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Shop What's New Sign inSolution Manual for A Transition to Advanced Mathematics, 7th Edition
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CONTENTS
Chapter 1.1 Propositions and Connectives
1
1.2 Conditionals and Biconditionals
6
1.3 Quantifiers
13
1.4 Basic Proof Methods I
17
1.5 Basic Proof Methods II
22
1.6 Proofs Involving Quantifiers
26
1.7 Additional Examples of Proofs
29
Chapter 2.1 Basic Concepts of Set Theory
38
2.2 Set Operations
40
2.3 Extended Set Operations and Indexed Families of Sets
2.4 Mathematical Induction
49
2.5 Equivalent Forms of Induction
59
2.6 Principles of Counting
62
Chapter 3.1 Cartesian Products and Relations
3.2 Equivalence Relations
70
3.3 Partitions
75
3.4 Ordering Relations
78
3.5 Graphs
82
46
67
Chapter 4.1 Functions as Relations
85
4.2 Constructions of Functions
88
4.3 Functions That Are Onto; One-to-One Functions
92
4.4 One-to-One Correspondences and Inverse Functions
95
4.5 Images of Sets
199
4.6 Sequences
102
Chapter 5.1 Equivalent Sets; Finite Sets
105
5.2 Infinite Sets
108
5.3 Countable Sets
111
5.4 The Ordering of Cardinal Numbers
114
5.5 Comparability of Cardinal Numbers and the Axiom of Choice
Chapter 6.1 Algebraic Structures
119
6.2 Groups
122
6.3 Subgroups
126
6.4 Operation Preserving Maps
6.5 Rings and Fields
131
128
Chapter 7.1 Completeness of the Real Numbers
133
7.2 The Heine-Borel Theorem
136
7.3 The Bolzano-Weierstrass Theorem
139
7.4 The Bounded Monotone Sequence Theorem
7.5 Equivalents of Completeness
143
140
117
1
Logic and Proofs
1.1
Propositions and Connectives
1.
(a) true
(e) false
(b) false
(f) false
(c) true
(g) false
(d) false
(h) false
2. (a) Not a proposition
(b) False proposition
(c) Not a proposition. It would be a proposition if a value for x had been
assigned.
(d) Not a proposition. It would be a proposition if values for x and y had been
assigned.
(e) False proposition
(f) True proposition
(g) False proposition
(h) True proposition
(i) False proposition
(j) Not a proposition. It is neither true nor false.
3. (a)
(b)
(c)
(d)
(e)
(f)
P
T
F
โผP
F
T
Pโง โผ P
T
F
P
T
F
โผP
F
T
Pโจ โผ P
T
T
P
T
F
T
F
Q
T
T
F
F
โผQ
F
F
T
T
Pโง โผ Q
F
F
T
F
P
T
F
T
F
Q
T
T
F
F
โผQ
F
F
T
T
Qโจ โผ Q
T
T
T
T
P
T
F
T
F
Q
T
T
F
F
โผQ
F
F
T
T
P โงQ
T
F
F
F
P
T
F
T
F
Q
T
T
F
F
P โงQ
T
F
F
F
P โง (Qโจ โผ Q)
T
F
T
F
(P โง Q)โจ โผ Q
T
F
T
T
โผ (P โง Q)
F
T
T
T
1
1
LOGIC AND PROOFS
(g)
(h)
(i)
(j)
4.
2
โผQ
F
F
T
T
F
F
T
T
Pโจ โผ Q
T
F
T
T
T
F
T
T
(P โจ โผ Q) โง R
T
F
T
T
F
F
F
F
P
T
F
T
F
T
F
T
F
Q
T
T
F
F
T
T
F
F
R
T
T
T
T
F
F
F
F
P
T
F
T
F
Q
T
T
F
F
โผP
F
T
F
T
โผQ
F
F
T
T
โผ Pโง โผ Q
F
F
F
T
P
T
F
T
F
T
F
T
F
Q
T
T
F
F
T
T
F
F
R
T
T
T
T
F
F
F
F
QโจR
T
T
T
T
T
T
F
F
P โง (Q โจ R)
T
F
T
F
T
F
F
F
P
T
F
T
F
T
F
T
F
Q
T
T
F
F
T
T
F
F
R
T
T
T
T
F
F
F
F
P โงQ
T
F
F
F
T
F
F
F
P โงR
T
F
T
F
F
F
F
F
(a) false
(e) false
(i) true
(b) true
(f) false
(j) true
(P โง Q) โจ (P โง R)
T
F
T
F
T
F
F
F
(c) true
(g) false
(k) false
(d) true
(h) false
(1) false
5. (a) No solution.
(b)
P Q P โจQ QโจP
T T
T
T
F T
T
T
T F
T
T
F F
F
F
Since the third and fourth columns are the same, the propositions are
equivalent.
1
LOGIC AND PROOFS
(c)
(d)
(e)
(f)
(g)
3
P Q P โงQ QโงP
T T
T
T
F T
F
F
T F
F
F
F F
F
F
Since the third and fourth columns are the same, the propositions are
equivalent.
P Q R Q โจ R P โจ (Q โจ R) P โจ Q (P โจ Q) โจ R
T T T
T
T
T
T
F T T
T
T
T
T
T F T
T
T
T
T
F F T
T
T
F
T
T T F
T
T
T
T
F T F
T
T
T
T
T F F
F
T
T
T
F F F
F
F
F
F
Since the ๏ฌfth and seventh columns are the same, the propositions are
equivalent.
P Q R Q โง R P โง (Q โง R) P โง Q (P โง Q) โง R
T T T
T
T
T
T
F T T
T
F
F
F
T F T
F
F
F
F
F F T
F
F
F
F
T T F
F
F
T
F
F T F
F
F
F
F
T F F
F
F
F
F
F F F
F
F
F
F
Since the ๏ฌfth and seventh columns are the same, the propositions are
equivalent.
P Q R Q โจ R P โง (Q โจ R) P โง Q P โง R (P โง Q) โจ (P โง R)
T T T
T
T
T
T
T
F T T
T
F
F
F
F
T F T
T
T
F
T
T
F F T
T
F
F
F
F
T T F
T
T
T
F
T
F T F
T
F
F
F
F
T F F
F
F
F
F
F
F F F
F
F
F
F
F
Since the ๏ฌfth and eighth columns are the same, the propositions are
equivalent.
P
T
F
T
F
T
F
T
F
Q
T
T
F
F
T
T
F
F
R
T
T
T
T
F
F
F
F
QโงR
T
T
F
F
F
F
F
F
P โจ (Q โง R)
T
T
T
F
T
F
T
F
P โจQ
T
T
T
F
T
T
T
F
P โจR
T
T
T
T
T
F
T
F
(P โจ Q) โง (P โจ R)
T
T
T
F
T
F
T
F
1
LOGIC AND PROOFS
4
Since the ๏ฌfth and eighth columns are the same, the propositions are
equivalent.
(h) No solution.
(i)
P Q P โจ Q โผ (P โจ Q) โผ P โผ Q โผ P โง โผ Q
T T
T
F
F
F
F
F T
T
F
T
F
F
T F
T
F
F
T
F
F F
F
T
T
T
T
Since the fourth and eighth columns are the same, the propositions are
equivalent.
6.
(a) equivalent
(c) equivalent
(e) equivalent
(g) not equivalent
(b) equivalent
(d) equivalent
(f) not equivalent
(h) not equivalent
7.
(a) โผ P , true
(c) P Q, true
(b) P โง Q, true
(d) P โจ Q โจ R, true
8. (a) Since P is equivalent to Q, P has the same truth table as Q. Therefore, Q
has the same truth table as P , so Q is equivalent to P .
(b) Since P is equivalent to Q, P and Q have the same truth table. Since Q is
equivalent to R, Q and R have the same truth table. Thus, P and R have
the same truth table so P is equivalent to R.
(c) Since P is equivalent to Q, P and Q have the same truth table. That is, the
truth table for P has value true on exactly the same lines that the truth
table for Q has value true. Therefore the truth table for โผ Q has value false
on exactly the same lines that the truth table for โผ P has the value false.
Thus โผ Q and โผ P have the same truth table.
9. (a) (P โง Q) โจ (โผ P โง โผ Q) is neither.
P
T
F
T
F
Q
T
T
F
F
โผP
F
T
F
T
โผQ
F
F
T
T
P โงQ
T
F
F
F
โผ Pโง โผ Q
F
F
F
T
(P โง Q) โจ (โผ P โง โผ Q)
T
F
F
T
(b) โผ (P โง โผ P ) is a tautology.
P
T
F
โผP
F
T
Pโง โผ P
F
F
โผ (P โง โผ P )
T
T
(c) (P โง Q) โจ (โผ P โจ โผ Q) is a tautology.
P
T
F
T
F
Q
T
T
F
F
โผP
F
T
F
T
โผQ
F
F
T
T
P โงQ
T
F
F
F
โผ Pโจ โผ Q
F
T
T
T
(P โง Q) โจ (โผ P โจ โผ Q)
T
T
T
T
(d) (A โง B) โจ (Aโง โผ B) โจ (โผ A โง B) โจ (โผ Aโง โผ B) is a tautology.
1
LOGIC AND PROOFS
A
T
F
T
F
B
T
T
F
F
โผA
F
T
F
T
โผB
F
F
T
T
5
AโงB
T
F
F
F
Aโง โผ B
F
F
T
F
โผAโงB
F
T
F
F
โผ Aโง โผ B
F
F
F
T
(A โง B) โจ (Aโง โผ B)โจ
(โผ A โง B) โจ (โผ Aโง โผ B)
T
T
T
T
(e) (Qโง โผ P )โง โผ (P โง R) is neither.
P
T
F
T
F
T
F
T
F
Q
T
T
F
F
T
T
F
F
R
T
T
T
T
F
F
F
F
โผP
F
T
F
T
F
T
F
T
Qโง โผ P
F
T
F
F
F
T
F
F
P โงR
T
F
T
F
F
F
F
F
โผ (P โง R)
F
T
F
T
T
T
T
T
(Qโง โผ P )โง โผ (P โง R)
F
T
F
F
F
T
F
F
(f) P โจ [(โผ Q โง P ) โง (R โจ Q)] is neither.
P
T
F
T
F
T
F
T
F
10.
Q
T
T
F
F
T
T
F
F
R
T
T
T
T
F
F
F
F
โผQ
F
F
T
T
F
F
T
T
โผQโงP
F
F
T
F
F
F
T
F
(a) contradiction
(c) tautology
RโจQ
T
T
T
T
T
T
F
F
[(โผ Q โง P ) โง (R โจ Q)]
F
F
T
F
F
F
F
F
P โจ [(โผ Q โง P ) โง (R โจ Q)]
T
F
T
F
T
F
T
F
(b) tautology
(d) tautology
11. (a) x is not a positive integer.
(b) Cleveland will lose the ๏ฌrst game and the second game. Or, Cleveland will
lose both games.
(c) 5 < 3
(d) 641,371 is not composite. Or 641,371 is prime.
(e) Roses are not red or violets are not blue.
(f) T is bounded and T is not compact.
(g) M is not odd or M is not one-to-one.
(h) The function.f does not have a positive ๏ฌrst derivative at x or does not
have a positive second derivative at x.
(i) The function g does not have a relative maximum at x = 2 (deleted comma)
and does not have a relative maximum at x = 4, or else g does not have a
relative minimum at x = 3.
(j) z 4) โจ (n > 10)
(i) (x is Cauchy) โ (x is convergent)
(j) (limxโx0 f (x) = f (x0 )) โ (f is continuous at x0 )
(k) [(f is di๏ฌerentiable at x0 ) โง (f is strictly increasing at x0 )] โ (f (x0 ))
11. (a) Let S be โI go to the storeโ and R be โIt rains.โ The preferred translation:
is โผ S โ R (or, equivalently, โผ R โ S). This could be read as โIf it
doesnโt rain, then I go to the store.โ
The speaker might mean โI go to the store if and only if it doesnโt rain
(S โโผ R) or possibly โIf it rains, then I donโt go to the storeโ (R โโผ S).
(b) There are three nonequivalent ways to translate the sentence, using the
symbols D: โThe Dolphins make the playo๏ฌsโ and B: โThe Bears win all
the rest of their games.โ The ๏ฌrst translation is preferred, but the speaker
may have intended any of the three.
โผ B โโผ D or, equivalently, D โ B
โผ D โโผ B or, equivalently, B โ D
โผ B โโผ D or, equivalently, B โ D
(c) Let G be โYou can go to the gameโ and H be โYou do your homework
๏ฌrst.โ
It is most likely that a student and parent both interpret this statement as
a biconditional, G โ H.
(d) Let W be โYou win the lotteryโ and T be โYou buy a ticket.โ Of the three
common interpretations for the word โunless,โ only the form โผ T โโผ W
(or, equivalently, W โ T ) makes sense here.
12. (a)
(b)
P Q R P โจ Q (P โจ Q) โ R โผ P โง โผ Q โผ R โ (โผ P โง โผ Q)
T T T
T
T
F
T
F T T
T
T
F
T
T F T
T
T
F
T
F F T
F
T
T
T
T T F
T
F
F
F
F T F
T
F
F
F
T F F
T
F
F
F
F F F
F
T
T
T
Since the ๏ฌfth and seventh columns are the same, (P โจ Q) โ R and
โผ R โ (โผ P โง โผ Q) are equivalent.
P
T
F
T
F
T
F
T
F
Q
T
T
F
F
T
T
F
F
R
T
T
T
T
F
F
F
F
P โงQ
T
F
F
F
T
F
F
F
(P โง Q) โ R
T
T
T
T
F
T
T
T
โผQ
F
F
T
T
F
F
T
T
โผR
F
F
F
F
T
T
T
T
Pโง โผ R
F
F
F
F
T
F
T
F
(P โง โผ R) โโผ Q
T
T
T
T
F
T
T
T
Since the ๏ฌfth and ninth columns are the same, the propositions (P โง Q) โ
R and (P โง โผ R) โโผ Q are equivalent.
1
LOGIC AND PROOFS
(c)
(d)
(e)
(f)
12
P Q R Q โง R P โ (Q โง R) โผ Qโจ โผ R (โผ Qโจ โผ R) โโผ P
T T T
T
T
F
T
F T T
T
T
F
T
T F T
F
F
T
F
F F T
F
T
T
T
T T F
F
F
T
F
F T F
F
T
T
T
T F F
F
F
T
F
F F F
F
T
T
T
Since the ๏ฌfth and seventh columns are the same, the propositions P โ
(Q โง R) and (โผ Qโจ โผ R) โโผ P are equivalent.
P Q R Q โจ R P โ (Q โจ R) P โง โผ R (P โง โผ R) โ Q
T T T
T
T
F
T
F T T
T
T
F
T
T F T
T
T
F
T
F F T
T
T
F
T
T T F
T
T
T
T
F T F
T
T
F
T
T F F
F
F
T
F
F F F
F
T
F
T
Since the ๏ฌfth and seventh columns are the same, the propositions P โ
(Q โจ R) and (P โง โผ R) โ Q are equivalent.
P Q R P โ Q (P โ Q) โ R P โง โผ Q (P โง โผ Q) โจ R
T T T
T
T
F
T
F T T
T
T
F
T
T F T
F
T
T
T
F F T
T
T
F
T
T T F
T
F
F
F
F T F
T
F
F
F
T F F
F
T
T
T
F F F
T
F
F
F
Since the ๏ฌfth and seventh columns are the same, the propositions (P โ
Q) โ R and (P โง โผ Q) โจ R are equivalent.
P Q P โ Q โผ P โจ Q โผ Q โจ P (โผ P โจ Q) โง (โผ Q โจ P )
T T
T
T
T
T
F T
F
T
F
F
T F
F
F
T
F
F F
T
T
T
T
Since the third and sixth columns are the same, the propositions P โ Q
and (โผ P โจ Q) โง (โผ Q โจ P ) are equivalent.
13. (a) If 6 is an even integer, then 7 is an odd integer.
(b) If 6 is an odd integer, then 7 is an odd integer.
(c) This is not possible.
(d) If 6 is an even integer, then 7 is an even integer. (Any true conditional
statement will work here.)
14. (a) If 7 is an odd integer, then 6 is an odd integer.
(b) This is not possible.
1
LOGIC AND PROOFS
13
(c) This is not possible.
(d) If 7 is an odd integer, then 6 is an odd integer. (Any false conditional
statement will work here.)
15. (a) Converse: If f (x0 ) = 0, then f has a relative minimum at x0 and is
di๏ฌerentiable at x0 . False: f (x) = x3 has ๏ฌrst derivative 0 but no minimum
at x0 = 0.
Contrapositive: If f (x0 ) = 0, then f either has no relative minimum at x0
or is not di๏ฌerentiable at x0 . True.
(b) Converse: If n = 2 or n is odd, then n is prime. False: 9 is odd but not
prime.
Contrapositive: If n is even and not equal to 2, then n is not prime. True.
(c) Converse: If x is irrational, then x is real and not rational. True
Contrapositive: If x is not irrational, then x is not real or x is rational. True
(d) Converse: If |x| = 1, then x = 1 or x = โ1. True.
Contrapositive: If |x| = 1, then x = 1 and x = โ1. True.
16.
(a) tautology
(d) neither
(g) contradiction
(j) neither
17. (a)
(b) tautology
(e) tautology
(h) tautology
(k) tautology
(c) contradiction
(f) neither
(i) contradiction
(l) neither
P Q P โ Q โผ P โผ Q โผ P โโผ Q
T T
T
F
F
T
F T
T
T
F
F
T F
F
F
T
T
F F
T
T
T
T
Comparison of the third and sixth columns of the truth table shows that
P โ Q and โผ P โโผ Q are not equivalent.
(b) We see from the truth table in part (a) that both propositions P โ Q and
โผ P โโผ Q are true only when P and Q have the same truth value.
(c) The converse of P โ Q is Q โ P . The contrapositive of the inverse of
P โ Q is โผโผ Q โโผโผ P , so the converse and the contrapositive of the
inverse are equivalent.
The inverse of the contrapositive of P โ Q is also โผโผ Q โโผโผ P , so it
too is equivalent to the converse.
1.3
Quanti๏ฌers
1. (a) โผ (โx)(x is preciousโ x is beautiful) or (โ x)(x is precious and x is not
beautiful)
(b) (โx)(x is preciousโ x is not beautiful)
(c) (โ x)(x is isoscelesโงx is a right triangle)
(d) (โx)(x is a right triangleโ x is not isosceles) or โผ (โ x)(x is a right
triangleโงx is isosceles)
(e) (โx)(x is honest)โจ โผ (โ x)(x is honest)
(f) (โ x)(x is honest) โง (โ x)(x is not honest)
(g) (โx)(x = 0 โ (x > 0 โจ x โ4 โจ x โ4 โจ x < 6)
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