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Reflexive relation

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In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.

Contents

In mathematical notation, this is:

a X ( a R a )

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.

A relation that is irreflexive, or anti-reflexive, is a binary relation on a set where no element is related to itself. An example is the "greater than" relation (x>y) on the real numbers. Note that 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 (i.e., neither all nor none are). For example, the binary relation "the product of x and y is even" is reflexive on the set of even numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.

A relation ~ on a set S is called quasi-reflexive if every element that is related to some element is also related to itself, formally: if ∀x,yS: x~yx~xy~y. An example is the relation "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.

The reflexive closure ≃ of a binary relation ~ on a set S is the smallest reflexive relation on S that is a superset of ~. Equivalently, it is the union of ~ and the identity relation on S, formally: (≃) = (~) ∪ (=). For example, the reflexive closure of x<y is xy.

The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set S is the smallest relation ≆ such that ≆ shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalent to the complement of the identity relation on S with regard to ~, formally: (≆) = (~) \ (=). That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of xy is x<y.

Examples

Examples of reflexive relations include:

  • "is equal to" (equality)
  • "is a subset of" (set inclusion)
  • "divides" (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" (for the integers>1, since 1 is coprime to itself)
  • "is a proper subset of"
  • "is greater than"
  • "is less than"
  • Number of reflexive relations

    The number of reflexive relations on an n-element set is 2n2n.

    Philosophical logic

    Authors in philosophical logic often use deviating designations. A reflexive and a quasi-reflexive relation in the mathematical sense is called a totally reflexive and a reflexive relation in philosophical logic sense, respectively.

    References

    Reflexive relation Wikipedia