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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.
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 values ...
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, 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).
A relation is quasi-reflexive if, and only if, it is both left and right quasi-reflexive. The previous 6 alternatives are far from being exhaustive; e.g., the binary relation xRy defined by y = x 2 is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair (0, 0), and (2, 4), but not (2, 2), respectively. The latter two ...
Reflexive closure. 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 ...
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 partial orders, but not quite, as they are not necessarily antisymmetric.
In mathematics, a binary relation R is called well-founded (or wellfounded or foundational[1]) on a set or, more generally, a class X if every non-empty subset S ⊆ X has a minimal element with respect to R; that is, there exists an m ∈ S such that, for every s ∈ S, one does not have s R m.