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] Formally, this may be written ∀x ∈ X : 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.

A binary relation is called irreflexive, or anti-reflexive, if it doesn't relate any element to itself. An example is the "greater than" relation (x > y) on the real numbers. 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 X is called quasi-reflexive if every element that is related to some element is also related to itself, formally: ∀ x, y ∈ X : x ~ y ⇒ (x ~ x ∧ y ~ 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.

A relation ~ on a set X is called coreflexive if for all x and y in X it holds that if x ~ y then x = y.[3] An example of a coreflexive relation is the relation on integers 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 and a transitive relation is always transitive.

A reflexive relation on a nonempty set X can neither be irreflexive, nor asymmetric, nor antitransitive.

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

The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set X 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 X with regard to ~, formally: (≆) = (~) \ (=). That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of (≤) is (<).

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 2n2−n.[4]

Number of n-element binary relations of different types
nalltransitivereflexivepreorderpartial ordertotal preordertotal orderequivalence relation
011111111
122111111
21613443322
35121716429191365
46553639944096355219752415
n2n22n2−nΣn
k=0  k! S(n, k)		n!Σn
k=0  S(n, k)
OEISA002416A006905A053763A000798A001035A000670A000142A000110

Philosophical logic

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.[5][6]

See also

Notes

  1. ↑ Levy 1979:74
  2. ↑ Relational Mathematics, 2010
  3. ↑ Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions to Hash Tables. In Mathematics of Program Construction (p. 337).
  4. ↑ On-Line Encyclopedia of Integer Sequences A053763
  5. ↑ Alan Hausman; Howard Kahane; Paul Tidman (2013). Logic and Philosophy — A Modern Introduction. Wadsworth. ISBN 1-133-05000-X. Here: p.327-328
  6. ↑ 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

References

  • Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover. ISBN 0-486-42079-5
  • Lidl, R. and Pilz, G. (1998). Applied abstract algebra, Undergraduate Texts in Mathematics, Springer-Verlag. ISBN 0-387-98290-6
  • Quine, W. V. (1951). Mathematical Logic, Revised Edition. Reprinted 2003, Harvard University Press. ISBN 0-674-55451-5
  • Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.
  • Hazewinkel, Michiel, ed. (2001) [1994], "Reflexivity", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.