Let $\sim$ be a binary relation on a set $X$. 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.
Notes.
- Irreflexive is also known as strict.
- Left-unique is also known as one-to-many and injective.
- Right-unique is also known as many-to-one and functional. In fact, a right-unique relation is just a function.
Let $\sim$ be a binary relation on two sets $X$ 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 $\im{\sim} = Y$, i.e.
\[\forall y \in Y \, \exists x \in X : x \sim y.\]
Note. Right-total is also known as onto and surjective.