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It encodes the common concept of relation: an element is related to an element , if and only if the pair (,) belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the "divides" relation over the set of prime numbers and the set of integers, in which each prime is related to each integer that is ...
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 ...
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.
The set of all homogeneous relations () over a set X is the set 2 X×X, which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on B ( X ) {\displaystyle {\mathcal {B}}(X)} , it forms a monoid with involution where the identity element ...
A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if: [1], (), where the notation aRb means that (a, b) ∈ R. An example is the relation "is equal to", because if a = b is true then b = a is also true.
Orders are special binary relations. Suppose that P is a set and that ≤ is a relation on P ('relation on a set' is taken to mean 'relation amongst its inhabitants', i.e. ≤ is a subset of the cartesian product P x P).
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets. It is an important tool in combinatorial mathematics and theoretical computer science.
The set of binary relations on a set (that is, relations from to ) together with (left or right) relation composition forms a monoid with zero, where the identity map on is the neutral element, and the empty set is the zero element.