Suppose that is a function from A to B where A and If are finite sets. Explain why | f(S) | = | S | for all subsets S of A if and only if f is one-to-one.Suppose that f is a function from A to B. We define the function Sf from p(A) to p(B) by the rule Sf(X) = f(X) for each subset X of A. Similarly, we define the function Sf?1 from p(B) to p(A) by the rule Sf?1(Y) = f?1(Y) for each subset Y of B. Here, we are using Definition 4, and the definition of the inverse image of a set found in the preamble to Exercise 42, both in Section 2.3.

SolutionStep 1In this problem, we have to show that function A to B where A and B finite sets.and we have to explain that why f(S) = |S| for all subsets S to A if and only if f is one to one function.Step 2One to one functionA function is said to be one to one if and if one object of the domain have exactly one range of that object.Bijective functionThe function is said to be bijective if and only if each object of one set is joined with exactly one object of the other set and vice versa.CardinalityIt is the number of element in a set.Step 3Proof by contradictionLet us assume that f is onto function then by the pigeonhole...