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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.

https://brainmass.com/math/discrete-math/equivalence-relation-vs-equivalence-class-506758

#### Solution Preview

Question 1: What is an equivalence relation?
Solution:
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.

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