# Equivalence Relation vs. Equivalence Class

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.

© BrainMass Inc. brainmass.com October 10, 2019, 5:29 am ad1c9bdddfhttps://brainmass.com/math/discrete-math/equivalence-relation-vs-equivalence-class-506758

#### Solution Preview

Please see the attached file for your help.

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.