Irreflexive Relation

A Irreflexive Relation is a relation that is false when all the relation arguments are identical to each other.

• AKA: Strict Relation, Antireflexive, Distinct Relation.
• Context:
  • • ∀ s In S: ¬R(s,s)
• Example(s):
  • Any Subsumption Relation.
  • GreaterThan(1,2) ⇒ True; because if x is greater than y, then x cannot be y.
  • ProperSubset().
  • HeadquarterLocation(): because the Parameter Types are different.
  • IsSiblingTo(): because nobody is a sibling to themselves.
  • IsParentTo(): because nobody is a parent to themselves.
  • PartOf(): because nothing is a part of itself.
  • NotEqualTo(): because nothing can be not equalt to itself.
  • http://www.gabormelli.com/images/IrreflexiveRelationFig1.gif Notice how every relation of each node and itself is FALSE.
  • …
• Counter-Example(s):
  • Equality Relation: by definition.
  • Set Inclusion Relation: because a set is its own subset (but not a "proper" subset).
  • GreaterThanOrEqualTo(X,Y): because if x is greater or equal to y then it can be true that x has identical value y.
• See: Reflexive Relation, Antisymmetric Relation.

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References

• http://www.cs.odu.edu/~toida/nerzic/content/relation/property/property.html
  • Definition(irreflexive relation): A relation R on a set A is called irreflexive if and only if <a, a> R for every element a of A.
  • Example 3: The relation > (or <) on the set of integers {1, 2, 3} is irreflexive. In fact it is irreflexive for any set of numbers.
  • Example 4: The relation {< 1, 1 >, < 1, 2 >, < 1, 3 >, < 2, 3>, < 3, 3 > } on the set of integers {1, 2, 3} is neither reflexive nor irreflexive.
• (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:

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