Transitive relation

A relation

R

is transitive when

x R y

and

y R z

imply

x R z

.

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- Transitive closure (relation) Transitive relation

Transitive closure (relation) ( $R^{+} = ⋃_{n \geq 1} R^{n}$ )

The transitive closure of a relation connects

x

to

y

when a finite nonempty chain of

R

-steps leads from

x

to

y

. Concatenation proves transitivity, and induction along a chain shows that any transitive relation containing

R

contains

R^{+}

. This concerns closure of a relation, rather than the transitive closure of a set under membership.

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Transitive relation by

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A relation \( R \) on a set is called a transitive relation if, for all elements \( a, b, c \) in that set, whenever \( a \) is related to \( b \) (denoted \( aRb \)) and \( b \) is related to \( c \) (denoted \( bRc \)), then \( a \) must also be related to \( c \) (denoted \( aRc \)).

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