Let f be a function from the set A to the set B. Let S be a subset of B. We define the inverse image of S to be the subset of A whose elements are precisely all pre - images of all elements of S. We denote the inverse image of S by , so {a A |}. (Beware: the notation is used in two different ways. Do not confuse the notation introduced here with the notation for the value at y of the inverse of the invertible function f. Notice also that , the inverse image of the set S, makes sense for all functions f, not just invertible functions.)

Let f be a function from A to B. Let S be a subset of B. Show that

Solution:

Step 1

In this problem we are asked to show that where the inverse image of S is denoted by and {a A |}.