Reflexive relation

(Redirected from Irreflexive relation)

In mathematics, a binary relation on a set is reflexive if it relates every element of to itself.[1][2]

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.

Definitions edit

Let   be a binary relation on a set   which by definition is just a subset of   For any   the notation   means that   while "not  " means that  

The relation   is called reflexive if   for every   or equivalently, if   where   denotes the identity relation on   The reflexive closure of   is the union   which can equivalently be defined as the smallest (with respect to  ) reflexive relation on   that is a superset of   A relation   is reflexive if and only if it is equal to its reflexive closure.

The reflexive reduction or irreflexive kernel of   is the smallest (with respect to  ) relation on   that has the same reflexive closure as   It is equal to   The reflexive reduction of   can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of   For example, the reflexive closure of the canonical strict inequality   on the reals   is the usual non-strict inequality   whereas the reflexive reduction of   is  

Related definitions edit

There are several definitions related to the reflexive property. The relation   is called:

irreflexive, anti-reflexive or aliorelative
[3] if it does not relate any element to itself; that is, if   holds for no   A relation is irreflexive if and only if its complement in   is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.
left quasi-reflexive
if whenever   are such that   then necessarily  [4]
right quasi-reflexive
if whenever   are such that   then necessarily  
if every element that is part of some relation is related to itself. Explicitly, this means that whenever   are such that   then necessarily   and   Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation   is quasi-reflexive if and only if its symmetric closure   is left (or right) quasi-reflexive.
if whenever   are such that   then necessarily  
if whenever   are such that   then necessarily  [5] A relation   is coreflexive if and only if its symmetric closure is anti-symmetric.

A reflexive relation on a nonempty set   can neither be irreflexive, nor asymmetric (  is called asymmetric if   implies not  ), nor antitransitive (  is antitransitive if   implies not  ).

Examples edit

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" on the integers larger than 1
  • "is a proper subset of"
  • "is greater than"
  • "is less than"

An example of an irreflexive relation, which means that it does not relate any element to itself, is the "greater than" relation ( ) 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 (that is, neither all nor none are). For example, the binary relation "the product of   and   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.

An example of a quasi-reflexive relation   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 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 relation and a transitive relation on the same set is always transitive.

Number of reflexive relations edit

The number of reflexive relations on an  -element set is  [6]

Number of n-element binary relations of different types
Elem­ents Any Transitive Reflexive Symmetric Preorder Partial order Total preorder Total order Equivalence relation
0 1 1 1 1 1 1 1 1 1
1 2 2 1 2 1 1 1 1 1
2 16 13 4 8 4 3 3 2 2
3 512 171 64 64 29 19 13 6 5
4 65,536 3,994 4,096 1,024 355 219 75 24 15
n 2n2 2n(n−1) 2n(n+1)/2 n
k!S(n, k)
n! n
S(n, k)
OEIS A002416 A006905 A053763 A006125 A000798 A001035 A000670 A000142 A000110

Note that S(n, k) refers to Stirling numbers of the second kind.

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.[7][8]

Notes edit

  1. ^ Levy 1979, p. 74
  2. ^ Schmidt 2010
  3. ^ This term is due to C S Peirce; see Russell 1920, p. 32. Russell also introduces two equivalent terms to be contained in or imply diversity.
  4. ^ The Encyclopedia Britannica calls this property quasi-reflexivity.
  5. ^ Fonseca de Oliveira & Pereira Cunha Rodrigues 2004, p. 337
  6. ^ On-Line Encyclopedia of Integer Sequences A053763
  7. ^ Hausman, Kahane & Tidman 2013, pp. 327–328
  8. ^ Clarke & Behling 1998, p. 187

References edit

  • Clarke, D.S.; Behling, Richard (1998). Deductive Logic – An Introduction to Evaluation Techniques and Logical Theory. University Press of America. ISBN 0-7618-0922-8.
  • Fonseca de Oliveira, José Nuno; Pereira Cunha Rodrigues, César de Jesus (2004), "Transposing relations: from Maybe functions to hash tables", Mathematics of Program Construction, Springer: 334–356
  • Hausman, Alan; Kahane, Howard; Tidman, Paul (2013). Logic and Philosophy – A Modern Introduction. Wadsworth. ISBN 1-133-05000-X.
  • Levy, A. (1979), Basic Set Theory, Perspectives in Mathematical Logic, Dover, ISBN 0-486-42079-5
  • Lidl, R.; 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
  • Russell, Bertrand (1920). Introduction to Mathematical Philosophy (PDF) (2nd ed.). London: George Allen & Unwin, Ltd. (Online corrected edition, Feb 2010)
  • Schmidt, Gunther (2010), Relational Mathematics, Cambridge University Press, ISBN 978-0-521-76268-7

External links edit