Transitivity (mathematics)

binary relation R with the property that xRy and yRz implies xRz

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. [1]
  • 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  .[2]



  1. "Transitivity". Retrieved 2020-10-12.
  2. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-10-12.