Transitive relation

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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.

To write this in predicate logic:

<math>\forall a, b, c \in X,\ a \,R\, b \and b \,R\, c \; \Rightarrow a \,R\, c</math>

1 Counting transitive relations
2 Examples
3 Other properties that require transitivity
4 See also
5 External links
6 References

[edit] Counting transitive relations

Unlike other relation properties, no general formula that counts the number of transitive relations on a finite set (sequence A006905 in OEIS) is known.<ref>Steven R. Finch, "Transitive relations, topologies and partial orders", 2003.</ref> However, there is a formula for finding the number of relations which are simultaneously reflexive, symmetric, and transitive (sequence A000110 in OEIS), those which are symmetric and transitive, those which are symmetric, transitive, and antisymmetric, and those which are total, transitive, and antisymmetric. Pfeiffer<ref>Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.</ref> has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one is difficult.

[edit] Examples

For example, "is greater than" and "is equal to" are transitive relations: if a = b and b = c, then a = c.

On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire.

Examples of transitive relations include:

"is equal to" (equality)
"is a superset of"
"is a subset of" (set inclusion)
"is less than" and "is less than or equal to" (inequality)
"divides" (divisibility)
"implies" (implication)

[edit] Other properties that require transitivity

partial order - a preorder that is antisymmetric
preorder - a transitive relation that is also reflexive
total preorder - a preorder that is total
equivalence relation - a preorder that is symmetric.
strict weak ordering - a strict partial order in which incomparability is an equivalence relation
total ordering - a relation is a total order if it is transitive, antisymmetric, and total