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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 Summary

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

    $2.19

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