   Chapter 1.2, Problem 27E

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# Let f : A → B , where A and B are nonempty. Prove that f has the property that f − 1 ( f ( S ) ) = S for every subset S of A if and only if f is one-to-one. (Compare with Exercise 15 b.).b. For the mapping f , show that if S = { 1 , 2 } , then f − 1 ( f ( S ) ) ≠ S .

To determine

To prove: f has the property that f1(f(S))=S for every subset S of A if f is one-to-one.

Explanation

Given information:

Here f:AB, where A and B are nonempty sets.

Proof:

Suppose f1(f(S))=f(S) for all SA and need to show that f is one-to-one.

Let, S={a}, a set with single element. Then {a}=f1{f(a)}.

If f is not one-to-one, then there exists a1a2Asuchthatf(a1)=f(a2).

Now,

{a1}{a2}and{f(a1)}={f(a2)}

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