.
To solve problems like this, you need to know and to understand simple basic formulas related to subsets of a finite set.
If M and D are two subsets of a finite set S, then the following formulas are true related to the numbers of elements in subsets:
n((M U D) = n(M) + n(D) - n(M & D), (1) and
n(M & D) = n(M) + n(D) - n((M U D). (2)
where n(X) denotes the number of elements of a subset X.
The proof of these formulas is elementary, and three lines are enough.
Formula (1) is true, because to count the elements in the union of two subsets, we should take the sum of elements of each subset
n(M) and n(D) and then subtract the number of element in the intersection, sice we counted them twice, when we added n(M) and n(D).
Thus the formula (1) is just proved in this way. The formula (2) is the formal algebraic consequense of the formula (1).
After this introduction, we are ready to solve the problem.
Let S be the set of all 16 students; the subset M is the set of the students violated the money regulation (8 students)
and the subset D is the set of students violated the dresses regulation (10 students).
(a) What was the maximum number of students breaking both rules ?
Answer to this question is 8.
It happens when the subset M is the part of the subset D.
(b) what was the minimum number of students breaking both rules ?
They ask you what is the minimum number of elements in the intersection (M & D).
The number of elements in the intersection (M & D) is minimal, when the subsets M and D cover the entire set S.
In this case, n(M U D) = n(S) = 16, and formula (2) gives then
2 = 10 + 8 - 16.
It is the answer to question (b).
(c) What can you say about the number of students breaking neither rules ?
The maximum is 16-10 = 6 (when the set M is the part (is inside) of the set D;
The minimum is 0 (when the intersection of sets M and D is minimal comprising of 2 students)
Solved.
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To see other similar solved problems, look into the lesson
- Counting elements in sub-sets of a given finite set
in this site.
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The lesson to learn from my post is THIS :
a) to solve problems like this, you need to know and to understand the formulas (1) and (2).
b) Problems like this have two extreme cases: one when the smaller set is the part of the larger set, and the other case,
when the subsets have minimal intersection. The last case happens when two subsets together cover the entire set.