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 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.
(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(For the congruence questions, give proofs based on the definition of congruence -- do not quote the results proven in Section 4.2.)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.)