Search results
Results From The WOW.Com Content Network
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the ...
An operation of arity zero, called a nullary operation, is simply an element of the codomain Y. An n-ary operation can also be viewed as an (n + 1)-ary relation that is total on its n input domains and unique on its output domain. An n-ary partial operation ω from X n to X is a partial function ω: X n → X.
A relation algebra (L, ∧, ∨, −, 0, 1, •, I, ˘) is an algebraic structure equipped with the Boolean operations of conjunction x∧y, disjunction x∨y, and negation x −, the Boolean constants 0 and 1, the relational operations of composition x•y and converse x˘, and the relational constant I, such that these operations and constants satisfy certain equations constituting an ...
The relational algebra uses set union, set difference, and Cartesian product from set theory, and adds additional constraints to these operators to create new ones.. For set union and set difference, the two relations involved must be union-compatible—that is, the two relations must have the same set of attributes.
A binary relation is also called a heterogeneous relation when it is not necessary that =. Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets.
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.
A relation is transitive if it is closed under this operation, and the transitive closure of a relation is its closure under this operation. A preorder is a relation that is reflective and transitive. It follows that the reflexive transitive closure of a relation is the smallest preorder containing it. Similarly, the reflexive transitive ...
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.