MA 240, Theory of Proof, Section 2
Worcester State University, Fall 2018
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.)