Page 158 - HOW TO PROVE IT: A Structured Approach, Second Edition

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144 Proofs
p d8. Prove that for any sets A and B,if P (A) ∪ P (B) = P (A ∪ B) then
either A ⊆ B or B ⊆ A.
9. Suppose x and y are real numbers and x = 0. Prove that y + 1/x =
1 + y/x iff either x = 1or y = 1.
2
10. Prove that for every real number x,if |x − 3| > 3 then x > 6x. (Hint:
According to the definition of |x − 3|,if x − 3 ≥ 0 then |x − 3|= x −
3, and if x − 3 < 0 then |x − 3|= 3 − x. The easiest way to use this
fact is to break your proof into cases. Assume that x − 3 ≥ 0 in case 1,
and x − 3 < 0 in case 2.)
∗ 11. Prove that for every real number x, |2x − 6| > x iff |x − 4| > 2. (Hint:
Read the hint for exercise 10.)
12. (a) Prove that for all real numbers a and b, |a|≤ b iff −b ≤ a ≤ b.
(b) Prove that for any real number x, −|x|≤ x ≤|x|. (Hint: Use
part (a).)
(c) Prove that for all real numbers x and y, |x + y|≤|x|+|y|. (This is
called the triangle inequality. One way to prove this is to combine
parts (a) and (b), but you can also do it by considering a number of
cases.)
2
13. Prove that for every integer x, x + x is even.
4
14. Prove that for every integer x, the remainder when x is divided by 8 is
either 0 or 1.
∗
15. Suppose F and G are nonempty families of sets.
p d(a) Prove that ∪(F ∪ G) = (∪F) ∪ (∪G).
(b) Can you discover and prove a similar theorem about ∩(F ∪ G)?
16. Suppose F is a nonempty family of sets and B is a set.
p d(a) Prove that B ∪ (∪F) =∪(F ∪{B}).
(b) Prove that B ∪ (∩F) =∩ A∈F (B ∪ A).
(c) Can you discover and prove a similar theorem about B ∩ (∩F)?
17. Suppose F, G, and H are nonempty families of sets and for every A ∈ F
and every B ∈ G, A ∪ B ∈ H. Prove that ∩H ⊆ (∩F) ∪ (∩G).
p
d18. Suppose A and B are sets. Prove that ∀x(x ∈ A B ↔ (x ∈ A ↔
x /∈ B)).
d 19. Suppose A, B, and C are sets. Prove that A B and C are disjoint iff
p ∗
A ∩ C = B ∩ C.
p
d20. Suppose A, B, and C are sets. Prove that A B ⊆ C iff A ∪ C =
B ∪ C.
p d21. Suppose A, B, and C are sets. Prove that C ⊆ A B iff C ⊆ A ∪ B
and A ∩ B ∩ C =∅.
d 22. Suppose A, B, and C are sets.
p ∗
(a) Prove that A \ C ⊆ (A \ B) ∪ (B \ C).
(b) Prove that A C ⊆ (A B) ∪ (B C).