Converse relation


In mathematics, the converse relation, or transpose, 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 and are sets and is a relation from to, then is the relation defined so that if and only if. In set-builder notation,.
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 commutes with the order-related operations of the calculus of relations, that is it commutes with union, intersection, and complement.
The converse relation is also called the 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, or the inverse of the original relation, or the reciprocal L° of the relation L.
Other notations for the converse relation include,,, or.

Examples

For the usual order relations, the converse is the naively expected "opposite" order, for examples,
A relation may be represented by a logical matrix such as
Then the converse relation is represented by its transpose matrix:
The converse of kinship relations are named: "A is a child of B" has converse "B is a parent of A". "A is a nephew or niece of B" has converse "B is an uncle or aunt of A". The relation "A is a sibling of B" is its own converse, since it is a symmetric relation.
In set theory, one presumes a universe U of discourse, and a fundamental relation of set membership x ∈ A when A is a subset of U. The power set of all subsets of U is the domain of the converse

Properties

In the monoid of binary endorelations on a set, the converse relation does not satisfy the definition of an inverse from group theory, i.e. if L is an arbitrary relation on X, then does not equal the identity relation on X in general. The converse relation does satisfy the axioms of a semigroup with involution: and.
Since one may generally consider relations between different sets, in this context the converse relation conforms to the axioms of a dagger category. A relation equal to its converse is a symmetric relation; in the language of dagger categories, it is self-adjoint.
Furthermore, the semigroup of endorelations on a set is also a partially ordered structure, and actually an involutive quantale. Similarly, the category of heterogeneous relations, Rel is also an ordered category.
In the calculus of relations, conversion commutes with other binary operations of union and intersection. Conversion also commutes with unary operation of complementation as well as with taking suprema and infima. Conversion is also compatible with the ordering of relations by inclusion.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder, or an equivalence relation, its converse is too.

Inverses

If I represents the identity relation, then a relation R may have an inverse as follows:

Converse relation of a function

A function is invertible if and only if its converse relation is a function, in which case the converse relation is the inverse function.
The converse relation of a function is the relation defined by.
This is not necessarily a function: One necessary condition is that f be injective, since else is multi-valued. This condition is sufficient for being a partial function, and it is clear that then is a function if and only if f is surjective. In that case, i.e. if f is bijective, may be called the inverse function of f.
For example, the function has the inverse function.
However, the function has the inverse relation, which is not a function, being multi-valued.