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).

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- Nov 3rd 2009, 02:18 PMl888l888l888Surjective 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). - Nov 4th 2009, 02:23 AMHallsofIvy
- Nov 4th 2009, 06:53 AMl888l888l888
f(g(y))= y because f(g(x))=x

- Nov 4th 2009, 03:51 PMDrexel28
More explicitly (like HallsOfIvy said)

: Suppose that $\displaystyle f:X\mapsto Y$ and $\displaystyle g:Y\mapsto X$ are functions such that $\displaystyle f\circ g=\iota_Y$ (identity mapping on Y). Prove that $\displaystyle f:X\mapsto Y$ is surjective.**Problem**

Let $\displaystyle y\in Y$. Since $\displaystyle g:Y\mapsto X$ we know that $\displaystyle g(y)\in X$. Therefore $\displaystyle f(g(y))=y$ since $\displaystyle f\circ g=\iota_Y$. And since $\displaystyle y$ was arbitrary this proves surjectivity.**Proof:**