What are Equivalence Classes?

An equivalence class is the name that we give to the subset of S which includes all elements that are equivalent to each other. “Equivalent” is dependent on a specified relationship, called an equivalence relation. If there’s an equivalence relation between any two elements, they’re called equivalent.

Equivalence Class Definition

Given an equivalence relation on a set S, an equivalence class with respect to an element a in S is the set of all elements in S that are related to a i.e.,

[a] OR {x ϵ S| x is related to a}

For example, consider the set of integers ℤ and the equivalence relation defined by congruence modulo n. Two integers a and b are considered equivalent (denoted as (a ≡ b mod(n) if they have the same remainder when divided by n. In this case, the equivalence class of an integer a is the set of all integers that have the same remainder as a when divided by n.

What is Equivalence Relation?

Any relation R, is said to be Equivalence Realtion if and only if, it satisfy the following three condition:

• Reflexivity: For any element a, a is related to itself.
• Symmetry: If a is related to b, then b is related to a.
• Transitivity: If a is related to b, and b is related to c, then a is related to c.

Read more about Equivalence Relation.

Some examples of equivalence relation are:

Equality on a Set: Let X be any set, and define a relation R on X such that a R b if and only if a = b for a, b ϵ X.

• Reflexivity: For every a ϵ X, a = a (trivially true).
• Symmetry: If a = b, then b = a (trivially true).
• Transitivity: If a = b and b = c, then a = c (trivially true).

Congruence modulo n: Let n be a positive integer, and define a relation R on the integers ℤ such that a R b if and only if a – b is divisible by n.

• Reflexivity: For every a ϵ ℤ, a – a = 0 is divisible by n.
• Symmetry: If a – b is divisible by n, then -(a – b) = b – a is also divisible by n.
• Transitivity: If a – b is divisible by n and b – c is divisible by n, then a – c is also divisible by n.

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.