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.)