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. 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.
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: 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 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.
|Number of n-element binary relations of different types|
|n||all||transitive||reflexive||preorder||partial order||total preorder||total order||equivalence relation|
Authors in philosophical logic often use deviating designations. A reflexive and a quasi-reflexive relation in the mathematical sense is called a totally reflexive and a reflexive relation in philosophical logic sense, respectively.
- Levy 1979:74
- Relational Mathematics, 2010
- On-Line Encyclopedia of Integer Sequences A053763
- Alan Hausman, Howard Kahane, Paul Tidman (2013). Logic and Philosophy — A Modern Introduction. Wadsworth. ISBN 1-133-05000-X.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> Here: p.327-328
- 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.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> Here: p.187
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- Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.