Find least number of element that an intersection can hold in a given set

Question

In a class, there are 200 students, at least 140 of students like Maths(Set A), at least 150 like Science(Set B) and at least 160 like English(Set C). What is the minimum and maximum number of students who like all three subjects?

I am able to find the maximum value that an intersection of three set can hold. My approach is, Maximum value for set A,B,C can be 200. So when they completely overlap i.e Each student among 200 student likes all three subject that maximum value of (A ∩ B ∩ C) can be 200.

But I am not able to figure out the minimum value of (A ∩ B ∩ C).

Approach that I tried finding the minimum value of (A ∩ B ∩ C).

As $(A \cup B \cup C)=A+B+C-(A ∩ B)-(A ∩ C)-(B ∩ C)+(A ∩ B ∩ C)$ Considering the minimum value of A,B,C then the Maximum value of $(A ∩ B)=(A ∩ C)=(B ∩ C)$ should be the size of set with minimum value of student i.e $140$ so $(A ∩ B ∩ C)=-140-150-160+3*140+200$ But this approach is not giving the right answer?

What I am doing wrong?

Note: In case someone else is looking hint for some similar problem then please refer following links

https://www.handakafunda.com/set-theory-maximum-and-minimum-values/

https://www.handakafunda.com/solve-questions-based-least-n-set-theory-cat-exam/

Hope it Helps :-)

Answer 1

On the maximum value of the intersection, note that only 140 students like maths (for shame!), so even if all these students like Science and English as well, you'll have at most 140 students liking both maths, science and English, i.e. $\max(\# A\cap B\cap C)=140$.

For the minimum number it might be useful to consider the problem from a different angle. We denote by $A^{c}$ the complement of $A$, i.e. the number of students not liking maths. Using laws of sets note that $$A\cap B\cap C=(A^{c}\cup B^{c}\cup C^{c})^{c}$$ so finding the minimum number of students in $A\cap B\cap C$ is the same problem as finding the maximum of students in $A^{c}\cup B^{c}\cup C^{c}$. In the case that Lord Shark the Unknown described, where the 60 students not liking maths like science and English, the 50 student not liking science like maths and English and the 40 students not linking English like maths and science, we find that $\max(\# A^{c}\cup B^c\cup C^c)=150$, so $$\max(\# A\cap B\cap C)=200-\max(\# A^{c}\cup B^c\cup C^c)=50.$$