Inverse relation

In mathematics, the inverse relation of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms, if $X$ and $Y$ are sets and $L \subseteq X \times Y$ is a relation from $X$ to $Y$ , then $L -1$ is the relation defined so that $y L -1 x$ if and only if $x L y$ . In set-builder notation, $L -1 = {(y, x) \in Y \times X | (x, y) \in L$ }.

The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique inverse. Despite the notation and terminology, the inverse relation is not an inverse in the sense of group inverse. However, the unary operation that maps a relation to the inverse 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 inverse (sometimes called inversion) commutes with the order-related operations of relation algebra, i.e., it commutes with union, intersection, complement etc.

The inverse relation is also called the converse relation or transpose relation— the latter in view of its similarity with the transpose of a matrix. It has also been called the opposite or dual of the original relation. Other notations for the inverse relation include $L C, L T, L ~$ or ${\breve {L}}$ or $L °$ or $L \lor$ .

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