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.[1][2] Formally, this may be written xX : x R x.

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.

Related terms[edit]

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: x, yS : x ~ y ⇒ (x ~ 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:

  • "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[edit]

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

Number of n-element binary relations of different types
n all transitive reflexive preorder partial order total preorder total order equivalence relation
0 1 1 1 1 1 1 1 1
1 2 2 1 1 1 1 1 1
2 16 13 4 4 3 3 2 2
3 512 171 64 29 19 13 6 5
4 65536 3994 4096 355 219 75 24 15
n 2n2 2n2n Σn
k! S(n, k)
n! Σn
S(n, k)
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110

Philosophical logic[edit]

Authors in philosophical logic often use different terminology. Reflexive relations in the mathematical sense are called totally reflexive in philosophical logic, and quasi-reflexive relations are called reflexive.[4][5]

See also[edit]


  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[edit]