In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself. In other words, a relation ~ on a set S is reflexive when x ~ x holds true for every x in S, formally: when ∀x∈S: x~x holds.^{[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.
Contents

Related terms 1

Examples 2

Number of reflexive relations 3

Philosophical logic 4

See also 5

Notes 6

References 7

External links 8
Related terms
A relation that is irreflexive, or antireflexive, 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 quasireflexive if every element that is related to some element is also related to itself, formally: if ∀x,y∈S: 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.
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 x≤y.
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 x≤y is x<y.
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 nelement set is 2^{n2−n}.^{[3]}
Philosophical logic
Authors in philosophical logic often use deviating designations. A reflexive and a quasireflexive relation in the mathematical sense is called a totally reflexive and a reflexive relation in philosophical logic sense, respectively.^{[4]}^{[5]}
See also
Notes

^ Levy 1979:74

^ Relational Mathematics, 2010

^ OnLine Encyclopedia of Integer Sequences A053763

^ Alan Hausman, Howard Kahane, Paul Tidman (2013). Logic and Philosophy — A Modern Introduction. Wadsworth. Here: p.327328

^ D.S. Clarke, Richard Behling (1998). Deductive Logic — An Introduction to Evaluation Techniques and Logical Theory. University Press of America. Here: p.187
References

Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, SpringerVerlag. Reprinted 2002, Dover. ISBN 0486420795

Lidl, R. and Pilz, G. (1998). Applied abstract algebra, Undergraduate Texts in Mathematics, SpringerVerlag. ISBN 0387982906

Quine, W. V. (1951). Mathematical Logic, Revised Edition. Reprinted 2003, Harvard University Press. ISBN 0674554515

Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 9780521762687.
External links

Hazewinkel, Michiel, ed. (2001), "Reflexivity",
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