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The equivalence kernel of a function $f$ is the equivalence relation $\sim$ defined by $$x\sim y \iff f(x) = f(y)\;.$$ The equivalence kernel of an injection is the identity relation.

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    This is very straightforward; what have you tried?2012-06-22
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    $f$ injective iff for every $x,y$ with $f(x)=f(y)$ one has $x=y$. Therefore for $f$ injective $f(x)=f(y) \iff x=y$.2012-06-22

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