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Chapter 3: Methods of Proof
Exercises for Section 3.1. Quantified Statements
1. (a) For every real number x, (x − 1)2 > 0.
(b) There exists a real number x such that (x − 1)2 > 0.
2. For every integer n, n3 + n + 1 is an odd integer.
3. If n is an odd integer, then 7n + 4 is odd.
4. ∃n ∈ Z, |n − 2| + |n − 1| = |n|.
5. For every odd integer n, n2 + 1 is even.
There exists an odd integer n such that n2 is even.
6. Let S = N and R(x): (x2 + 5x + 6)/2 is even.
For every integer x, (x2 + 5x + 6)/2 is even.
There exists an integer x such that (x2 + 5x + 6)/2 is even.
7. (a) There exists some set A such that A ∩ A ̸= ∅.
(b) For every set A, A ̸⊆ A.
8. (a) There exists a rational number r such that 1/r is not rational.
(b) For every rational number r, r2 ̸= 2.
9. (a) There exists an irrational number s such that for every rational number r, rs ̸= 0.
(b) There exists a rational number r such that for every irrational number s, rs ̸= 0.
(c) For every even integer a there exists an integer b such that ab is odd.
(d) For every odd integer a there exists an integer b such that ab is even.
10. (a) ∀a ∈ Z, ∃b ∈ Z, |a − b| = 1.
(b) ∃a ∈ Z, ∀b ∈ Z, |a − b| = 1.
11. (a) There exists an even integer a such that either a2 or a + 2 is odd.
(b) For every real number x, x + 3 ≤ 0.
12. (a) There is a successful man such that there is no woman who supports him.
(b) There is a successful woman such that there is no man who supports her.
(c) For every woman W, there is a man M such that W is not richer than M.
(d) For every woman W, there is a man M such that W is not older than M.
(e) There is a movie that no one likes.
13. (a) There was never a time when any man owned a singing frog.
(b) No odd integer is the sum of two odd integers.
√
(c) 2 is a rational number.
(d) There is a smallest positive real number.
(e) There are only finitely many primes.
(f) There is a rational number that cannot be expressed as the sum of two rational numbers.
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14. Let S be the set of even integers and T the set of odd integers. Furthermore, let
P (a, b): a + b is odd.
(a) ∀a ∈ S, ∀b ∈ T , P (a, b).
(b) ∃a ∈ S, ∃b ∈ T , ∼ P (a, b).
(c) There exist an even integer a and an odd integer b such that a + b is even.
15. Let S be the set of even integers and T the set of odd integers. Furthermore, let
P (a, b): (a + 2)2 + (b + 3)2 = 0.
(a) ∃a ∈ S, ∃b ∈ T , P (a, b).
(b) ∀a ∈ S, ∀b ∈ T , ∼ P (a, b).
(c) For every even integer a and every odd integer b, (a + 2)2 + (b + 3)2 ̸= 0.
16. Let P (x, y): y < x2 .
(a) ∀x ∈ R, ∃y ∈ R+ , P (x, y).
(b) ∃x ∈ R, ∀y ∈ R+ , ∼ P (x, y).
(c) There exists a real number x such that for every positive real number y, y ≥ x2 .
17. Let P (a, b): ab ≥ 0.
(a) ∃a ∈ Z, ∀b ∈ Z, P (a, b).
(b) ∀a ∈ Z, ∃b ∈ Z, ∼ P (a, b).
(c) For every integer a, there exists some integer b such that ab < 0.
18. There exists an integer a such that for every integer b, | a+1
2 − b| > 1.
19. There exists an integer a such that for every integer b, | 2a+1
− b| ≥ 21 .
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Exercises for Section 3.2. Direct Proof
1. (a) R(2): 1 is even. (F)
R(4): 10 is even. (T)
R(6): 35 is even. (F)
n3 − n
is even. (F)
6
n3 − n
(c) There exists n ∈ S such that
is even. (T)
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(b) For each n ∈ S,
2. (a) P (0): 0 is even. (T)
P (3): 14 is even. (T)
P (4): 30 is even. (T)
n(n + 1)(2n + 1)
(b) For each n ∈ S,
is even. (T)
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n(n + 1)(2n + 1)
is even. (T)
(c) There exists n ∈ S such that
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3. Proof. Let x be a real number such that (x− 1)2 = 0. Thus x− 1 = 0 and so x = 1. Therefore,
x3 − 1 = 13 − 1 = 0.