Transitive relation

In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.

Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x = y and y = z then x = z.

Definition

A homogeneous relation R on the set X is a transitive relation if,

for all a, b, c ∈ X, if a R b and b R c, then a R c.

Or in terms of first-order logic:

∀

a

,

b

,

c

∈

X

:

(

a

R

b

∧

b

R

c

)

⇒

a

R

c

{\displaystyle \forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc}

,

where a R b is the infix notation for (a, b) ∈ R.

Examples

As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy is also an ancestor of Carrie.

On the other hand, "is the birth mother of" is not a transitive relation, because if Alice is the birth mother of Brenda, and Brenda is the birth mother of Claire, then it does not follow that Alice is the birth mother of Claire. In fact, this relation is antitransitive: Alice can never be the birth mother of Claire.

Non-transitive, non-antitransitive relations include sports fixtures (playoff schedules), 'knows' and 'talks to'.

The examples "is greater than", "is at least as great as", and "is equal to" (equality) are transitive relations on various sets.