A relation $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ is injective if, for each $a,a'\in A$, the following equivalent conditions are satisfied:
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If there exists some $b\in B$ such that $a\sim _{R}b$ and $a'\sim _{R}b$, then $a=a'$.
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If $R(a)\cap R(a')\neq \text{Ø}$, then $a=a'$.
Proof of the Equivalences in Definition 10.1.3.1.1.
Since we have $R(a)\cap R(a')\neq \text{Ø}$ iff $a\sim _{R}b$ and $a'\sim _{R}b$ for some $b\in B$, the result follows.
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Injective and total relations are monomorphisms in $\mathsf{Rel}$, but there are monomorphisms which are neither injective nor total; see Chapter 8: Relations, Section 8.5.10.
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