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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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True or False

Label each of the following statements as either true or false.

The set of all bijections from A to A is closed with respect to the binary operation of composition defined on the set of all mappings from A to A .

To determine

Whether the statement, “The set of all bijections from A to A is closed with respect to the binary operation of composition defined on the set of all mappings from A to A” is true or false.

Explanation

Suppose A is any non-empty set.

Let,

G=setofallbijectivefunctionsfromAtoA={f:AA|fisbijective}

Define binary operation * on G such that

For f,gG, *:G×GG as

*(f,g)=fg

To show that the composition of two bijective functions fandg from A to A is a bijective function from A to A.

gf is defined as

gf:AA such that gf=g[f(x)]

1. gf is one-one:

Let x,yA such that x=y

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