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

### Homework 10, due Wednesday, 12/5:

**Chapter 10:** 9, **10** (construct an enumeration of \(A \times B\)),
23, **24** (exhibit a bijection from one set to the other), 29.

**(hand in):** Let \(A\) and \(B\) be countable sets. Prove that \(A \cup B\) is countable.

[Consider first the special case where \(A \cap B = \varnothing\) (see exercise 10.3),
then adjust your argument for the general case.]

**(hand in):** Let \(A_1, A_2, \ldots, A_n\) be countable sets.
Prove that \(A_1 \cup A_2 \cup \cdots \cup A_n\) is countable.

[You may use the results of the previous exercise.]

**(hand in):** Let \(B\) be a countable set and \(f:A \to B\) a one-to-one function.
Prove that \(A\) is countable.

[Construct a bijection from \(A\) to a subset of \(B\).]

(Five exercises to hand in.)

### Homework 9, due Friday, 11/16:

**Chapter 5:** 40.

**Chapter 6:** 5, 6(b), **10** (\(a\) is a constant), 13, 21, **48**, **50**.

**(hand in):** Recall that if \(f\) and \(g\) are differentiable functions,
then \[\frac{d}{dx} \left(f(x) + g(x)\right) = f'(x) + g'(x).\]

Prove: If \(f_1, f_2, \ldots, f_n\) are differentiable functions, then
\[\frac{d}{dx}\left(f_1(x) + f_2(x) + \cdots + f_n(x)\right) = f_1'(x) + f_2'(x) + \cdots + f_n'(x)\]
for any natural number \(n\).

(Four exercises to hand in.)

### Homework 8, due Wednesday, 11/7:

**Chapter 9:** 33, 37, 41, **42**,
43 (for part (a), only prove it once - not "using as many of the
following proof techniques as possible"),
**58**.

**Chapter 5:** **4**, 10, 11, 13, 14, **16**, 23.

A single sentence is all that is required for exercise 4.
For number 10, *largest* refers to the usual ordering of real numbers,
so, for example, -1 is larger than -5.

(Four exercises to hand in.)

### Homework 7, due Wednesday, 10/31:

**Chapter 9:** 3, 7, 8, **12a,b,c**, **18**, 19, 23, 25,
**26**, 28.

**(hand in):**
Let \(a,b \in \mathbb{R}\) with \(a \neq 0\).
Prove that the function \(f : \mathbb{R} \to \mathbb{R}\)
defined by \(f(x) = ax+b\) is a bijection.

Exercises 23 and 25 are good questions that require some thought - it's a shame that
the answers are in the back of the book. We will discuss these in class.

(Four exercises to hand in.)

### 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.)