   Chapter 1.3, Problem 12E

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# Let f : A → B and g : B → A . Prove that f is one-to-one and onto if f ∘ g is one to-one and g ∘ f onto.

To determine

To prove: The function f is one-to-one and onto if fg is one to-one and gf onto.

Explanation

Given Information:

The function f:AB, the function g:BA.

Proof:

Let aA, bB and A=B

=Z, such that f(a)=b and g(b)=a.

Here, Z is the set of integers.

Assume, f(x)=|x| and g(x)=x

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