Transitive Relation

A Transitive Relation is a Binary Relation where two elements are related if they are already related by an intermediary element.

• AKA: Transitive, Transitive Binary Relation.
• Context:
  • More formally [math]\displaystyle{ R }[/math] is a Transitive Relation IF ∀ a,b,c ∈ S: IF R(a,b) =>True ∧ R(b,c) =>True THEN R(a,c) =>True.
• Example(s):
  • If R(a,b), R(b,c), R(c,c), ¬R(a,a), and ¬R(c,d) then R(a,c) and ¬R(b,d) ; while R(b,b), R(d,d), R(b,a), R(a,d), R(d,a), R(d,b), R(d,c) are Undefined Relation Instances.
  • LessThan, GreaterThan, ≥, ...
  • If PartOf(X,Y), PartOf(Y,Z) then PartOf(X,Z).
  • If TypeOf(X,Y), TypeOf(Y,Z) then TypeOf(X,Z).
• See: Intransitive Relation, Order Relation, Equivalence Relation.

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References

2009

• http://www.cs.odu.edu/~toida/nerzic/content/relation/property/property.html
  • Definition(transitive relation): A relation [math]\displaystyle{ R }[/math] on a set A is called transitive if and only if for any a, b, and c in A, whenever [math]\displaystyle{ R }[/math] <a, b> R, and <b, c> R, <a, c> [math]\displaystyle{ R }[/math] .
  • Example 8: The relation on the set of integers {1, 2, 3} is transitive, because for <1, 2> and <2, 3> in, <1, 3> is also in, for <1, 1> and <1, 2> in, <1, 2> is also in, and similarly for the others. As a matter of fact on any set of numbers is also transitive. Similarly and = on any set of numbers are transitive.

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