1. Define the relation R: Any two elements X and Y in a have a relation R if and only if X and Y belong to the same partition. So r = {

2. Use inclusive exclusion principle or Venn diagram. Let a, b and c represent the number of elements in the set {1, 2, 3, … 1000} that can be divisible by 4, 5 and 6 respectively. Then |A|=[ 1000/4]=250, |B|=[ 1000/5]=200, |C|=[ 1000/6]= 166. Here [] stands for integer function.

So it is the number of elements in the set A∪B∪C, so | A ∪ B ∪ C | = (| A |+B |+C |)-(| A ∪ B |+A ∪ C |+| B \.

3, for any x, y, z∈A, because (x-x)/3=0∈A, so

if

if

So p is equivalent.

Found partition block. Two integers A and B are in the same partition block, if and only if

[0]={x|x=3n, where n is an integer}

[1]={x|x=3n+ 1, where n is an integer}

[2]={x|x=3n+2, where n is an integer}

So the division corresponding to the relation ρ is {[0], [1], [2]}.

This is the same as the second question.

Let a, b and c respectively represent the number of integers in 1 ~ 300 that can be divisible by 3, 5 and 8. Then |A|=[300/3]= 100, |B|=[300/5]=60, |C|=[300/8]=37, | a ∩ b | = [300//kloc-

Therefore, in order to get the number of elements of the complement of the set A∪B∪C, first find | A ∪ B ∪ C | = (| A |+| B |+| C |)-(| A ∪ B |+| A ∪ C |)