DEF-RP.
Let $\sim$ be a binary relation on $X$. We define the following properties of $\sim$:
Property Condition Reflexive $x\sim x$ Irreflexive / Strict $x\not\sim x$ Symmetric $x\sim y\Rightarrow y\sim x$ Antisymmetric $(x\sim y\land y\sim x)\Rightarrow x=y$ Asymmetric $x\sim y\Rightarrow y\not\sim x$ Transitive $(x\sim y\land y\sim z)\Rightarrow x\sim z$ Connected $x\neq y\Rightarrow(x\sim y\lor y\sim x)$ Strongly connected $x\sim y\lor y\sim x$ Left-unique / One-to-many / Injective $(x\sim z\land y\sim z)\Rightarrow x=y$ Right-unique / Many-to-one / Functional $(z\sim x\land z\sim y)\Rightarrow x=y$ The statements under “Condition” are meant to hold either for all $x\in X$; all $x$, $y\in X$; or all $x$, $y$, $z\in X$, accordingly.
DEF-RP-T.
Let $\sim$ be a relation on $X$ and $Y$. $\sim$ is
left-total if $\operatorname{dom}{\sim}=X$, namely
\[\forall x\in X\,\exists y\in Y : x\sim y.\]right-total / onto / surjective if $\operatorname{ran}{\sim}=Y$, namely
\[\forall y\in Y\,\exists x\in X : x\sim y.\]
REM-RP-T.
Notice that the left-total and right-total properties are not intrinsic, as they depend on the choices of $X$ and $Y$.