The Transitivity (mathematics) reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Transitivity (mathematics)

In mathematics, the word transitive admits at least two distinct meanings:

• A group G acts transitively on a set S if for any x, y ∈ S, there is some g ∈ G such that gx = y. See group action. A somewhat related meaning is explained at ergodic theory.

• A binary relation is transitive if whenever A is related to B and B is related to C, then A is related to C, for all A, B, and C in the domain of the relation.

In notation, this is:

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.

Example of transitive relations include:

• "is equal to" -- equality
• "is a subset of"

If a transitive relation is also reflexive and symmetric, then it is said to be an equivalence relation.

See also transitive closure, Intransitivity

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