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Equivalence Classes of an Equivalence Relation

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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 provided by:
Thokchom Sarojkumar Sinha, MSc
Education
  • BSc, Manipur University
  • MSc, Kanpur University
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