   Chapter 7.3, Problem 30ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Let f : X → Y and g : Y → Z . Is the following property true or false? For every subset C in Z. ( g ∘ f ) − 1 ( C ) = f − 1 ( g − 1 ( C ) ) . Justify your answer.

To determine

To check:

Whether the property (gof)1(C)=f1(g1(C)) is true or false, for all subsets C in Z.

Explanation

Given information:

Let f:XY and g:YZ are one-to-one and onto functions.

Concept used:

A function F is said to be one-to-one function if, and only if, the distinct elements in its domain are mapped with the distinct elements in its co-domain.

A function is said to be onto function if, and only if, each element in its codomain is the image of at least one element in its domain.

Calculation:

Let f:XY and  g:YZ are one-to-one and onto functions, because the inverse of both function are defined.

Let C be any element in Z, such that.

C=g(y)

Where, yY,CZ

Then,

y=g1(C)

Also let y=f(x)

Where, yY,xX

Then,

x=f1(y)

(gf)(x)=g[f(x)]=g(y)=C

Since gf is one-to-one and onto

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