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Encyclopedia > Reflexive relation

In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ...

  • A reflexive relation R on set X is one where for all a in X, a is R-related to itself. In mathematical notation, this is:
forall a in X, a R a
  • An irreflexive (or aliorelative) relation R is one where for all a in X, a is never R-related to itself. In mathematical notation, this is:
forall a in X, lnot (a R a).

Note: A common misconception is that reflexive and irreflexive are opposites, they are not. Irreflexivity is a stronger condition than failure of reflexivity, so a binary relation may be reflexive, irreflexive, or neither. The strict inequalities "less than" and "greater than" are irreflexive relations whereas the inequalities "less than or equal to" and "greater than or equal to" are reflexive. However, if we define a relation R on the integers such that a R b iff a = -b, then it is neither reflexive nor irreflexive, because 0 is related to itself. Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ... The feasible regions of linear programming are defined by a set of inequalities. ... The feasible regions of linear programming are defined by a set of inequalities. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...


Properties containing the reflexive property

Preorder - A reflexive relation that is also transitive. Varieties of preorders such as partial orders and equivalence relations are, therefore, also reflexive. This article is about the mathematics concept. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...


Examples

Examples of reflexive relations include:

Image:GreaterThanOrEqualTo.png

Examples of irreflexive relations include: In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Image File history File links GreaterThanOrEqualTo. ...

  • "is greater than":
Image:GreaterThan.png

  Results from FactBites:
 
Reflexive relation - Wikipedia, the free encyclopedia (205 words)
A reflexive relation R on set X is one where for all a in X, a is R-related to itself.
The strict inequalities "less than" and "greater than" are irreflexive relations whereas the inequalities "less than or equal to" and "greater than or equal to" are reflexive.
However, if we define a relation R on the integers such that a R b iff a = -b, then it is neither reflexive nor irreflexive, because 0 is related to itself.
Equivalence Relation (2634 words)
Equivalence relation, a mathematical concept, is a type of relation on a given set that provides a way for elements of that set to be identified with (meaning considered equivalent to for some present purpose) other elements of that set.
Equivalence relation is defined in a branch of mathematics called set theory, a vital branch underpinning all branches of mathematics and those fields that use mathematics.
Equivalence relations are so ubiquitous in mathematics and other fields that use mathematics because they enable the user to partition a set in a particular way of the user’s design.
  More results at FactBites »

 
 

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