Prove the following Given any set X and given any functions

Prove the following. Given any set X and given any functions f: X --> X, g: X-->X, and h: X-->X, if h is onto and (f º h) = (g º h) (the dot is supposed to represent the composition of functions) , then f = g.

Solution

Suppose f and g are injective. Let x, y A be given, and assume h(x) = h(y). Since h = g f, this means that g(f(x)) = g(f(y)), by the definition of the composition of functions. Since g is injective, this implies f(x) = f(y). Since f is injective, it follows that x = y. Summarizing, we have shown that, for any x, y A, h(x) = h(y) implies x = y. Thus, h is injective, by the definition of injectivity.