   Chapter 7.3, Problem 20ES ### 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 function and g ∘ f is onto, must f be onto? Prove or give a counterexample.

To determine

To check:

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

Explanation

Given information:

f:XY and g:YZ are two functions.

Concept used:

A function is said to be onto if every element in codomain is mapped with atleast one element in domain.

Calculation:

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

Consider the composite function gf is onto.

The objective is to prove or provide counter example for the result f must be onto.

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

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

Counter example:

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

Find the arrow diagram for functions f and g.

From the arrow diagram of the functions the composition gf is onto one but the functions f is not onto

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