Page 149 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Proofs Involving Conjunctions and Biconditionals 135
Proof. Suppose x ∈∪(F \ G). Then we can choose some A ∈ F \
G suchthat x ∈ A.Since A ∈ F \ G, A ∈ F and A /∈ G.Since x ∈ A
and A ∈ F, x ∈∪F. Since x ∈ A and A /∈ G, x /∈∪G. Therefore
x ∈ (∪F) \ (∪G).

(c) Prove that ∪(F \ G) ⊆ (∪F) \ (∪G)iff ∀A ∈ (F \ G)∀B ∈ G(A ∩
B = ∅).
(d) Find an example of families of sets F and G for which ∪(F \ G) =
(∪F) \ (∪G).
p ∗
d 21. Suppose F and G are families of sets. Prove that if ∪F ⊆∪G, then there
is some A ∈ F such that for all B ∈ G, A ⊆ B.
22. Suppose B is a set, {A i | i ∈ I} is an indexed family of sets, and I =∅.
(a) What proof strategies are used in the following proof that B ∩
(∪ i∈I A i ) =∪ i∈I (B ∩ A i )?
Proof. Let x be arbitrary. Suppose x ∈ B ∩ (∪ i∈I A i ). Then x ∈ B
.
and x ∈∪ i∈I A i , so we can choose some i 0 ∈ I such that x ∈ A i 0
. Therefore x ∈∪ i∈I (B ∩
Since x ∈ B and x ∈ A i 0 , x ∈ B ∩ A i 0
A i ).
Now suppose x ∈∪ i∈I (B ∩ A i ). Then we can choose some i 0 ∈
. Since x ∈
I such that x ∈ B ∩ A i 0 . Therefore x ∈ B and x ∈ A i 0
, x ∈∪ i∈I A i . Since x ∈ B and x ∈∪ i∈I A i , x ∈ B ∩ (∪ i∈I A i ).
A i 0
Since x was arbitrary, we have shown that ∀x[x ∈ B ∩
(∪ i∈I A i ) ↔ x ∈∪ i∈I (B ∩ A i )], so B ∩ (∪ i∈I A i ) =∪ i∈I (B ∩ A i ).

(b) Prove that B \ (∪ i∈I A i ) =∩ i∈I (B \ A i ).
(c) Can you discover and prove a similar theorem about B \ (∩ i∈I A i )?
(Hint: Try to guess the theorem, and then try to prove it. If you can’t
finish the proof, it might be because your guess was wrong. Change
your guess and try again.)
*23. Suppose {A i | i ∈ I} and {B i | i ∈ I} are indexed families of sets and
I =∅.
(a) Prove that ∪ i∈I (A i \ B i ) ⊆ (∪ i∈I A i ) \ (∩ i∈I B i ).
(b) Find an example for which ∪ i∈I (A i \ B i ) = (∪ i∈I A i ) \ (∩ i∈I B i ).
24. Suppose {A i | i ∈ I} and {B i | i ∈ I} are indexed families of sets.
(a) Prove that ∪ i∈I (A i ∩ B i ) ⊆ (∪ i∈I A i ) ∩ (∪ i∈I B i ).
(b) Find an example for which ∪ i∈I (A i ∩ B i ) = (∪ i∈I A i ) ∩ (∪ i∈I B i ).
25. Prove that for all integers a and b there is an integer c such that a | c
and b | c.
26. (a) Prove that for every integer n, 15 | n iff 3 | n and 5 | n.
(b) Prove that it is not true that for every integer n, 60 | n iff 6 | n and
10 | n.