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Concept
Reflexive relation
Summary
In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself.
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. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
Let be a binary relation on a set which by definition is just a subset of
For any the notation means that while "not " means that
The relation is called if for every or equivalently, if where denotes the identity relation on
The of is the union which can equivalently be defined as the smallest (with respect to ) reflexive relation on that is a superset of A relation is reflexive if and only if it is equal to its reflexive closure.
The or of is the smallest (with respect to ) relation on that has the same reflexive closure as It is equal to The reflexive reduction of can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of
For example, the reflexive closure of the canonical strict inequality on the reals is the usual non-strict inequality whereas the reflexive reduction of is
There are several definitions related to the reflexive property.
The relation is called:
or If it does not relate any element to itself; that is, if not for every A relation is irreflexive if and only if its complement in is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.
If whenever are such that then necessarily
If whenever are such that then necessarily
If every element that is part of some relation is related to itself. Explicitly, this means that whenever are such that then necessarily Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation is quasi-reflexive if and only if its symmetric closure is left (or right) quasi-reflexive.
Official source
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