Reflexive relation

In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.[1][2]

In mathematical notation, this is:

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.

Related terms

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 of reflexive relations include:

Examples of irreflexive relations include:

Number of reflexive relations

The number of reflexive relations on an n-element set is 2n2n.[3]

Number of n-element binary relations of different types
nalltransitivereflexivepreorderpartial ordertotal preordertotal orderequivalence relation
n2n2 2n2-n Σn
k! S(n,k)
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110

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.[4][5]

See also


  1. Levy 1979:74
  2. Relational Mathematics, 2010
  3. On-Line Encyclopedia of Integer Sequences A053763
  4. Alan Hausman; Howard Kahane; Paul Tidman (2013). Logic and Philosophy — A Modern Introduction. Wadsworth. ISBN 1-133-05000-X. Here: p.327-328
  5. D.S. Clarke; Richard Behling (1998). Deductive Logic — An Introduction to Evaluation Techniques and Logical Theory. University Press of America. ISBN 0-7618-0922-8. Here: p.187


External links

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