# Set Theory and Counting

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1. List the ordered pairs in the equivalence relations produced by these partitions of {0,1,2,3,4,5}

a) {0}, {1,2}, {3,4,5}

b) {0,1}, {2,3}, {4,5}

c) {0,1,2}, {3,4,5}

d) {0}, {1}, {2}, {3}, {4}, {5}

2. Which of these collections of subsets are partitions of the set of integers?

a) the set of even integers and the set of odd integers

b) the set of positive integers and the set of negative integers

4. Which of these collections of subsets are partitions of {1,2,3,4,5,6}?

a) {1,2}, {2,3,4}, {4,5,6}

b) {1}, {2,3,6}, {4}, {5}

c) {2,4,6}, {1,3,5}

d) {1,4,5}, {2,6}.

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The solution discusses the set theory and counting.

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Solution to Q1.

(a)

- As {0} is an equivalence class, 0 is related to 0. Hence, (0,0) is the only ordered pair corresponding to the equivalence class {0}.

- As {1,2} is an equivalence class, there are four ordered pairs (1,1), (1,2), (2,1), (2,2).

- As {3,4,5} is an equivalence class, there are nine ordered pairs (3,3), (3,4), (4,3), (4,4), (3,5), (5,3), (4,5),(5,4), (5, 5). So, the ordered pairs in the equivalence relation are {(0,0), (1,1), (1,2), (2,1), (2,2), (3,3), (3,4), (4,3), (4,4), (3,5), (5,3), (4,5),(5,4), (5, 5)}

(b)

- As {0,1} is an equivalence class, there are four ordered pairs (0, 0), (0,1), (1,0), (1,1).

- ...

###### 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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