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Equivalence Class


An equivalence class is defined as a subset of the form {x in X:xRa}, where a is an element of X and the notation "xRy" is used to mean that there is an equivalence relation between x and y. It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of X. For all a,b in X, we have aRb iff a and b belong to the same equivalence class.

A set of class representatives is a subset of X which contains exactly one element from each equivalence class.

For n a positive integer, and a,b integers, consider the congruence a=b (mod n), then the equivalence classes are the sets {...,-2n,-n,0,n,2n,...}, {...,1-2n,1-n,1,1+n,1+2n,...} etc. The standard class representatives are taken to be 0, 1, 2, ..., n-1.


See also

Congruence, Coset, Equivalence Relation

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References

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 56-57, 1993.

Referenced on Wolfram|Alpha

Equivalence Class

Cite this as:

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

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