Do all of the indicated exercises. Hand in *only* the **bold** exercises.

### Homework 6, due Wednesday , 10/24:

**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.)

### Homework 5, due Wednesday, 10/10:

**Chapter 4:** **60**, **61**, **65**, 67.

(Three exercises to hand in.)

### Homework 4, due Wednesday, 10/3:

**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:

(a) {1, 3, 5, 9, 13, 15}

(b) {9}

(c) {1, 5, 13}

(d) {3, 15}

(e) {3, 7, 11, 15}

(f) {1, 5, 13}

**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do not quote the results proven in Section 4.2.)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.)

### Homework 3, due Wednesday, 9/26:

**(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.)

### Homework 2, due Wednesday, 9/19:

**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.]

### Homework 1, due Wednesday, 9/12:

**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.)