Page 175 - HOW TO PROVE IT: A Structured Approach, Second Edition

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More Examples of Proofs 161
Exercises

d 1. Suppose F is a family of sets. Prove that there is a unique set A that has
p ∗
the following two properties:
(a) F ⊆ P (A).
(b) ∀B(F ⊆ P (B) → A ⊆ B).
(Hint: First try an example. Let F ={{1, 2, 3}, {2, 3, 4}, {3, 4, 5}}. Can
you find the set A that has properties (a) and (b)?)
p d2. Suppose A and B are sets. What can you prove about P (A \ B) \ (P (A) \
P (B))? (No, it’s not equal to ∅. Try some examples and see what you
get.)
p d3. Suppose that A, B, and C are sets. Prove that the following statements
are equivalent:
(a) (A C) ∩ (B C) =∅.
(b) A ∩ B ⊆ C ⊆ A ∪ B.
(c) A C ⊆ A B.
∗
4. Suppose {A i | i ∈ I} is a family of sets. Prove that if P (∪ i∈I A i ) ⊆
∪ i∈I P (A i ), then there is some i ∈ I such that ∀ j ∈ I(A j ⊆ A i ).
5. Suppose F is a nonempty family of sets. Let I =∪F and J =∩F.
Suppose also that J =∅, and notice that it follows that for every X ∈ F,
X =∅, and also that I =∅. Finally, suppose that {A i | i ∈ I} is an
indexed family of sets.
(a) Prove that ∪ i∈I A i =∪ X∈F (∪ i∈X A i ).
(b) Prove that ∩ i∈I A i =∩ X∈F (∩ i∈X A i ).
(c) Prove that ∪ i∈J A i ⊆∩ X∈F (∪ i∈X A i ). Is it always true that ∪ i∈J A i =
∩ X∈F (∪ i∈X A i )? Give either a proof or a counterexample to justify
your answer.
(d) Discover and prove a theorem relating ∩ i∈J A i and ∪ X∈F (∩ i∈X A i ).
2
6. Prove that lim 3x −12 = 12.
x→2 x−2
∗
7. Prove that if lim f (x) = L and L > 0, then there is some number δ> 0
x→c
such that for all x,if0 < |x − c| <δ then f (x) > 0.
8. Prove that if lim f (x) = L then lim 7 f (x) = 7L.
x→c x→c
∗
9. Consider the following putative theorem.
b
Theorem. There are irrational numbers a and b such that a is rational.
Is the following proof correct? If so, what proof strategies does it use?
If not, can it be fixed? Is the theorem correct? (Note: The proof uses the
√
fact that 2 is irrational, which we’ll prove in Chapter 6.)