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

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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 provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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