Reflexive Relation

A Reflexive Relation is a Relation that is True when the Relation Arguments are Identical.

• AKA: Reflexive.
• Context:
  • Forall s in S: R(s,s)
  • If x in S then R(x,x) is True.
• Example(s):
  • Equality Relation "is equal to"
  • Set Inclusion Relation "is a subset of"
  • GreaterThanOrEqualTo(X,Y)
• Counter-Example(s):
  • GreaterThanOrEqualTo(X,Y): TRUE by definition.
  • ProperSubset(X,Y).
  • IsSiblingTo(A,B): because nobody is a sibling to themselves.
  • IsParentTo(A,B): because nobody is a parent to themselves.
  • OrganizationHeadquarterLocation(X,Y): because they Parameter Types are different.
• See: Irreflexive Relation, Symmetric Relation.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Irreflexive_relation
  • In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity.
  • At least in this context, (binary) relation (on X) always means a subset of X × X.
  • If a relation is reflexive, all elements in the set are related to themselves. For example, the relations "is not greater than" and "is equal to" are reflexive over the set of all real numbers. Since no real number is greater than itself, if you compare any number to itself, you will find "is not greater than" to be true. Since every real number is equal to itself, if you compare any number to itself, you will find "is equal to" to be true.
  • A reflexive relation is on set X. This means that all elements in a set are related to themselves by the relation. There are relations which are reflexive on certain sets but not reflexive on the set of real numbers. Say the relation is:
• http://www.cs.odu.edu/~toida/nerzic/content/relation/property/property.html
  • Definition(reflexive relation): A relation R on a set A is called reflexive if and only if < a, a > R for every element a of A.
  • Example 1: The relation on the set of integers {1, 2, 3} is {<1, 1>, <1, 2>, <1, 3>, <2, 2>, <2, 3>, <3, 3>} and it is reflexive because <1, 1>, <2, 2>, <3, 3> are in this relation. As a matter of fact on any set of numbers is also reflexive. Similarly and = on any set of numbers are reflexive. However, < (or >) on any set of numbers is not reflexive.
  • Example 2: The relation on the set of subsets of {1, 2} is { <, >, <, {1} >, <, {2} >, <, {1, 2} >, < {1}, {1} >, < {1}, {1, 2} >, < {2}, {2} >, < {2}, {1, 2} >, < {1, 2}, {1, 2} > }

and it is reflexive. In fact relation on any collection of sets is reflexive.

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