True or False? Given any set X and given any function f : X → X , g : X → X , and h : X → X , if h is one-to-one and h ∘ f = h ∘ g , 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 h o f=h o g then f=g ” is true or false.

Given information:

h:one-to-one

h o f=h o g.

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:

The given statement is true.

Since h is a one-to-one function from X to X.

So for every x,y in X ,

h(x)=h(y)x=y

Also, consider that h o f=h o g.

Therefore (h o f)(a)=(h o g)(a), for all a in X