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
changeIn 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]
Related pages
changeReferences
change- ↑ "Transitivity". nrich.maths.org. Retrieved 2020-10-12.
- ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-10-12.
