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Mathematics-Online lexicon:

Properties of Relations


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A (binary) relation $ R\subseteq A^2$ on a set $ A$ is called

A reflexive, symmetric and transitive relation is called an equivalence relation, usually symbolized by $ a\sim b$ instead of $ a
\operatorname{R}b$ . An equivalence relation divides a set $ A$ in disjoint subsets (equivalence classes), with any two elements of a particular subset being related (equivalent) to each other, while two elements of distinct subsets are not related to one another.

A reflexive, asymmetric and transitive relation is called a partial order, symbolized as $ a \leq b$ instead of $ a
\operatorname{R}b$. If a partial order is complete, it is called a (total) order; $ A$ is then ordered by $ \leq$.

(Authors: Hörner/Abele)

see also:


[Examples]

  automatically generated 6/19/2007