True or False? Given any set X and given any functions f : X → X , g : X → X , and h : X → X , if h is one-to-one and f ∘ h = g ∘ h , then f = g . Justify your answer.

To check:

Whether the given statement, “Given any set X and given any functions f:X→X, g:X→X and h:X→X if h is one-to-one and f o h=g o h then f=g ” is true or false.

Given information:

f:X→X, g:X→X and h:X→X be three functions.

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.

Calculation:

Let X be any set and the functions f:X→X, g:X→X and h:X→X.

Consider the statement,

Any set X and any functions f:X→X, g:X→X and h:X→X if h is one-to-one and f o h=g o h then f=g.

The objective is to identify whether it is true or false.

This statement is false.

Find the below counterexample:

Let X be the natural number set N.

Define the function f:N→N by,

f(x)={x, if x is even1,  if x is odd

Define the function g:N→N by,

g(x)={x, if x is even3,  if x is odd