nLab
opposite relation

Definition

Given a binary relation $R$ from $X$ to $Y$, its opposite relation (or dual, or inverse, or converse, etc) is a relation $R^{\mathrm{op}}$ from $Y$ to $X$ as follows:

• $b$ is $R^{\mathrm{op}}$-related to $a$ if and only if $a$ is $R$-related to $b$.

Note that $(R^{\mathrm{op}}{)}^{\mathrm{op}}=R$.

The operation $\mathrm{op}$ is part of the requirements for Rel to be an allegory.

Examples

If $f$ is a function thought as a functional entire relation, then $f^{\mathrm{op}}$ is also a function if and only if $f$ is a bijection; in that case, $f^{\mathrm{op}}$ is the inverse of $f$.

More generally, we have the following:

If $R$ is …		then $R^{\mathrm{op}}$ is …
functional		injective
entire		surjective
injective		functional
surjective		entire