equivalence class

mathematics

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June 26, 2026

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In mathematics, an equivalence class is a set formed by elements that share an equivalence relation with a specific element. An equivalence relation is a type of mathematical association that is reflexive, symmetric, and transitive.123

For example, if we consider the relation "is parallel to" between lines, all lines that are parallel to a given line form its equivalence class. This means that every pair of elements within an equivalence class has the defined equivalence relation between them. Equivalence classes are fundamental in partitioning a larger set into distinct, non-overlapping subsets, where each subset contains elements that are equivalent to each other.231

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set theory
Relations in set theory
…form what is called the equivalence class of a. For example, the equivalence class of a line for the relation “is parallel to” consists of the set of all lines parallel to it.
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…form what is called the equivalence class of a. For example, the equivalence class of a line for the relation “is parallel to” consists of the set of all lines parallel to it.
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metalogic
Ultrafilters, ultraproducts, and ultrapowers
…of B are divided into equivalence classes with the help of D : f ≡ g if and only if {i|f(i) = g(i)} ∊ D—in other words, the set of indices i such that f(i) = g(i) belong to D [or f(i) and g(i) are equal “almost everywhere”].
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…of B are divided into equivalence classes with the help of D : f ≡ g if and only if {i|f(i) = g(i)} ∊ D—in other words, the set of indices i such that f(i) = g(i) belong to D [or f(i) and g(i) are equal “almost everywhere”].
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