Page 185 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Relations 171
7. If A has m elements and B has n elements, how many elements does
A × B have?
d 8. Is it true that for any sets A, B, and C, A×(B\C) = (A×B)\(A×C)?
p ∗
Give either a proof or a counterexample to justify your answer.
p d9. Prove that for any sets A, B, C, and D,(A × B) \ (C × D) = [A ×
(B \ D)] ∪ [(A \ C) × B].
p d10. Prove that for any sets A, B, C, and D,if A × B and C × D are disjoint,
then either A and C are disjoint or B and D are disjoint.
11. 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) For each (i, j) ∈ I × I let C (i, j) = A i × B j , and let P = I × I.
Prove that ∪ p∈P C p = (∪ i∈I A i ) × (∪ i∈I B i ).
12. This problem was suggested by Prof. Alan Taylor of Union College.
∗
Consider the following putative theorem.
Theorem? For any sets A, B, C, and D, if A × B ⊆ C × D then A ⊆ C
and B ⊆ D.

Is the following proof correct? If so, what proof strategies does it use?
If not, can it be fixed? Is the theorem correct?

Proof. Suppose A × B ⊆ C × D. Let a be an arbitrary element of A
and let b be an arbitrary element of B. Then (a, b) ∈ A × B, so since
A × B ⊆ C × D, (a, b) ∈ C × D. Therefore a ∈ C and b ∈ D. Since
a and b were arbitrary elements of A and B, respectively, this shows that
A ⊆ C and B ⊆ D.

4.2. Relations

Suppose P(x, y) is a statement with two free variables x and y. Often such a
statement can be thought of as expressing a relationship between x and y. The
truth set of the statement P(x, y) is a set of ordered pairs that records when this
relationship holds. In fact, it is often useful to think of any set of ordered pairs
in this way, as a record of when some relationship holds. This is the motivation
behind the following definition.

Definition 4.2.1. Suppose A and B are sets. Then a set R ⊆ A × B is called a
relation from A to B.
If x ranges over A and y ranges over B, then clearly the truth set of
any statement P(x, y) will be a relation from A to B. However, note that