  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 set  and an equivalence relation  on , the equivalence class of an element  in  is the subset of all elements in  which are equivalent to . it follows from the definition of the equivalence relations that the equivalence classes form a partition of . the set of equivalence classes is sometimes called the quotient set or the quotient space of  by  and is denoted by . when  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 quotient categories. ==notation and formal definition== an equivalence relation is a binary relation  satisfying three properties: *for every element  in ,  (reflexivity), *for every two elements  and  in , if , then  (symmetry) *for every three elements , , and  in , if  and , then  (transitivity). the equivalence class of an element  is denoted  and is defined as the set :[a] = \{ x \in x \mid a \sim x \} of elements that are related to  by&nbsp;. an alternative notation  can be used to denote the equivalence class of the element , specifically with respect to the equivalence relation . this is said to be the -equivalence class of . the set of all equivalence classes in  with respect to an equivalence relation  is denoted as  and called  modulo  (or the quotient set of  by {{m
Mathematical_relations