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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 ...
A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z. In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold. For example, the relation defined by xRy if xy is an even number is intransitive, [13] but not antitransitive. [14]
Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle. Each relation that is both reflexive and left (or right) Euclidean is also an equivalence relation.
Definitions, for all , ... In mathematics, a binary relation associates elements of one set called the domain with elements of another set called the codomain. [1]
An equivalence relation is a mathematical relation that generalizes the idea of similarity or sameness. It is defined on a set X {\displaystyle X} as a binary relation ∼ {\displaystyle \sim } that satisfies the three properties: reflexivity , symmetry , and transitivity .
This set-theoretic definition is based on the fact that a function establishes a relation between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets X and Y is a subset of the set of all ordered pairs ( x , y ) {\displaystyle (x,y)} such that x ∈ X {\displaystyle x ...
A finitary or n-ary relation is a set of n-tuples. Specific types of relations include: Relation (mathematics) (an elementary treatment of binary relations) Binary relation (or diadic relation – a more in-depth treatment of binary relations) Equivalence relation; Homogeneous relation; Reflexive relation; Serial relation
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.. More precisely, if , …, are elements of a (left) module M over a ring R (the case of a vector space over a field is a special case), a relation between , …, is a sequence (, …,) of elements of R such that