Quotients

Quotients are constructions formed by partitioning a set by an equivalence relation; the quotient set X/~ consists of equivalence classes. The natural projection q maps each element to its class.

In algebra, quotient groups: if G is a group and H a normal subgroup, the cosets gH partition G; the set G/H inherits a group structure. Similarly quotient rings R/I by ideal I; quotient modules M/N; quotient vector spaces V/W.

In arithmetic, quotient and remainder: for integers a, b ≠ 0, a = bq + r, where q is the quotient and r the remainder; arithmetic quotients are often written as a/b with floor behavior in division algorithm.

In topology and geometry: quotient spaces arise by identifying points per an equivalence relation. Example: gluing endpoints of [0,1] gives a circle; identifying opposite edges of a square yields a torus; RP^n as quotient of S^n by antipodal identification. The projection map is continuous, and the quotient topology is the coarsest making q continuous.

Universal property: any map constant on equivalence classes factors uniquely through the quotient map.