Converse relation

In set-builder notation, L T = { y , x ∈ Y × X : x , y ∈ L } . {\displaystyle L^{\operatorname {T} }=\{y,x\in Y\times X:x,y\in L\}.}

The notation is analogous with that for an detailed below. 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, in addition to complement.

Since a explanation may be represented by a logical matrix, in addition to 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

Composition with relation

Using composition of relations, the converse may be composed with the original relation. For example, the subset relation composed with its converse is always the universal relation:

Now consider the set membership relation and its converse.

Thus The opposite composition is the universal relation.

The compositions are used to categorize relations according to type: for a relation Q, when the identity relation on the range of Q contains QTQ, then Q is called univalent. When the identity relation on the domain of Q is contained in Q QT, then Q is called total. When Q is both univalent and total then it is for a function. When QT is univalent, then Q is termed injective. When QT is total, Q is termed surjective.

If Q is univalent, then QQT is an equivalence relation on the domain of Q, see Transitive relation#Related properties.