# Equivalence Classes of an Equivalence Relation

Modern Algebra

Set Theory (II)

Equivalence Classes of an Equivalence Relation

The distinct equivalence classes of an equivalence relation on a set A provide us with a decomposition of A as a union of mutually disjoint subsets.

Conversely, given a decomposition of A as a union of mutually disjoint, nonempty subsets, we can define an equivalence relation on A for which these subsets are the distinct equivalence classes.

Or, An equivalence relation over a set induces a partition of the set. Conversely, a partition of a set defines an equivalence relation.

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Modern Algebra

Set Theory (II)

Equivalence Classes of an Equivalence Relation

By:- Thokchom Sarojkumar Sinha

The distinct equivalence classes of an equivalence relation on a set provide us with a decomposition of as a union of mutually

disjoint subsets.

Conversely, given a decomposition of as a union of mutually disjoint, nonempty subsets, we can define an equivalence relation

on for which these subsets are the distinct equivalence classes.

Or,

An equivalence relation over a set induces a partition of the set. Conversely, a partition of a set defines an equivalence relation.

Solution:- Let ~ be an equivalence relation on a set .

For any ,

We have to prove that the equivalence relation ~ over the set decomposes the set as a union of mutually disjoint

...

#### Solution Summary

This problem defines the properties of Equivalence Classes of an Equivalence Relations. The solution is detailed and well presented.