Chapter 8: 11, 12, 13, 14, 17, 18, 23, 25, 26, 28, 29, 30, 31, 39, 63.

(hand in): Let \(A = \{1,2,3\}\). Define a relation \(R : A \to \mathcal{P}(A)\) by \( x \, R \, Y \) if \(x \in Y\). What is \(R\)?
(Your answer should be an explicit definition of the set \(R\), listing all its elements.)

(Six exercises to hand in.)

Chapter 4: 60, 61, 65, 67.

(Three exercises to hand in.)

Chapter 1: 1, 3, 15, 16, 22, 29.

Check your answers to the odd-numbered questions in the back of the text.
For 16, the cardinality is 4.
For 22 the answers are:

Chapter 4: 14, 15, 18, 19, 42, 45, 75.
(For the congruence questions, give proofs based on the definition of congruence -- do not quote the results proven in Section 4.2.)

Observe for number 15 that \(a \equiv b \pmod n\) if and only if \(b \equiv a \pmod n\), so you are in fact proving the very important result that congruence is transitive: If \(a \equiv b \pmod n\) and \(b \equiv c \pmod n\), then \(a \equiv c \pmod n\).

(Four exercises to hand in.)

(hand in): Prove the following Lemma: Let \(a,b \in \mathbb{Z}\). If \(a\) is odd and \(ab\) is even, then \(b\) is even.

Chapter 4: 1, 2, 5, 6 (use the lemma above), 8, 10, 73.

(Five exercises to hand in.)

Chapter 2: 65, 68, 69, 70.
Chapter 3: 8, 9, 10, 16, 18, 19 (see the proof on pages 88 and 89 that 5x-7 odd implies 9x+2 even), 45, 46.

(Six exercises to hand in.)

[Answers to Chapter 2, 70: (a) True, (b) True, (c) False, (d) True, (e) True, (f) False, (g) True, (h) False.]

Chapter 2: 13, 15, 19, 21, 25, 26, 30, 34, 35, 46, 52, 58, 59, and:
(hand in): Show that \(P \Rightarrow Q\) is logically equivalent to \((\sim\! Q) \Rightarrow (\sim\! P)\).

(Seven exercises to hand in.)