quotiëntverzamelingen

Quotient sets, also known as quotient spaces or factor sets, are a fundamental concept in mathematics, particularly in set theory and abstract algebra. A quotient set is formed by partitioning a given set into equivalence classes based on an equivalence relation. An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. When an equivalence relation is defined on a set, it divides the set into disjoint subsets, where each subset contains elements that are equivalent to each other. These subsets are precisely the equivalence classes. The quotient set is then the set of all these distinct equivalence classes.

For example, consider the set of integers and the equivalence relation "congruence modulo n". Two integers are congruent modulo n if their difference is divisible by n. This relation partitions the integers into n distinct equivalence classes, each represented by a remainder when divided by n (0, 1, ..., n-1). The quotient set in this case is the set of these n remainders, often denoted as Z/nZ. Quotient sets generalize this idea to any set and any equivalence relation. They are crucial for constructing new mathematical structures from existing ones and for understanding symmetry and equivalence in various mathematical contexts.