Equivalence Classes and Partition

Groups of elements in a set related by an equivalence relation, whereas a collection of these equivalence classes, covering the entire set with no overlaps are called partition.

Difference between Equilavalence Classes and Partition

The key difference between Equilavalence Classes and Partition are given in the following table:

Feature	Equivalence Classes	Partitions
Definition	Sets of elements that are considered equivalent under a relation.	A collection of non-empty, pairwise disjoint subsets such that their union is the entire set.
Notation	If A is an equivalence class, it is often denoted as [a] or [a]R, where a is a representative element and R is the equivalence relation.	A partition of a set X is denoted as {B₁, B₂, … ,B_n}, where B_i are the disjoint subsets in the partition.
Relationship	Equivalence classes form a partition of the underlying set.	A partition may or may not arise from an equivalence relation.
Cardinality	Equivalence classes may have different cardinalities.	All subsets in the partition have the same cardinality.
Example	Consider the set of integers and the equivalence relation “having the same remainder when divided by 5.” Equivalence classes are {…,−5,0,5,…}, {…,−5,0,5,…}, {…,−4,1,6,…}, and {…,−4,1,6,…}, etc.	Consider the set of integers partitioned into even and odd numbers: {…,−4,−2,0,2,4,…}, and {…,−3,−1,1,3,5,…}.
Intersection of Classes	Equivalence classes are either disjoint or identical.	Partitions consist of disjoint subsets.

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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