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In mathematics, when the elements of some set S {\displaystyle S} have a notion of equivalence , then one may naturally split the set S {\displaystyle S} into equivalence classes. These equivalence classes are constructed so that elements a {\displaystyle a} and b {\displaystyle b} belong to the same equivalence class if, and only if, they are equivalent.
Formally, given a set S {\displaystyle S} and an equivalence relation ∼ {\displaystyle \,\sim \,} on S , {\displaystyle S,} the equivalence class of an element a {\displaystyle a} in S , {\displaystyle S,} denoted by , {\displaystyle ,} is the set
When the set S {\displaystyle S} has some structure and the equivalence relation ∼ {\displaystyle \,\sim \,} is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.