ækvivalensklasser

Ækvivalensklasser, also known as equivalence classes, are a fundamental concept in abstract algebra and set theory. An equivalence class is a subset of a larger set, where all elements within that subset share a common property defined by an equivalence relation. An equivalence relation is a binary relation that is reflexive, symmetric, and transitive.

Reflexivity means that every element in the set is related to itself. Symmetry implies that if element 'a' is related to element 'b', then 'b' is also related to 'a'. Transitivity states that if 'a' is related to 'b' and 'b' is related to 'c', then 'a' must also be related to 'c'.

When an equivalence relation is defined on a set, it partitions the set into disjoint equivalence classes. This means that every element of the original set belongs to exactly one equivalence class. The union of all equivalence classes is the original set, and the intersection of any two distinct equivalence classes is empty.

For example, consider the set of integers and the relation "has the same remainder when divided by 3." This relation is an equivalence relation. The equivalence classes would be {..., -6, -3, 0, 3, 6, ...}, {..., -5, -2, 1, 4, 7, ...}, and {..., -4, -1, 2, 5, 8, ...}. Each class contains integers that, when divided by 3, yield the same remainder (0, 1, or 2 respectively). The concept of equivalence classes is crucial for constructing quotient sets and understanding modular arithmetic.