+
||
 +
<!-- * Topics -->
 +
* [[Quantifiers]]
 +
* [[Mathematical Proofs]]
 +
* [[Proofs:Direct]]
 +
||
 +
<!-- * Prerequisites -->
 +
* [[Sentential Logic]]
 +
* [[Deductive Rules]]
 
||
 
||
−
* direct proofs
+
<!-- * Outcomes -->
−
* ''modus ponens''
 
−
* proofs by contradiction
 
−
 
 
 
|-
 
|-
−
|4
+
| <!-- * Week -->
−
|| Set theory
+
4.
 
||
 
||
−
* Basic concepts
+
<!-- * Sections -->
−
* Operations and constructions with sets
+
* 2.3-2.4
−
|| 2.1-3
 
−
|| Basic concepts of set theory
 
−
|| How to start working with sets
 
 
||
 
||
−
* notation
+
<!-- * Topics -->
−
* subsets
+
* [[Proofs:Contraposition]]
−
* proving sets are equal
+
* [[Proofs:Contradiction]]
−
* unions, intersections, complements
+
* [[Proofs:Cases]]
−
 
+
||
 +
<!-- * Prerequisites -->
 +
* [[Mathematical Proofs]]
 +
* [[Proofs:Direct]]
 +
||
 +
<!-- * Outcomes -->
 
|-
 
|-
−
|5
+
| <!-- * Week -->
−
|| Induction and counting
+
5.
 
||
 
||
−
* Mathematical induction
+
<!-- * Sections -->
−
* Counting principles
+
* 2.4-2.6
−
||2.4-6
 
−
||Natural numbers
 
 
||
 
||
−
* Learn constructive proofs and reasoning.
+
<!-- * Topics -->
−
* Learn basic counting principles of discrete mathematics.
+
* [[Proofs:Biconditionals]]
 +
* [[Proofs:Quantifiers]]
 +
* [[Writing Mathematics]]
 
||
 
||
−
* sums of consecutive powers
+
<!-- * Prerequisites -->
−
* other induction proofs
+
* [[Quantifiers]]
−
* well ordering principle
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* [[Mathematical Proofs]]
−
* inclusion-exclusion principle
+
||
 +
<!-- * Outcomes -->
 
|-
 
|-
−
|6
+
| <!-- * Week -->
 +
6.
 +
||
 +
<!-- * Sections -->
 +
* 3.1-3.3
 +
||
 +
<!-- * Topics -->
 +
* [[Sets:Definitions]]
 +
* [[Sets:Operations]]
 +
* [[Sets:Families]]
 +
||
 +
<!-- * Prerequisites -->
 
||
 
||
−
* Catch up and review
+
<!-- * Outcomes -->
−
* Midterm 1
 
−
 
 
 
|-
 
|-
−
|7
+
| <!-- * Week -->
−
||Relations 1
+
7.
 +
||
 +
<!-- * Sections -->
 +
||
 +
<!-- * Topics -->
 +
* Review of Chapters 1-3.
 +
* Midterm exam.
 
||
 
||
−
* Cartesian products and their subsets
+
<!-- * Prerequisites -->
−
* Equivalence relations
 
−
||3.1-3
 
−
||Set theory
 
−
||Gain basic concepts about relations.
 
 
||
 
||
−
* modular congruence
+
<!-- * Outcomes -->
−
* gluing sets
 
 
|-
 
|-
−
|8
+
| <!-- * Week -->
−
||Relations 2
+
8.
 +
||
 +
<!-- * Sections -->
 +
* 4.1-4.3
 +
||
 +
<!-- * Topics -->
 +
* [[Functions:Definition]]
 +
* [[Functions:Forward Image]]
 +
* [[Functions:Forward Image|Functions:Inverse Image]]
 +
* [[Functions:Composition]]
 
||
 
||
−
* Partial orders
+
<!-- * Prerequisites -->
−
* Graphs
+
* [[Sets:Definitions]]
−
||3.4-5
+
* [[Sets:Operations]]
−
||Relations 1
 
 
||
 
||
−
* Familiarize with ordering.
+
<!-- * Outcomes -->
−
* Learn how to use graph representations of relations.
 
−
|| partial ordering of the power set under inclusion
 
 
|-
 
|-
−
|9
+
| <!-- * Week -->
−
||Functions 1
+
9.
 +
||
 +
<!-- * Sections -->
 +
* 4.3-4.4
 
||
 
||
−
* Functions
+
<!-- * Topics -->
−
* Constructions with functions
+
* [[Functions:Inverses]]
−
||4.1-2
+
* [[Functions:Injective]]
 +
* [[Functions:Surjective]]
 +
* [[Functions:Bijective]]
 
||
 
||
−
* Relations
+
<!-- * Prerequisites -->
−
* Function sense (precalculus)
+
* [[Functions:Definition]]
−
|| Gain basic rigorous knowledge of functions.
+
* [[Functions:Composition]]
 
||
 
||
−
functional composition
+
<!-- * Outcomes -->
 
|-
 
|-
−
|10
+
| <!-- * Week -->
−
||Functions 2
+
10.
 
||
 
||
−
* One-to-one
+
<!-- * Sections -->
−
* Onto
+
* 5.1 & 5.3
−
* Compositional inverse
 
−
||4.3-4
 
−
||Functions 1
 
 
||
 
||
−
* Determine whether a function is one-to-one of onto, with proofs.
+
<!-- * Topics -->
−
* Finding inverses
+
* [[Relations]]
 +
* [[Functions as Relations]]
 +
* [[Equivalence Relations]]
 
||
 
||
−
* examples with finite sets
+
<!-- * Prerequisites -->
−
* many precalculus examples
+
* [[Sets:Definitions]]
−
 
+
* [[Sets:Operations]]
 +
||
 +
<!-- * Outcomes -->
 
|-
 
|-
−
|11
+
| <!-- * Week -->
−
||Functions 3
+
11.
 +
||
 +
<!-- * Sections -->
 +
* 6.1-6.3
 
||
 
||
−
* Images of subsets
+
<!-- * Topics -->
−
* Preimages of subsets
+
* [[Natural Numbers:Postulates]]
−
* Sequences
+
* [[Natural Numbers:Well-Ordering]]
−
||4.5-6
+
* [[Proofs:Induction]]
−
||Functions 2
 
−
||Find images and preimages of subsets under functions, with proofs.
 
 
||
 
||
−
* examples with finite sets
+
<!-- * Prerequisites -->
−
* many precalculus examples
+
* [[Sets:Definitions]]
−
 
+
* [[Functions:Definition]]
 +
* [[Relations]]
 +
||
 +
<!-- * Outcomes -->
 
|-
 
|-
−
|12
+
| <!-- * Week -->
 +
12.
 
||
 
||
−
* Catch up and review
+
<!-- * Sections -->
−
* Midterm 2
+
* 6.3-6.4
 +
||
 +
<!-- * Topics -->
 +
* [[Proofs:Induction|Induction:Variants]]
 +
* [[Recursion]]
 +
||
 +
<!-- * Prerequisites -->
 +
* [[Proofs:Induction]]
 +
* [[Functions:Definition]]
 +
||
 +
<!-- * Outcomes -->
 
|-
 
|-
−
|13
+
| <!-- * Week -->
−
||Cardinality 1
+
13.
 +
||
 +
<!-- * Sections -->
 +
* 6.5
 
||
 
||
−
* Finite and infinite sets
+
<!-- * Topics -->
−
* Equivalent sets
+
* [[Sets:Cardinality]]
−
||5.1-2
 
−
||Sets and functions
 
 
||
 
||
−
* Learn classification of sets by size.
+
<!-- * Prerequisites -->
−
* Generalizing the concept of size to infinite sets
+
* [[Sets:Definitions]]
 +
* [[Equivalence Relations]]
 +
* [[Functions:Injective]]
 +
* [[Functions:Bijective]]
 
||
 
||
−
 
+
<!-- * Outcomes -->
 
|-
 
|-
−
|14
+
| <!-- * Week -->
−
||Cardinality 2
+
14.
−
|| Countable and uncountable sets
 
−
||5.3-5
 
−
||Cardinality 1
 
−
||Learn properties of countable sets.
 
 
||
 
||
 +
<!-- * Sections -->
 +
* 6.6-6.7
 +
||
 +
<!-- * 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 -->
 +
* [[Sets:Cardinality]]
 +
* [[Natural Numbers:Postulates]]
 +
||
 +
<!-- * Outcomes -->
 
|-
 
|-
−
|15
+
| <!-- * Week -->
 +
15.
 +
||
 +
<!-- * Sections -->
 +
||
 +
<!-- * Topics -->
 +
* Catch-up and review for final exam.
 +
||
 +
<!-- * Prerequisites -->
 
||
 
||
−
*Catch up and review for final
+
<!-- * Outcomes -->
−
* Study days
 
 
|}
 
|}
  

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"