information:extraction, linguistic:paraphrasing
In mathematics, when the elements of some set S {\displaystyle S} have a notion of equivalence (formalized as an equivalence relation), 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} is denoted {\displaystyle } or, equivalently, ∼ {\displaystyle _{\sim }} to emphasize its equivalence relation ∼. {\displaystyle \sim.} The definition of equivalence relations implies that the equivalence classes form a partition of S, {\displaystyle S,} meaning, that every element of the set belongs to exactly one equivalence class.The set of the equivalence classes is sometimes called the quotient set or the quotient space of S {\displaystyle S} by ∼, {\displaystyle \,\sim \,,} and is denoted by S / ∼. {\displaystyle S/{\sim }.}

When the set S {\displaystyle S} has some structure (such as a group operation or a topology) 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.