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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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Let f : Α Β , where Α and Β are nonempty. Prove that f has the property that f ( f 1 ( T ) ) = T for every subset T of Β if and only if f is onto. (Compare with Exercise 15c.)

Exercise 15c.

c. For this same f and T = { 4 , 9 } , show that f ( f 1 ( T ) ) T .

To determine

To prove: f has the property that f(f1(T))=T for every subset T of B if and only if f is onto.

Explanation

Given information:

Let f:AB, where A and B are non-empty.

Proof:

Let f:AB, where A and B are non-empty.

Assume that f(f1(T))=T;TB

Let x be any element of set B.

Let T={x}

f(f1({x}))={x}

In other words, there exist aA such that f(a)=x

Since xB be any arbitrary element, there exists aA such that f(a)=x

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