Math 300: Introduction to Mathematical Reasoning

Spring 2022

Lectures: Final exam: Instructor: TAs:

Course content: We will introduce rigorous mathematical reasoning with the goal of teaching you how to write mathematical arguments and proofs. We will cover elementary set theory, elementary examples of functions and operations on functions, the principle of induction, counting, elementary number theory, elementary combinatorics, and recurrence relations.

Schedule: A very brief summary of the content of each lecture will be posted here. The sections of the textbook quoted below are only a rough approximation of what we actually covered in class.

Date Topics covered Due
Week 1
Mar 28 Introduction (Gerstein §1.1)
Mar 30 Logical operations (§1.2-1.3) Reflection 1
Apr 1 Proof strategies and logical equivalence (§1.3-1.4) HW 1, solutions
Week 2
Apr 4 Introduction to sets (§2.1-2.2)
Apr 6 Quantifiers (§2.3) Quiz 1
Apr 8 Set inclusion and operations on sets (§2.4-2.5) HW 2, solutions
Week 3
Apr 11 Index sets and power sets (§2.6-2.7)
Apr 13 Cartesian products and partitions (§2.8-2.9) Reflection 2
Apr 15 Relations (§2.9) HW 3, solutions
Week 4
Apr 18 More on relations and partitions (§2.9)
Apr 20 Induction (§2.10) Quiz 2
Apr 22 Induction -- Take two (§2.10) HW 4, solutions
Week 5
Apr 25 Introduction to functions (§ 3.1)
Apr 27 Injectivity, surjectivity and bijectivity (§ 3.2) Reflection 3
Apr 29 Composition of functions (§ 3.3) HW 5, solutions
Week 6
May 2 Cardinality (§ 4.1)
May 4 Cardinality continued (§ 4.1) Quiz 3
May 6 Class cancelled HW 6, solutions
Week 7
May 9 Finite vs infinite sets (§ 4.1-4.2)
May 11 Countable sets (§ 4.2-4.3) Reflection 4
May 13 Problem session
Week 8
May 16 Uncountable sets (§ 4.2-4.3), slides, video HW 7, solutions
May 18 Uncountability of the real numbers (§ 4.3), slides, video Quiz 4
May 20 Problem session
Week 9
May 23 Number theory--prime factorization, infinitely many primes and division algorithm (§ 6.3) HW 8, solutions
May 25 More number theory: modular arithmetic and discussion (§ 6.4) Reflection 5
May 27 Intro to combinatorics: permutations and counting number of k-subsets of {1,...,n} (§ 5.8)
Week 10
May 30 No class--Memorial Day
Jun 1 More combinatorics: Pascal's triangle and the binomial theorem (§ 5.8), slides, video
Jun 3 Review, slides, video HW 9

Homework assignments: There will be 9 weekly homework assignments each submitted on Canvas. Homeworks will usually be due by class time on Friday. The homework assignments constitute a large component of the class. Homework provides you an opportunity to engage directly with the course material and reinforce ideas from the lectures and textbook.

A selection of the homework problems will be graded and returned to you. Your homework average contributes to 45% of your final grade.

You should expect to spend a considerable amount of time working on the homework. Typically you may want to spend at least six hours outside of class on each homework.

The lowest homework score will be dropped. Homework extensions will not be granted except for extraordinary circumstances.

Reflections: Self-reflections can serve as important educational devices. Every other week, you are required to submit through Canvas a short self-reflection journal entry documenting your mathematical journey over the previous two weeks. Self-reflections are due by Wednesday at 11:59 pm. These entries should be at least one or two paragraphs but you are welcome to write more. Reflections will be graded based on completion with a score of either 0 or 1. Your are allowed to miss one self-reflection.

The content of each self-reflection is up to you but here are some ideas for topics you may want to address:

Quizzes: There will be 5 quizzes during the course each administered during your own time on Canvas. Each quiz is designed to take 20 minutes but you will be given a total of 30 minutes to compensate for the additional time needed to download/upload the quiz. Quizzes need to be completed by Wednesday at 11:59 pm. The lowest quiz score will be dropped.

The content of each quiz will be directly based on the previous one or two homework assignments that have already been graded.

Course policies: