Note that first you should do the exercises from Hammock, and then the challenge problems provided. If you can, then attempt the questions from Munker’s topology. Do read the Behnke book, it contains good discussions. While the notation used is a bit outdated, the discussion is very illuminating.

Readings

1. Book of proofs, Hammock — Chapters 11 and 12
2. Naive Set Theory, Halmos — Chapters 7,8,9
3. Topology, Munkres — Sections 6,7
4. Fundamentals of Mathematics, Behnke — Chapter 1 (1-8)

Book Exercises

• Chapter 1 (Hammock) — Section 1.1 (B & C) , Section 1.2 (A) Section 1.3, 1.4, 1.5 , 1.8
• Chapter 11 (Hammock) — Section 11.1 (1-11) , Section 11.2 (1-8 , 10, 13-15) , Section 11.4
• Chapter 12 (Hammock) — Section 12.1 , Section 12.2
• Section 7 (Munkres) — Exercises of section 7

Warm Up Problems

1. Suppose $f : A \rightarrow B$ is a function and it is surjective. What can you say about the cardinality of $A$ and $B$ assuming both are finite?
2. Suppose $f : A\rightarrow B$ is a function and it is injective. What can you say about the cardinality of $A$ and $B$ assuming both are finite?
3. Suppose $f : A \rightarrow B$ is a bijective function. What can you say about cardinality of $A$ and $B$ assuming both are finite?
4. Show that equality is an equivalence relation.
5. Prove De-Morgan’s laws for both logical statements and sets.
6. Show that the composition of a function with its inverse is the identity function $Id : A \rightarrow A$ defined by $Id(x) = x$

Challenge Problems