Equivalence class

In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:

Related Topics:
Mathematics - Set - Equivalence relation - Subset

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: = { x ∈ X | x ~ a }

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The notion of equivalence classes is useful for constructing sets out of already constructed ones. The set of all equivalence classes in X given an equivalence relation ~ is usually denoted as X / ~ and called the quotient set of X by ~. This operation can be thought of (very informally indeed) as the act of "dividing" the input set by the equivalence relation, hence both the name "quotient", and the notation, which are both reminiscent of division.

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In cases where X has some additional structure preserved under ~, the quotient becomes an object of the same type in a natural fashion; the map that sends a to is then an epimorphism. See congruence relation.

Related Topics:
Map - Epimorphism - Congruence relation

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