Difference between revisions of "MAT3013"
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D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole | D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole | ||
− | ==Topics List== | + | ==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 | ||
+ | * Ideas of proofs and logic | ||
+ | * Axioms and 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 | ||
+ | * Truth values | ||
+ | * Truth tables | ||
+ | * Quantifiers | ||
+ | * nan | ||
+ | || | ||
+ | |||
+ | || | ||
+ | * Gain the prerequisites for writing and evaluating proofs. | ||
+ | |- | ||
+ | |3.0 | ||
+ | || | ||
+ | * 1.4-6 | ||
+ | || | ||
+ | * Methods for proofs | ||
+ | || | ||
+ | * Propositional logic | ||
+ | || | ||
+ | * Start proving elementary results. | ||
+ | |- | ||
+ | |4.0 | ||
+ | || | ||
+ | * 2.1-3 | ||
+ | || | ||
+ | * Basic concepts | ||
+ | * Operations and constructions with sets | ||
+ | || | ||
+ | * Basic concepts of set theory | ||
+ | || | ||
+ | * How to start working with sets | ||
+ | |- | ||
+ | |5.0 | ||
+ | || | ||
+ | * 2.4-6 | ||
+ | || | ||
+ | * Mathematical induction | ||
+ | * Counting principles | ||
+ | || | ||
+ | * Natural numbers | ||
+ | || | ||
+ | * Learn constructive proofs and reasoning. | ||
+ | * Learn basic counting principles of discrete mathematics. | ||
+ | |- | ||
+ | |6.0 | ||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | |- | ||
+ | |7.0 | ||
+ | || | ||
+ | * 3.1-3 | ||
+ | || | ||
+ | * Cartesian products and their subsets | ||
+ | * Equivalence relations | ||
+ | || | ||
+ | * Set theory | ||
+ | || | ||
+ | * Gain basic concepts about relations. | ||
+ | |- | ||
+ | |8.0 | ||
+ | || | ||
+ | * 3.4-5 | ||
+ | || | ||
+ | * Partial orders | ||
+ | * Graphs | ||
+ | || | ||
+ | * Relations 1 | ||
+ | || | ||
+ | * Familiarize with ordering. | ||
+ | * Learn how to use graph representations of relations. | ||
+ | * nan | ||
+ | |- | ||
+ | |9.0 | ||
+ | || | ||
+ | * 4.1-2 | ||
+ | || | ||
+ | * Functions | ||
+ | * Constructions with functions | ||
+ | || | ||
+ | * Relations | ||
+ | * Function sense (precalculus) | ||
+ | || | ||
+ | * Gain basic rigorous knowledge of functions. | ||
+ | |- | ||
+ | |10.0 | ||
+ | || | ||
+ | * 4.3-4 | ||
+ | || | ||
+ | * One-to-one | ||
+ | * Onto | ||
+ | * Compositional inverse | ||
+ | || | ||
+ | * Functions 1 | ||
+ | || | ||
+ | * Determine whether a function is one-to-one of onto, with proofs. | ||
+ | * Finding inverses | ||
+ | |- | ||
+ | |11.0 | ||
+ | || | ||
+ | * 4.5-6 | ||
+ | || | ||
+ | * Images of subsets | ||
+ | * Preimages of subsets | ||
+ | * Sequences | ||
+ | || | ||
+ | * Functions 2 | ||
+ | || | ||
+ | * Find images and preimages of subsets under functions, with proofs. | ||
+ | |- | ||
+ | |12.0 | ||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | |- | ||
+ | |13.0 | ||
+ | || | ||
+ | * 5.1-2 | ||
+ | || | ||
+ | * Finite and infinite sets | ||
+ | * Equivalent sets | ||
+ | || | ||
+ | * Sets and functions | ||
+ | || | ||
+ | * Learn classification of sets by size. | ||
+ | * Generalizing the concept of size to infinite sets | ||
+ | |- | ||
+ | |14.0 | ||
+ | || | ||
+ | * 5.3-5 | ||
+ | || | ||
+ | * Countable and uncountable sets | ||
+ | || | ||
+ | * Cardinality 1 | ||
+ | || | ||
+ | * Learn properties of countable sets. | ||
+ | |- | ||
+ | |15.0 | ||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | || | ||
+ | |||
+ | |} | ||
+ | |||
+ | ==Topics List B== | ||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples | ! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples |
Revision as of 07:58, 16 July 2020
Foundations of Mathematics (3-0) 3 Credit Hours
Contents
Course Catalog
MAT 3013. Foundations of Mathematics. (3-0) 3 Credit Hours.
Prerequisite: MAT 1214. 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 MAT 2243. Credit cannot be earned for MAT 3013 and MAT 2243.) 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
D. Smith, M. Eggen, R. St. Andre, A Transition to Advanced Mathematics (7e), Brooks/Cole
Topics List A
Date | Sections | Topics | Prerequisite Skills | Student Learning Outcomes |
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1.0 |
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2.0 |
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3.0 |
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4.0 |
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5.0 |
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6.0 | ||||
7.0 |
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8.0 |
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9.0 |
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10.0 |
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11.0 |
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12.0 | ||||
13.0 |
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14.0 |
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15.0 |
Topics List B
Week | Session | Topics | Section | Prerequisite skills | Learning outcomes | Examples |
---|---|---|---|---|---|---|
1 | Introduction |
|
1.1 |
mathematics from a historical perspective
and why this course is going | ||
2 | Propositional logic |
|
1.2-3 | Gain the prerequisites for
writing and evaluating proofs. |
| |
3 | Proof methods | Methods for proofs | 1.4-6 | Propositional logic | Start proving elementary results. |
|
4 | Set theory |
|
2.1-3 | Basic concepts of set theory | How to start working with sets |
|
5 | Induction and counting |
|
2.4-6 | Natural numbers |
|
|
6 |
| |||||
7 | Relations 1 |
|
3.1-3 | Set theory | Gain basic concepts about relations. |
|
8 | Relations 2 |
|
3.4-5 | Relations 1 |
|
partial ordering of the power set under inclusion |
9 | Functions 1 |
|
4.1-2 |
|
Gain basic rigorous knowledge of functions. |
functional composition |
10 | Functions 2 |
|
4.3-4 | Functions 1 |
|
|
11 | Functions 3 |
|
4.5-6 | Functions 2 | Find images and preimages of subsets under functions, with proofs. |
|
12 |
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13 | Cardinality 1 |
|
5.1-2 | Sets and functions |
|
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14 | Cardinality 2 | Countable and uncountable sets | 5.3-5 | Cardinality 1 | Learn properties of countable sets. | |
15 |
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