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).
Last edited by mr fantastic; November 3rd 2009 at 10:59 PM. Reason: Removed shouting
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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.
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