Editing Reflexive relation (section)

== Examples ==
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Examples of reflexive relations include:
* "is equal to" ([[Equality (mathematics)|equality]])
* "is a [[subset]] of" (set inclusion)
* "divides" ([[Divisor|divisibility]])
* "is greater than or equal to"
* "is less than or equal to"
Examples of irreflexive relations include:
* "is not equal to"
* "is [[coprime]] to" on the integers larger than 1
* "is a [[proper subset]] of"
* "is greater than"
* "is less than"

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An example of an irreflexive relation, which means that it does not relate any element to itself, is the "greater than" relation (<math>x > y</math>) on the [[real number]]s. Not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not (that is, neither all nor none are). For example, the binary relation "the product of <math>x</math> and <math>y</math> is even" is reflexive on the set of [[even number]]s, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of [[natural number]]s. 

An example of a quasi-reflexive relation <math>R</math> is "has the same limit as" on the set of sequences of real numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same limit as some sequence, then it has the same limit as itself. 
An example of a left quasi-reflexive relation is a left [[Euclidean relation]], which is always left quasi-reflexive but not necessarily right quasi-reflexive, and thus not necessarily quasi-reflexive. 

An example of a coreflexive relation is the relation on [[integer]]s in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation. The union of a coreflexive relation and a transitive relation on the same set is always transitive.