Page 136 - HOW TO PROVE IT: A Structured Approach, Second Edition

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122 Proofs
2. Prove that if A and B \ C are disjoint, then A ∩ B ⊆ C.
∗
3. Prove that if A ⊆ B \ C then A and C are disjoint.
p d4. Suppose A ⊆ P (A). Prove that P (A) ⊆ P (P (A)).
5. The hypothesis of the theorem proven in exercise 4 is A ⊆ P (A).
(a) Can you think of a set A for which this hypothesis is true?
(b) Can you think of another?
6. Suppose x is a real number.
(a) Prove that if x = 1 then there is a real number y such that y+1 = x.
y−2
(b) Prove that if there is a real number y such that y+1 = x, then x = 1.
y−2
7. Prove that for every real number x,if x > 2 then there is a real number
∗
1
y such that y + = x.
y
p d8. Prove that if F is a family of sets and A ∈ F, then A ⊆∪F.
∗ 9. Prove that if F is a family of sets and A ∈ F, then ∩F ⊆ A.
10. Suppose that F is a nonempty family of sets, B is a set, and ∀A ∈ F(B ⊆
A). Prove that B ⊆∩F.
11. Suppose that F is a family of sets. Prove that if ∅ ∈ F then ∩F =∅.
p ∗
d 12. Suppose F and G are families of sets. Prove that if F ⊆ G then ∪F ⊆
∪G.
13. Suppose F and G are nonempty families of sets. Prove that if F ⊆ G
then ∩G ⊆∩F.
14. Suppose {A i | i ∈ I} is an indexed family of sets. Prove that
∗
∪ i∈I P (A i ) ⊆ P (∪ i∈I A i ). (Hint: First make sure you know what all
the notation means!)
15. Suppose {A i | i ∈ I} is an indexed family of sets and I = ∅. Prove that
∩ i∈I A i ∈∩ i∈I P (A i ).
p d16. Prove the converse of the statement proven in Example 3.3.5. In other
words, prove that if F ⊆ P (B) then ∪F ⊆ B.
17. Suppose F and G are nonempty families of sets, and every element of
∗
F is a subset of every element of G. Prove that ∪F ⊆∩G.
18. In this problem all variables range over Z, the set of all integers.
(a) Prove that if a | b and a | c, then a | (b + c).
(b) Prove that if ac | bc and c = 0, then a | b.
19. (a) Prove that for all real numbers x and y there is a real number z such
that x + z = y − z.
(b) Would the statement in part (a) be correct if “real number” were
changed to “integer”? Justify your answer.
20. Consider the following theorem:
∗
2
Theorem. For every real number x, x ≥ 0.
What’s wrong with the following proof of the theorem?