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In mathematics, a binary relation on a set is reflexive if it relates every element of to itself. [1][2] An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity.
Binary relation. In mathematics, a binary relation associates elements of one set called the domain with elements of another set called the codomain. [1] Precisely, a binary relation over sets and is a set of ordered pairs where is in and is in . [2]
In mathematics, the reflexive closure of a binary relation on a set is the smallest reflexive relation on that contains A relation is called reflexive if it relates every element of to itself. For example, if is a set of distinct numbers and means " is less than ", then the reflexive closure of is the relation " is less than or equal to ".
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric).
Partial orders. A reflexive, weak, [1] or non-strict partial order, [2] commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive. That is, for all it must satisfy: Reflexivity: , i.e. every element is related to itself.
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 binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order. A group with a compatible total order is a totally ordered group. There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation.
Equivalence classes (sets of elements such that x R y and y R x) are shown together as a single node. The relation on equivalence classes is a partial order. In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant to suggest that preorders are almost ...