   Chapter 7.3, Problem 19ES ### 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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# If + f : X → Y and g : Y → Z are functions and g ∘ f is one-to-one, must g be onr-to-one? Prove or given a counterexample.

To determine

To check:

When f:XY and g:YZ are two functions and gf is one-to-one, then whether g must be one-to-one or not.

Explanation

Given information:

f:XY and g:YZ are two functions.

Concept used:

A function F is said to be one-to-one function if and only if each element in the codomain of F is the image of at most one element in the domain of F.

Calculation:

Let X,Y and Z be any sets and the functions are f:XY and g:YZ.

The objective is to prove or provide counter example for the result if gf is one-to-one then g must be one-to-one.

Consider the statement, if f:XY and g:YZ are functions and gf is one-to-one, then g must be one-to-one.

This statement is not true for all sets with the reference of the counter example.

Consider example:

Let X={1,2},Y={a,b,c},Z={x,y}.

Find the arrow diagram for function f and g

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