In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of elements that are related to one another, forming what are called equivalence classes. Notationally, given a setX and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in Xwhich are equivalent to a. It follows from the definition of the equivalence relations that the equivalence classes form a partition of X. The set of equivalence classes is sometimes called the quotient set of X by ~ and is denoted by X / ~.When X has some structure, and the equivalence relation is defined with some connection to this structure, the quotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and the quotient category.