An Inverse Relation is a Binary Relation between Binary Relations where the Variable order is switched.
- (Wikipedia, 2009) http://en.wikipedia.org/wiki/Inverse_relation
- In mathematics, the inverse relation of a binary relation is the relation that occurs when you switch the order of the elements in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms, if
- L : X \to Y is a binary relation with \operatorname{graph}\,L\subset X\times Y
- then the inverse relation is
- L^{-1} : Y \to X defined by y\,L^{-1}\,x\iff x\,L\,y ,
- i.e. with \operatorname{graph}\,L^{-1} = \{(y, x)\in Y\times X\mid (x, y) \in \operatorname{graph}\, L\}.
- The notation comes by analogy with that for an inverse function.
- The inverse relation is also called the converse relation or transpose relation (in view of its similarity with the transpose of a matrix: these are the most familiar examples of dagger categories), and may be written as LC, LT, or \breve{L}.
- Note that, despite the notation, the converse relation is not an inverse in the sense of composition of relations: L \circ L^{-1} \neq \mathrm{id} in general.
- Properties
- A relation equal to its inverse is a symmetric relation (in the language of dagger categories, it is self-adjoint).
- If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its inverse is too.
- However, if a relation is extendable, this need not be the case for the inverse.
- The operation of taking a relation to its inverse gives the category of relations Rel the structure of a dagger category.