   Chapter 1.5, Problem 4E

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# Let f : A → A , where A is nonempty. Prove that f a has left inverse if and only if f − 1 ( f ( S ) ) = S for every subset S of A .

To determine

To prove: The function f:AA has a left inverse if and only if f1(f(S))=S for every subset S of A.

Explanation

Proof:

Consider the statements

p: The function f:AA has a left inverse.

q: f1(f(S))=S for every subset S of A.

pq

Let aA.

Since f has a left inverse, therefore

f1(f(a))=a

Let sSA

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