Converse relation

In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if ${\displaystyle X}$ and ${\displaystyle Y}$ are sets and ${\displaystyle L\subseteq X\times Y}$ is a relation from ${\displaystyle X}$ to ${\displaystyle Y,}$ then ${\displaystyle L^{\operatorname {T} }}$ is the relation defined so that ${\displaystyle yL^{\operatorname {T} }x}$ if and only if ${\displaystyle xLy.}$ In set-builder notation,

Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, the inverse of the original relation, or the reciprocal ${\displaystyle L^{\circ }}$ of the relation ${\displaystyle L.}$

Other notations for the converse relation include ${\displaystyle L^{\operatorname {C} },L^{-1},{\breve {L}},L^{\circ },}$ or ${\displaystyle L^{\vee }.}$^{[citation needed]}

The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations as detailed below. As a unary operation, taking the converse (sometimes called conversion or transposition)^{[citation needed]} commutes with the order-related operations of the calculus of relations, that is it commutes with union, intersection, and complement.

For the usual (maybe strict or partial) order relations, the converse is the naively expected "opposite" order, for examples, ${\displaystyle {\leq ^{\operatorname {T} }}={\geq },\quad {<^{\operatorname {T} }}={>}.}$

A relation may be represented by a logical matrix such as $${\displaystyle {\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}}.}$$

Then the converse relation is represented by its transpose matrix: $${\displaystyle {\begin{pmatrix}1&0&0&0\\1&1&0&0\\1&0&1&0\\1&1&0&1\end{pmatrix}}.}$$

The converse of kinship relations are named: "${\displaystyle A}$ is a child of ${\displaystyle B}$" has converse "${\displaystyle B}$ is a parent of ${\displaystyle A}$". "${\displaystyle A}$ is a nephew or niece of ${\displaystyle B}$" has converse "${\displaystyle B}$ is an uncle or aunt of ${\displaystyle A}$". The relation "${\displaystyle A}$ is a sibling of ${\displaystyle B}$" is its own converse, since it is a symmetric relation.