Equivalence Relation

An equivalence relation on a set X is a subset of X×X, i.e., a collection R of ordered pairs of elements of X, satisfying certain properties. Write "xRy" to mean (x,y) is an element of R, and we say "x is related to y," then the properties are

1. Reflexive: aRa for all a in X,

2. Symmetric: aRb implies bRa for all a,b in X

3. Transitive: aRb and bRc imply aRc for all a,b,c in X,

where these three properties are completely independent. Other notations are often used to indicate a relation, e.g., a=b or a∼b.

See also

Equivalence Class, Teichmüller Space

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Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 18, 1990.Stewart, I. and Tall, D. The Foundations of Mathematics. Oxford, England: Oxford University Press, 1977.

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Equivalence Relation

Cite this as:

Weisstein, Eric W. "Equivalence Relation." From MathWorld--A Wolfram Web Resource.

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