Transitivity (mathematics)

In logic and mathematics, transitivity is a property of a binary relation. It is a prerequisite of an equivalence relation and of a partial order.

 Definition and examples 
In general, given a set with a relation, the relation is transitive if whenever a is related to b and b is related to c, then a is related to c. For example:

Size is transitive: if A>B and B>C, then A>C. 
Subsets are transitive: if A is a subset of B and B is a subset of C, then A is a subset of C. 
Height is transitive: if Sidney is taller than Casey, and Casey is taller than Jordan, then Sidney is taller than Jordan.
Rock, paper, scissors is not transitive: rock beats scissors, and scissors beats paper, but rock doesn't beat paper. This is called an intransitive relation.

Given a relation , the smallest transitive relation containing  is called the transitive closure of , and is written as .

 Related pages 

 Symmetric relation

 References 

Category:Mathematics