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    Equivalence Relation vs. Equivalence Class

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    Concerning discrete math, I am very confused as to the relationship between an equivalence relation and an equivalence class.

    I would very much appreciate it if someone could explain this relationship and give examples of each such that the relationship (or difference) is clear.

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    Solution Preview

    Please see the attached file for your help.

    Question 1: What is an equivalence relation?
    A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in A:
    • a ~ a. (Reflexivity)
    • if a ~ b then b ~ a. (Symmetry)
    • if a ~ b and b ~ c then a ~ c. (Transitivity)
    Now, we consider a binary relation ~ on the set of integers below:
    For any integers a and b, a~b if and only if 2|(a-b) .............................(1)
    To prove that the relation ~ is an equivalence relation on Z (the set of integers), we need to show that ~ is reflexive, symmetric and transitive.
    (A) To prove that the relation ~ is reflexive, we need to show that for every integer ...

    Solution Summary

    This solution helps with problems regarding equivalence relation and equivalence class.