Thread: Surjective functions

let f: From A to B. let g: From B to A.

How do I prove f(g(x)) = x (indentity function) implies f is onto (surjective).

Originally Posted by l888l888l888

let f: From A to B. let g: From B to A.

How do I prove f(g(x)) = x (indentity function) implies f is onto (surjective).

Suppose f is NOT surjective. Then there exist y in B such that, for all x in A, . Now, what is f(g(y))?

f(g(y))= y because f(g(x))=x

More explicitly (like HallsOfIvy said)

Problem: Suppose that and are functions such that (identity mapping on Y). Prove that is surjective.

Proof: Let . Since we know that . Therefore since . And since was arbitrary this proves surjectivity.