Quotienttijoukko

Quotienttijoukko is a mathematical construction that arises from collapsing a set by an equivalence relation. Given a set X equipped with an equivalence relation ~, the quotienttijoukko X/~ is the set of all equivalence classes with respect to ~. Each equivalence class [x] is defined as [x] = { y in X | y ~ x }. The natural projection map π: X → X/~ sends every element x to its class [x], and every element of X belongs to exactly one class. Thus X/~ partitions X into disjoint, nonempty blocks.

Key properties include that the quotienttijoukko encodes the partition of X into equivalence classes, and two elements of X lie in the same class if and only if they are related by ~. The number of elements of X/~ equals the number of equivalence classes; if X is finite, X/~ is finite as well. The universal property of the quotient map states that any function f: X → Y that is constant on equivalence classes factors uniquely through π as f = f̄ ∘ π for a unique function f̄: X/~ → Y.

Common examples illustrate the construction. For integers with a ≡ b (mod n), the quotienttijoukko Z/~ is Z/nZ, whose elements are the residue classes {0, 1, ..., n−1}. For the real numbers, the relation x ~ y if x − y ∈ Z yields the quotient R/Z, which is topologically a circle.

Quotienttijoukko is a foundational concept because it formalizes the idea of identifying elements that are considered equivalent. It underpins quotient structures in algebra, such as quotient groups or rings, when the relation is compatible with the relevant operations, and it also plays a central role in the construction of quotient spaces in topology.