# 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 ${\displaystyle R}$ , the smallest transitive relation containing ${\displaystyle R}$  is called the transitive closure of ${\displaystyle R}$ , and is written as ${\displaystyle R^{+}}$ .[2]

## References

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