Examples of Equivalence Class

The well-known example of an equivalence relation is the “equal to (=)” relation. In other words, two elements of the given set are equivalent to each other if they belong to the same equivalence class. The equivalence relationships can be explained in terms of the following examples:

Equivalence Relation on Integers

Equivalence Relation: Congruence modulo 5 (a ≡ b [mod(5)] )

Equivalence class of 0: [0] = {. . ., -10, -5, 0, 5, 10, . . .}
Equivalence class of 1: [1] = {. . ., -9, -4, 1, 6, 11, . . .}
Equivalence class of 2: [2] = {. . ., -8, -3, 2, 7, 12, . . .}
Equivalence class of 3: [3] = {. . ., -7, -2, 3, 8, 13, . . .}
Equivalence class of 4: [4] = {. . ., -6, -1, 4, 9, 14, . . .}

Equivalence Relation on Real Numbers

Equivalence Relation: Absolute difference (a ~ b if |a – b| < 1)

Equivalence class of 0: [0] = (-0.5, 0.5)
Equivalence class of 1: [1] = (0.5, 1.5)
Equivalence class of 2: [2] = (1.5, 2.5)
Equivalence class of 3: [3] = (2.5, 3.5)

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Equivalence Class

Equivalence Class are the group of elements of a set based on a specific notion of equivalence defined by an equivalence relation. An equivalence relation is a relation that satisfies three properties: reflexivity, symmetry, and transitivity. Equivalence classes partition the set S into disjoint subsets. Each subset consists of elements that are related to each other under the given equivalence relation.

In this article, we will discuss the concept of Equivalence Class in sufficient detail including its definition, example, properties, as well as solved examples.

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