What is equivalence class?

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Equivalence Relation Definition: An equivalence class is a concept from set theory and abstract algebra. It is a subset of a set formed by an equivalence relation. To understand what an equivalence class is, we first need to know what an equivalence relation is. An equivalence relation on a set is a binary relation that is reflexive, symmetric, and transitive. If we have an equivalence relation ~ on a set S, then for any element a in S, the equivalence class of a is the set of elements in S that are related to a by ~.Equivalence Class Definition: Let's denote the equivalence class of an element a in S by [a]. The definition of [a] is [a] = {x in S | x ~ a}. This means that the equivalence class of a consists of all elements in S that are equivalent to a under the relation ~.Example with Congruence Modulo: To give an example, consider the set of integers Z and the equivalence relation of congruence modulo n, denoted by ≡_n. Two integers a and b are said to be congruent modulo n if their difference a - b is divisible by n. The equivalence class of an integer a modulo n is the set of all integers that are congruent to a modulo n.Example with n = 3 and a = 2: For instance, if we take n = 3 and a = 2, then the equivalence class of 2 modulo 3, denoted by [2]_3, is the set of all integers x such that x - 2 is divisible by 3. This set would include ..., -4, -1, 2, 5, 8, ..., and so on, since all these numbers differ from 2 by a multiple of 3.Partition of Equivalence Classes: In general, the set of all equivalence classes of a set S under an equivalence relation ~ forms a partition of S. This means that every element of S is in exactly one equivalence class, and the union of all equivalence classes is the entire set S.Importance in Mathematics: It's important to note that the concept of an equivalence class is foundational in many areas of mathematics, including geometry, number theory, and algebra. It helps to classify objects that are similar in a certain sense into well-defined categories.

What is equivalence class?

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