Reflexive relation
Encyclopedia
In mathematics
, a reflexive relation is a binary relation
on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".
s.
A relation is called quasi-reflexive if every element that is related to some element is related to itself. 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 relation ~′ such that ~′ is a superset
of ~ and ~′ is reflexive on S. This is equivalent to the union of ~ and the identity relation on S. For example, the reflexive closure of x
The reflexive reduction 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 ~. That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of x≤y is x
Examples of irreflexive relations include:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a reflexive relation is a binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".
Related terms
An irreflexive, or anti-reflexive, relation is the opposite of a reflexive relation. It is a binary relation on a set where no element is related to itself. An example is the "greater than" relation (x>y). 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 related to themselves (i.e. neither all nor none). 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 on the set of natural numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s.
A relation is called quasi-reflexive if every element that is related to some element is related to itself. 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 relation ~′ such that ~′ is a superset
SuperSet
SuperSet Software was a group founded by friends and former Eyring Research Institute co-workers Drew Major, Dale Neibaur, Kyle Powell and later joined by Mark Hurst...
of ~ and ~′ is reflexive on S. This is equivalent to the union of ~ and the identity relation on S. For example, the reflexive closure of x
The reflexive reduction 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 ~. That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of x≤y is x
Examples
Examples of reflexive relations include:- "is equal to" (equality)
- "is a subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of" (set inclusion) - "divides" (divisibilityDivisorIn mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
) - "is greater/less than or equal to":
Examples of irreflexive relations include:
- "is not equal to"
- "is coprimeCoprimeIn number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
to"(for the integers>1, since 1 is coprime to itself) - "is a proper subset of"
- "is greater than"