Mathematical logic | Binary relations

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 . As a unary operation, taking the converse (sometimes called conversion or transposition) commutes with the order-related operations of the calculus of relations, that is it commutes with union, intersection, and complement. 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, or the inverse of the original relation, or the reciprocal of the relation Other notations for the converse relation include or (Wikipedia).

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Related pages

Composition of relations | Inverse function | If and only if | Symmetric relation | Unary operation | Asymmetric relation | Dagger category | Total order | Category of relations | Identity function | Connected relation | Binary relation | Transitive relation | Universe (mathematics) | Reflexive relation | Mathematics | Algebraic logic | Function (mathematics) | Set-builder notation | Set (mathematics) | Self-adjoint | Involution (mathematics) | Category (mathematics) | Semigroup with involution | Logical matrix | Equivalence relation | Naive Set Theory (book) | Transpose | Partial function | Quantale | Antisymmetric relation | Binary operation | Monoid