Quotient set

A quotient set is the set of equivalence classes formed by partitioning a set according to an equivalence relation. Each element of the quotient set is one entire class.

Equivalence relation: A binary relation that is reflexive, symmetric, and transitive. Its three properties ensure that the classes defining this set fit together without overlap.
Quotient map: The canonical surjection sending each element to its equivalence class. It turns the original set into the quotient by mapping every element to its class.
Integers modulo n: The finite set of congruence classes of integers modulo a positive integer n. Congruence modulo n groups integers into a standard arithmetic quotient.
Subsets: A set whose elements all belong to another set. A quotient is not a selected portion of the original set; its elements are classes of original elements.
Equivalence class: The set of all elements related to a given element by an equivalence relation. Each such class is one element of the quotient set.
Representative element: A chosen member of a set or equivalence class used to stand for it. Representatives let calculations use individual elements while the quotient identifies whole classes.
Projective space: The space of lines through the origin in a vector space, often represented by nonzero vectors up to scalar multiplication. It is formed by identifying nonzero vectors that lie on the same line.
Cartesian product: The set of ordered pairs formed by choosing one element from each of two sets. Products combine elements into tuples, whereas quotients identify elements into classes.
Partition of a set: A collection of nonempty, pairwise disjoint subsets whose union is the whole set. Equivalence classes partition the original set into the quotient's elements.
Well-defined function: A function whose output does not depend on arbitrary choices in its description. A function on classes must give the same result regardless of which representatives are chosen.

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