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Hot Alarm Clock V4.0.4.0 Incl Loader-SND [TorDigger] 64 BitQ:
Injective on surjective functions
I cannot understand the proof of an injective on surjective function.
Could someone explain? Thank you so much.
A:
We can use the contrapositive.
Let $f:X\to Y$ and $g:Y\to X$ be a pair of functions, and suppose that $f\circ g$ is surjective, but $g\circ f$ is not necessarily injective.
Then $f$ is not necessarily injective. If not injective, then $g$ is not necessarily surjective. If $g$ is surjective, then $f$ is not necessarily surjective, or $f\circ g$ is surjective, but $g\circ f$ need not be surjective.
Note that $f\circ g$ being surjective is equivalent to $g\circ f$ not being injective.
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