 TOPICS  # Equivalence Relation

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

1. Reflexive: for all ,

2. Symmetric: implies for all 3. Transitive: and imply for all ,

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

Equivalence Class, Teichmüller Space

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## References

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.

## Referenced on Wolfram|Alpha

Equivalence Relation

## Cite this as:

Weisstein, Eric W. "Equivalence Relation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EquivalenceRelation.html