Equivalence classes (Set theory)

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 [ a ] {\displaystyle [a]} or, equivalently, [ a ] ∼ {\displaystyle [a]_{\sim }} to emphasize its equivalence relation ∼ {\displaystyle \sim } , and is defined as the set of all elements in S {\displaystyle S} with which a {\displaystyle a} is ∼ {\displaystyle \sim } -related. 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.