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The identity element of this operation is the empty relation. For example, ≤ is the union of < and =, and ≥ is the union of > and =. Intersection [e] If R and S are relations over X then R ∩ S = { (x, y) | xRy and xSy} is the intersection relation of R and S. The identity element of this operation is the universal relation.
Another form of composition of relations, which applies to general -place relations for , is the join operation of relational algebra. The usual composition of two binary relations as defined here can be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle component.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation.The motivating example of a relation algebra is the algebra 2 X 2 of all binary relations on a set X, that is, subsets of the cartesian square X 2, with R•S interpreted as the usual composition of binary relations R and S, and with the ...
All operations defined in section § Operations also apply to homogeneous relations. Beyond that, a homogeneous relation over a set X {\displaystyle X} may be subjected to closure operations like: Reflexive closure
More complicated relations can exist; for example, the set {1} is both a member and a proper subset of the set {1, {1}}. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. [9] The following is a partial list of them:
In the particular case of groups, congruence relations can be described in elementary terms as follows: If G is a group (with identity element e and operation *) and ~ is a binary relation on G, then ~ is a congruence whenever: Given any element a of G, a ~ a (reflexivity); Given any elements a and b of G, if a ~ b, then b ~ a ;