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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Prove or given a counterexample: If f : X Y and g : Y X and function such that g f = I X and f g = L X , then f and g are both one-to-one and onto and g = f 1 .

To determine

To prove or provide a counterexample for the given statement.

Explanation

Given information:

Consider that f:XY and g:YX are function such that gf=IX and fg=IY then objective is to show that f and g are both one-to-one and onto and g=f1.

Concept used:

A function is said to be one-to-one function if the distinct elements in domain must be mapped with distinct elements in co-domain.

A function is said to be onto function if each element in co-domain is mapped with atleast one element in domain.

Proof:

First show that f is one-to-one, for this consider f(x1)=f(x2), where x1,x2X.

Now,

f(x1)=f(x2)

This implies that,

g(f(x1))=g(f(x2))

Or,

(gf)(x1)=(gf)(x2)

Since gf=IX therefore, (gf)(x1)=(gf)(x2) implies that IX(x1)=IX(x2).

This implies that x1=x2

This shows that f is one-to-one.

For this consider yY and objective is to show that there exists xX such that f(x)=y

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