PlanetMath: transitive relation

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transitive relation

(Definition)

A relation $\mathcal{R}$ on a set is transitive if and only if $\forall x,y,z \in A$ , $(x\mathcal{R}y \land y\mathcal{R}z) \rightarrow (x\mathcal{R}z)$ .

For example, the “is a subset of” relation $\subseteq$ on any set of sets is transitive. The “less than” relation on the set of real numbers is also transitive.

The “is not equal to” relation $\neq$ on the set of integers is not transitive, because $1\neq 2$ and $2\neq 1$ does not imply $1\neq 1$ .

"transitive relation" is owned by yark. [ full author list (3) | owner history (2) ]

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Also defines:

transitivity, transitive

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Cross-references: imply, integers, real numbers, relation
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This is version 10 of transitive relation, born on 2002-02-02, modified 2006-10-19.
Object id is 1669, canonical name is Transitive3.
Accessed 9205 times total.

Classification:

AMS MSC:

03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

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