What is Transitivity (mathematics)?
In [[logic]] and [[mathematics]], '''transitivity''' is a property of a binary [[Relation (mathematics)|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. <ref>{{Cite web|title=Transitivity|url=https://nrich.maths.org/1345|access-date=2020-10-12|website=nrich.maths.org}}</ref> *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 <math>R</math>, the smallest transitive relation containing <math>R</math> is called the transitive closure of <math>R</math>, and is written as <math>R^+</math>.<ref>{{Cite web|date=2020-03-25|title=Comprehensive List of Algebra Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/|access-date=2020-10-12|website=Math Vault|language=en-US}}</ref> == Related pages == * [[Symmetric relation]] == References == <references /> {{Math-stub}} [[Category:Mathematics]]