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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? – Brian M. Scott Jun 22 '12 at 13:36
$f$ injective iff for every $x,y$ with $f(x)=f(y)$ one has $x=y$. Therefore for $f$ injective $f(x)=f(y) \iff x=y$. – Mercy King Jun 22 '12 at 13:39


Remember the definition of an injective function: $f$ is injective if and only if for every $x,y$ in the domain of $f$ it holds that $x\neq y\implies f(x)\neq f(y)$.

(Generally speaking, it is best to always begin by examining the definitions of the terms you encounter in a problem. In this case, what does it mean to be injective and what does it mean that the kernel is the identity.)

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Another way of seeing that this is an equivalence relation is to note that it is induced by the partition $$\Bigl\{ f^{-1}(\{y\})\Bigm| y\in f(X)\Bigr\}.$$ Since $f$ is injective if and only if the inverse image of singletons is either empty or a singleton, the conclusion will follow.

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