Page 272 - HOW TO PROVE IT: A Structured Approach, Second Edition

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258 Functions

Figure 1

we get f (W) ={ f (2)}={5} and f (X) ={ f (3)}={5},so f (W) ∩ f (X) =
{5}∩{5}={5}. But f (W ∩ X) = f (∅) =∅,so f (W ∩ X) = f (W) ∩ f (X).
(If you’re not sure why f (∅) =∅, work it out using Definition 5.4.1!)
If you want to see an example in which W ∩ X = ∅, try W ={1, 2} and
X ={1, 3}.
This example shows that it would be incorrect to state a theorem saying that
f (W ∩ X) and f (W) ∩ f (X) are always equal. But our proof shows that the
following theorem is correct:

Theorem 5.4.2. Suppose f : A → B, and W and X are subsets of A.
Then f (W ∩ X) ⊆ f (W) ∩ f (X). Furthermore, if f is one-to-one, then
f (W ∩ X) = f (W) ∩ f (X).

Now, here are some questions for you to try to answer. In each case, try to
figure out as much as you can. Justify your answers with proofs and counterex-
amples.
Suppose f : A → B.
1. Suppose W and X are subsets of A.
(a) Will it always be true that f (W ∪ X) = f (W) ∪ f (X)?
(b) Will it always be true that f (W \ X) = f (W) \ f (X)?
(c) Will it always be true that W ⊆ X ↔ f (W) ⊆ f (X)?
2. Suppose that Y and Z are subsets of B.
(a) Will it always be true that f −1 (Y ∩ Z) = f −1 (Y) ∩ f −1 (Z)?
(b) Will it always be true that f −1 (Y ∪ Z) = f −1 (Y) ∪ f −1 (Z)?
(c) Will it always be true that f −1 (Y \ Z) = f −1 (Y) \ f −1 (Z)?
(d) Will it always be true that Y ⊆ Z ↔ f −1 (Y) ⊆ f −1 (Z)?