Let $\sim$ be a binary relation on $X$. We define the following properties of $\sim$:
Property Condition Reflexive \(x\sim x\) Irreflexive \(x\not\sim x\) Symmetric \(x\sim y\rimp y\sim x\) Antisymmetric \((x\sim y\land y\sim x)\rimp x=y\) Asymmetric \(x\sim y\rimp y\not\sim x\) Transitive \((x\sim y\land y\sim z)\rimp x\sim z\) Connected \(x\sim y\lor y\sim x\lor x=y\) Strongly connected \(x\sim y\lor y\sim x\) Left-unique \((x\sim z\land y\sim z)\rimp x=y\) Right-unique \((z\sim x\land z\sim y)\rimp x=y\) The statements in the Condition column are meant to hold either for all $x\in X$; all $x$, $y\in X$; or all $x$, $y$, $z\in X$, accordingly.
- Synonym of irreflexive: strict
- Synonyms of left-unique: one-to-many, injective
- Synonyms of right-unique: many-to-one, functional, a function
Let $\sim$ be a binary relation on $X$ and $Y$.
$\sim$ is left-total if $\dom{\sim}=X$, i.e.
\[\forall x\in X\,\exists y\in Y : x\sim y.\]$\sim$ is right-total if $\ran{\sim}=Y$, i.e.
\[\forall y\in Y\,\exists x\in X : x\sim y.\]
- Synonyms of right-total: onto, surjective