transitive relation

A relation $\mathcal{R}$ on a set $A$ is transitive if and only if $\forall x,y,z\in A$ , $(x\mathcal{R}y\land y\mathcal{R}z)\rightarrow(x\mathcal{R}z)$ .

For example, the “is a subset of” relation $\subseteq$ on any set of sets is transitive. The “less than” relation $<$ on the set of real numbers is also transitive.

The “is not equal to” relation $\neq$ on the set of integers is not transitive, because $1\neq 2$ and $2\neq 1$ does not imply $1\neq 1$ .

Title	transitive relation
Canonical name	TransitiveRelation
Date of creation	2013-03-22 12:15:52
Last modified on	2013-03-22 12:15:52
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	14
Author	yark (2760)
Entry type	Definition
Classification	msc 03E20
Related topic	Reflexive
Related topic	Symmetric
Related topic	Antisymmetric
Defines	transitivity
Defines	transitive