a) Give an example to show that the inclusion in part (b) in Exercise 40 may be proper________________b) Show that if f is one-to-one, the inclusion in part (b) in Exercise 40 is an equality.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 f-l (S). so f-1(S) {a.? A| f (a) ? S}. (Beware: The notation f-1 is used in two different ways. Do not confuse the notation introduced here with the notation f-1(y) for the value at y of the inverse of the invertible function f. Notice also that f-1(S), the inverse image of the set S, makes sense for all functions f, not just invertible functions.)

SolutionStep 1In the problem we have to show the inclusion part of f(S T) f (S) f(T)Let S = {3} and T = { 4} and f(3) = 5, f(4) = 5Then first we have to evaluate intersection of S and T Intersection of S and T = { }So, f(S) = {5} and f(T) = {5}Then, f (S) f(T) = {5} {5} = {5}f(S T) f (S) f(T)Hence, it is proved that it may be proper.