Yhtäläisyysluokkia

Yhtäläisyysluokkia, known in English as equivalence classes, are a fundamental concept in abstract algebra and set theory. They arise from an equivalence relation defined on a set. An equivalence relation is a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity. Reflexivity means that every element is related to itself. Symmetry means that if element a is related to element b, then element b is also related to element a. Transitivity means that if element a is related to element b, and element b is related to element c, then element a is also related to element c.

When an equivalence relation is defined on a set, it partitions the set into disjoint subsets called equivalence classes. Each element of the original set belongs to exactly one equivalence class. All elements within the same equivalence class are considered equivalent to each other under the defined relation. Conversely, elements from different equivalence classes are not equivalent.

A common example is the equivalence relation of congruence modulo n on the set of integers. Two integers a and b are congruent modulo n if their difference (a - b) is divisible by n. The equivalence classes in this case are the sets of integers that leave the same remainder when divided by n. For example, modulo 3, the integers are partitioned into three equivalence classes: the set of multiples of 3 (..., -3, 0, 3, 6, ...), the set of integers that leave a remainder of 1 when divided by 3 (..., -2, 1, 4, 7, ...), and the set of integers that leave a remainder of 2 when divided by 3 (..., -1, 2, 5, 8, ...). The set of these equivalence classes itself forms a new mathematical structure, often called a quotient set or factor set.