Foundations of Higher Mathematics (Math 300) — Winter 2019

Welcome to Math 300! As its name suggests, this course will introduce you to the fundamental mathematical concepts and skills that you will need to study mathematics at a higher level than a typical introductory calculus and linear algebra sequence. We will emphasise both problem-solving and proof-writing skills, including typesetting mathematics using LaTeX.

Most administrative aspects of the course will be handled using Canvas.

Time and place. Class meets Mo(Tu)WeFr 9:00–9:50am in Lunt Hall 103.

Syllabus. You can download the course syllabus here.

Textbook. Our textbook will be my book-in-progress, An infinite descent into pure mathematics, which can be downloaded for free here.

Homework. Homework assignments will be due at the beginning of class on Wednesdays—they can be downloaded from Canvas.

Examinations. There will be a midterm exam and a final exam.

• Midterm. During class on Wednesday 13th February 2019.
• Final. 9:00–11:00am on Monday 18th March 2019.

Handouts. What follows are the handouts from class and solutions to exercises.

1. Propositions, sets and divisibility — handout — solutions
2. Rational numbers, assumptions and goals — handout — solutions
3. Logical operators — handout — solutions
4. Variables and quantifiers — handout — solutions
5. Truth tables and logical equivalence — handout — solutions
6. LaTeX practical #1 — tex source — pdf output
7. Negation and de Morgan's laws — handout — solutions
8. Sets and set operations — handout — solutions
9. Power sets, products and indexed intersections — handout — solutions
10. Functions and sequences — handout — solutions
11. Images and preimages — handout — solutions
12. Injections, surjections and bijections I — handout — solutions
13. Injections, surjections and bijections II — handout — solutions
14. Definition by recursion and proof by weak induction — handout — solutions
15. Proof by strong induction — handout — solutions
16. LaTeX practical #2 — tex source — pdf output
17. Finite sets — handout — solutions
18. Counting principles I — handout — solutions
19. Counting principles II — handout — solutions
20. Countable sets I — handout — solutions
21. Countable sets II — handout — solutions
22. Relations — handout — solutions
23. Equivalence relations — handout — solutions
24. Discrete probability spaces — handout — solutions
25. Conditional probability — handout — solutions
26. Discrete random variables — handout — solutions