transitive relation

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. In mathematical notation, this is:

\forall a, b, c  \in X,\ a R b \and b R c \; \Rightarrow a R c

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 subset of" (set inclusion)
"is less than" and "is less than or equal to" (inequality)
"divides" (divisibility)

A transitive relation that is also reflexive is a preorder. A preorder that is antisymmetric is a partial order. A preorder that is symmetric, is an equivalence relation. See also transitive closure, Intransitivity

External Link

Transitivity in Action

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