Binary relation that relates every element to itself
In mathematics, a binary relation
on a set
is reflexive if it relates every element of
to itself.
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
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
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 ![{\displaystyle yRy.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- quasi-reflexive
- 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. - antisymmetric
- if whenever
are such that
then necessarily ![{\displaystyle x=y.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- coreflexive
- if whenever
are such that
then necessarily
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
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
The number of reflexive relations on an
-element set is
[6]
Note that S(n, k) refers to Stirling numbers of the second kind.
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.
Notes
- ^ 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.
- ^ The Encyclopedia Britannica calls this property quasi-reflexivity.
- ^ On-Line Encyclopedia of Integer Sequences A053763
References
- 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