Difference between revisions of "MAT3013"

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! 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
 
||
 
||
 +
<!-- * Topics -->
 
* [[Statements]]
 
* [[Statements]]
 
* [[Sentential Logic]]
 
* [[Sentential Logic]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 
||
 
||
 +
<!-- * Outcomes -->
 
* Identify syntactically correct formulas in sentential logic.
 
* Identify syntactically correct formulas in sentential logic.
 
* Translate compound statements in informal language to formal propositional sentences.
 
* Translate compound statements in informal language to formal propositional sentences.
 
* Find the interpretation of a sentential formula given interpretations of the propositional symbols therein.
 
* Find the interpretation of a sentential formula given interpretations of the propositional symbols therein.
 
|-
 
|-
−
|2.
+
| <!-- * Week -->
 +
2.
 
||
 
||
 +
<!-- * Sections -->
 
* 1.3-1.4
 
* 1.3-1.4
 
||
 
||
−
* [[Semantic Implication]]
+
<!-- * Topics -->
−
* [[Semantic Equivalence]]
+
* [[Logical Implication]]
 +
* [[Logical Equivalence]]
 
* [[Deductive Rules]]
 
* [[Deductive Rules]]
 
||
 
||
 +
<!-- * Prerequisites -->
 
* [[Sentential Logic]]
 
* [[Sentential Logic]]
 
||
 
||
 +
<!-- * Outcomes -->
 
* Express informally stated relations between sentences in terms of semantic implication and equivalence.
 
* 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.
 
* State and recognize basic rules of deductive reasoning and their correct application.
Line 59: Line 68:
 
* Distinguish between correct and incorrect applications of deductive rules.
 
* Distinguish between correct and incorrect applications of deductive rules.
 
|-
 
|-
−
|2/3
+
| <!-- * Week -->
 +
3.
 
||
 
||
 +
<!-- * Sections -->
 
* 1.5-2.2
 
* 1.5-2.2
 
||
 
||
 +
<!-- * Topics -->
 
* [[Quantifiers]]
 
* [[Quantifiers]]
 
* [[Mathematical Proofs]]
 
* [[Mathematical Proofs]]
 
* [[Proofs:Direct]]
 
* [[Proofs:Direct]]
 
||
 
||
 +
<!-- * Prerequisites -->
 
* [[Sentential Logic]]
 
* [[Sentential Logic]]
 
* [[Deductive Rules]]
 
* [[Deductive Rules]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|3.
+
| <!-- * Week -->
 +
4.
 
||
 
||
 +
<!-- * Sections -->
 
* 2.3-2.4
 
* 2.3-2.4
 
||
 
||
 +
<!-- * Topics -->
 
* [[Proofs:Contraposition]]
 
* [[Proofs:Contraposition]]
 
* [[Proofs:Contradiction]]
 
* [[Proofs:Contradiction]]
 
* [[Proofs:Cases]]
 
* [[Proofs:Cases]]
−
* [[Proofs:Biconditionals]]
 
 
||
 
||
 +
<!-- * Prerequisites -->
 
* [[Mathematical Proofs]]
 
* [[Mathematical Proofs]]
 
* [[Proofs:Direct]]
 
* [[Proofs:Direct]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|4.
+
| <!-- * Week -->
 +
5.
 
||
 
||
−
* 2.5-2.6
+
<!-- * Sections -->
 +
* 2.4-2.6
 
||
 
||
 +
<!-- * Topics -->
 +
* [[Proofs:Biconditionals]]
 
* [[Proofs:Quantifiers]]
 
* [[Proofs:Quantifiers]]
 
* [[Writing Mathematics]]
 
* [[Writing Mathematics]]
 
||
 
||
 +
<!-- * Prerequisites -->
 
* [[Quantifiers]]
 
* [[Quantifiers]]
 
* [[Mathematical Proofs]]
 
* [[Mathematical Proofs]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|5.
+
| <!-- * Week -->
 +
6.
 
||
 
||
 +
<!-- * Sections -->
 
* 3.1-3.3
 
* 3.1-3.3
 
||
 
||
 +
<!-- * Topics -->
 
* [[Sets:Definitions]]
 
* [[Sets:Definitions]]
 
* [[Sets:Operations]]
 
* [[Sets:Operations]]
 +
* [[Sets:Families]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|5.
+
| <!-- * Week -->
−
||
+
7.
−
* 3.4
 
 
||
 
||
−
* [[Sets:Families]]
+
<!-- * Sections -->
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|6.
 
 
||
 
||
 +
<!-- * Topics -->
 
* Review of Chapters 1-3.
 
* Review of Chapters 1-3.
 
* Midterm exam.
 
* Midterm exam.
 
||
 
||
−
 
+
<!-- * Prerequisites -->
 
||
 
||
−
* Prerequisites
+
<!-- * Outcomes -->
−
||
 
−
* Outcomes
 
 
|-
 
|-
−
|7.
+
| <!-- * Week -->
 +
8.
 
||
 
||
−
* 4.1-4.2
+
<!-- * Sections -->
 +
* 4.1-4.3
 
||
 
||
 +
<!-- * Topics -->
 
* [[Functions:Definition]]
 
* [[Functions:Definition]]
−
* [[Functions:ForwardImage]]
+
* [[Functions:Forward Image]]
−
* [[Functions:InverseImage]]
+
* [[Functions:Forward Image|Functions:Inverse Image]]
 +
* [[Functions:Composition]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sets:Definitions]]
 +
* [[Sets:Operations]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|8.
+
| <!-- * Week -->
 +
9.
 
||
 
||
 +
<!-- * Sections -->
 
* 4.3-4.4
 
* 4.3-4.4
 
||
 
||
−
* [[Functions:Composition]]
+
<!-- * Topics -->
 
* [[Functions:Inverses]]
 
* [[Functions:Inverses]]
 
* [[Functions:Injective]]
 
* [[Functions:Injective]]
Line 152: Line 177:
 
* [[Functions:Bijective]]
 
* [[Functions:Bijective]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Functions:Definition]]
 +
* [[Functions:Composition]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|9.
+
| <!-- * Week -->
 +
10.
 
||
 
||
 +
<!-- * Sections -->
 
* 5.1 & 5.3
 
* 5.1 & 5.3
 
||
 
||
 +
<!-- * Topics -->
 
* [[Relations]]
 
* [[Relations]]
−
* [[Functions as relations]]
+
* [[Functions as Relations]]
−
* [[Equivalence relations]]
+
* [[Equivalence Relations]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sets:Definitions]]
 +
* [[Sets:Operations]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|10.
+
| <!-- * Week -->
 +
11.
 
||
 
||
 +
<!-- * Sections -->
 
* 6.1-6.3
 
* 6.1-6.3
 
||
 
||
 +
<!-- * Topics -->
 
* [[Natural Numbers:Postulates]]
 
* [[Natural Numbers:Postulates]]
 
* [[Natural Numbers:Well-Ordering]]
 
* [[Natural Numbers:Well-Ordering]]
−
* [[Induction]]
+
* [[Proofs:Induction]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sets:Definitions]]
 +
* [[Functions:Definition]]
 +
* [[Relations]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|11.
+
| <!-- * Week -->
 +
12.
 
||
 
||
 +
<!-- * Sections -->
 
* 6.3-6.4
 
* 6.3-6.4
 
||
 
||
−
* [[Induction:Variants]]
+
<!-- * Topics -->
 +
* [[Proofs:Induction|Induction:Variants]]
 
* [[Recursion]]
 
* [[Recursion]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Proofs:Induction]]
 +
* [[Functions:Definition]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|12.
+
| <!-- * Week -->
 +
13.
 
||
 
||
 +
<!-- * Sections -->
 
* 6.5
 
* 6.5
 
||
 
||
−
* [[Cardinality of sets]]
+
<!-- * Topics -->
−
||
+
* [[Sets:Cardinality]]
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|13.
 
−
||
 
−
* 6.6-6.7
 
−
||
 
−
* [[Sets:Finite]]
 
−
* [[Sets:Countable]]
 
−
* [[Sets:Uncountable]]
 
−
* [[Cardinality:𝐍]]
 
−
* [[Cardinality:𝐙]]
 
−
* [[Cardinality:𝐐]]
 
−
* [[Cardinality:𝐑]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|15.0
 
−
||
 
−
 
 
−
||
 
−
 
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|}
 
−
==See also==
 
−
 
 
−
* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]
 
−
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==
 
−
 
 
−
* 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==
 
−
{| class="wikitable sortable"
 
−
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
 
−
|-
 
−
|1.
 
−
||
 
−
* 1.1-1.2
 
−
||
 
−
* [[Statements]]
 
−
* [[Sentential Logic]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* 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
 
−
||
 
−
* [[Semantic Implication]]
 
−
* [[Semantic Equivalence]]
 
−
* [[Deductive Rules]]
 
−
||
 
−
* [[Sentential Logic]]
 
−
||
 
−
* 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
 
−
||
 
−
* [[Quantifiers]]
 
−
* [[Mathematical Proofs]]
 
−
* [[Proofs:Direct]]
 
−
||
 
−
* [[Sentential Logic]]
 
−
* [[Deductive Rules]]
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|4.
 
−
||
 
−
* 2.3-2.4
 
−
||
 
−
* [[Proofs:Contraposition]]
 
−
* [[Proofs:Contradiction]]
 
−
* [[Proofs:Cases]]
 
−
* [[Proofs:Biconditionals]]
 
−
||
 
−
* [[Mathematical Proofs]]
 
−
* [[Proofs:Direct]]
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|5.
 
−
||
 
−
* 2.5-2.6
 
−
||
 
−
* [[Proofs:Quantifiers]]
 
−
* [[Writing Mathematics]]
 
−
||
 
−
* [[Quantifiers]]
 
−
* [[Mathematical Proofs]]
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|6.
 
−
||
 
−
* 3.1-3.3
 
 
||
 
||
 +
<!-- * Prerequisites -->
 
* [[Sets:Definitions]]
 
* [[Sets:Definitions]]
−
* [[Sets:Operations]]
+
* [[Equivalence Relations]]
−
* [[Sets:Families]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|7.
 
−
||
 
−
* Review of Chapters 1-3.
 
−
* Midterm exam.
 
−
||
 
−
 
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|8.
 
−
||
 
−
* 4.1-4.2
 
−
||
 
−
* [[Functions:Definition]]
 
−
* [[Functions:ForwardImage]]
 
−
* [[Functions:InverseImage]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|9.
 
−
||
 
−
* 4.3-4.4
 
−
||
 
−
* [[Functions:Composition]]
 
−
* [[Functions:Inverses]]
 
 
* [[Functions:Injective]]
 
* [[Functions:Injective]]
−
* [[Functions:Surjective]]
 
 
* [[Functions:Bijective]]
 
* [[Functions:Bijective]]
 
||
 
||
−
* Prerequisites
+
<!-- * Outcomes -->
−
||
 
−
* Outcomes
 
−
|-
 
−
|10.
 
−
||
 
−
* 5.1 & 5.3
 
−
||
 
−
* [[Relations]]
 
−
* [[Functions as relations]]
 
−
* [[Equivalence relations]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|11.
 
−
||
 
−
* 6.1-6.3
 
−
||
 
−
* [[Natural Numbers:Postulates]]
 
−
* [[Natural Numbers:Well-Ordering]]
 
−
* [[Induction]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
 
|-
 
|-
−
|12.
+
| <!-- * Week -->
−
||
+
14.
−
* 6.3-6.4
 
−
||
 
−
* [[Induction:Variants]]
 
−
* [[Recursion]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|13.
 
−
||
 
−
* 6.5
 
−
||
 
−
* [[Cardinality of sets]]
 
−
||
 
−
* Prerequisites
 
−
||
 
−
* Outcomes
 
−
|-
 
−
|14.
 
 
||
 
||
 +
<!-- * Sections -->
 
* 6.6-6.7
 
* 6.6-6.7
 
||
 
||
 +
<!-- * Topics -->
 
* [[Sets:Finite]]
 
* [[Sets:Finite]]
 
* [[Sets:Countable]]
 
* [[Sets:Countable]]
 
* [[Sets:Uncountable]]
 
* [[Sets:Uncountable]]
−
* [[Cardinality:𝐍]]
+
* [[Cardinality of important sets|Cardinality:𝐍]]
−
* [[Cardinality:𝐙]]
+
* [[Cardinality of important sets|Cardinality:𝐙]]
−
* [[Cardinality:𝐐]]
+
* [[Cardinality of important sets|Cardinality:𝐐]]
−
* [[Cardinality:𝐑]]
+
* [[Cardinality of important sets|Cardinality:𝐑]]
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 +
* [[Sets:Cardinality]]
 +
* [[Natural Numbers:Postulates]]
 
||
 
||
−
* Outcomes
+
<!-- * Outcomes -->
 
|-
 
|-
−
|15.
+
| <!-- * Week -->
 +
15.
 
||
 
||
−
 
+
<!-- * Sections -->
 
||
 
||
 +
<!-- * Topics -->
 
* Catch-up and review for final exam.
 
* Catch-up and review for final exam.
 
||
 
||
−
* Prerequisites
+
<!-- * Prerequisites -->
 
||
 
||
−
* Outcomes
+
<!-- * 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"