Page 273 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Images and Inverse Images: A Research Project 259
3. Suppose X ⊆ A. Will it always be true that f −1 ( f (X)) = X?
4. Suppose Y ⊆ B. Will it always be true that f ( f −1 (Y)) = Y?
5. Suppose g : B → C. Can you prove any interesting theorems about images
and inverse images of sets under g ◦ f ?
Note: An observant reader may have noticed an ambiguity in our notation
for images and inverse images. If f : A → B and Y ⊆ B, then we have used
the notation f −1 (Y) to stand for the inverse image of Y under f. But if f is
one-to-one and onto, then, as we saw in Section 5.3, f −1 is a function from
B to A. Thus, f −1 (Y) could also be interpreted as the image of Y under the
function f −1 . Fortunately, this ambiguity is harmless, as the next problem
shows.
6. Suppose f : A → B, f is one-to-one and onto, and Y ⊆ B. Show that the
inverse image of Y under f and the image of Y under f −1 are equal. (Hint:
First write out the definitions of the two sets carefully!)