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. 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.
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 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.
|n||all||transitive||reflexive||preorder||partial order||total preorder||total order||equivalence relation|
k=0 k! S(n, k)
k=0 S(n, k)
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.
- Coreflexive relation — a relation that satisfies ∀x,y: xRy ⇒ x=y
- Antisymmetric relation — a relation that satisfies ∀x,y: xRy ∧ yRx ⇒ x=y
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