# Introduction to Mathematical Proofs

Lectures: MW 17:35–19:40 in HW 611

Office hours: MW 17:00–17:30 in HE 924 or email me at miha.habic@hunter.cuny.edu for a different time

Course syllabus

The final will be on the last day of class, **July 13**.

To prepare for the final I recommend you go over the homeworks and make sure you understand those problems. Also have a look at the practice problems I’ve been assigning from the textbook. Additionally, here are some extra practice problems (and solutions).

Here is a sample final (and solutions). Take it as an example of the length and the type of problems that will appear on the actual final. The fact sheet on the sample is the same as the you will get on the actual final. Please do not expect the problems on the real final to be identical to the sample ones with the numbers changed.

## Homework sets and practice problems

July 6: Equipotency of sets, countable and uncountable sets, Cantor’s diagonal argument, the continuum hypothesis

June 29: The fundamental theorem of arithmetic, Euclid’s theorem on the infinitude of primes. Functions; injective, surjective, bijective functions (Sections 11.6, 9.1, 9.3)

- No homework today.

June 27: Bézout’s lemma, equivalent definitions of primes (I didn’t follow the book here, but much of this is in Sections 11.3 and 11.6)

- No homework today.
- Problem for honour and glory (Goldbach’s theorem on prime-producing formulas)

June 22: Sections 6.1, 6.2, 6.4

- Homework set #5
- Practice problems (don’t hand these in): 6.5, 6.9, 6.11, 6.16, 6.21, 6.42, 6.45

June 20: Sections 5.1, 5.2, 5.4, 5.5

- No homework today.
- Here is a guided “fill in the gaps” outline of a proof of the irrationality of
*e*. Have a look if you like. It needs a bit of familiarity with infinite series and is outside the scope of this course, but is a fun proof nonetheless. - Practice problems (don’t hand these in): 5.1, 5.3, 5.14, 5.17, 5.26, 5.28, 5.31

June 15: Sections 3.4, 4.1

- Homework set #4
- Please read Section 3.5 in the textbook.
- Practice problems (don’t hand these in): 3.16, 3.17, 3.20, 3.21, 3.29, 3.31, 3.35, 3.39, 3.40, 3.41, 3.53

June 13: Sections 3.2, 3.3

- Homework set #3
- Practice problems (don’t hand these in): 3.1, 3.3, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.13, 3.15

June 8: Sections 2.9, 2.10, 3.2

- Homework set #2 (pay close attention to problems 5 and 6 if you are into computer science or electrical engineering; there might be an honour and glory problem based on problem 4 in the future)
- Practice problems (don’t hand these in): 2.59, 2.68, 2.70, 2.71, 2.79, 2.88, 2.99

June 6: Sections 1.4, 1.6–2.8

- No homework today, but please do the practice problems (if you have the book). I may also transcribe some of the practice problems for those without the book on Tuesday.
- Practice problems (don’t hand these in): 2.3, 2.6, 2.13, 2.17, 2.21, 2.24, 2.29, 2.45, 2.48, 2.49, 2.54

June 1: Sections 1.1–1.4

- Read Chapter 0 in the textbook
- Homework set #1 (problem 6a is much like problem 4; do problem 6b if you are feeling inspired after doing those two)
- Problem for honour and glory (this is like problems 4 and 6 on the homework but much more tricky)
- Practice problems (don’t hand these in): 1.6, 1.10, 1.20, 1.21, 1.25, 1.29, 1.41, 1.45