   Chapter 1.5, Problem 9E

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Textbook Problem
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# Let f and g be permutations on A . Prove that ( f ∘ g ) − 1 = g − 1 ∘ f − 1 .

To determine

To prove: If f and g are two permutations on A, then (fg)1=g1f1.

Explanation

Formula Used:

If f:AA is invertible on A, then

ff1=IA=f1f

Explanation:

The given statement is f and g are two permutations on A.

Calculate (fg)(g1f1).

Use Associative Property of Composition,

(fg)(g1f1)=((fg)g1)f1=(f(gg1))f1=(fIA)f1=ff1=IA ……(1)

Now calculate (g1f1)(fg)

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