Inclusion & exclusion with 4 sets

Zermelo

Zermelo

Junior Member
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Jan 7, 2021
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148
Hello,
currently I'm working with the Inclusion and Exclusion principle in my Discrete Math class (didn't know where to put the thread, probability seemed the best choice, correct me if I'm wrong). The problem goes like this:
B = 12, M = 20, F = 20, H = 8, B&M = 5, B&F = 7, B&H = 4, M&F = 16, M&H = 4, H&F = 3, B&M&F = 3, B&M&H = 2, B&F&H = 3, M&F&H = 2, B&M&F&H = 2, Bc & Mc & Fc & Hc = 71.

The big letters represent number of elements in those sets, & represents the set intersection (the whole expression A&B&C is the number of elements in the set intersection), and c represents the complement (it's not mathematically correct, it would take a lot of time to print it out in LaTeX, I'm sorry, but you get what I meant).
The first question is, how many elements are there in total? I solved this easily, the Inclusion exclusion principle generalizes well, and I got S = 100 directly from the principle.
The problem is with the second question, "how many elements are there JUST in set B"? ; aka, B & Mc & Hc &Fc = ?
When I did this problem with 3 sets, I could easily draw a Venn diagram, and the answer would just pop up. For example, for 3 sets A,B,C, A* = A - A&B - A&C + A&B&C.
(A* is the set A&Bc&Cc, aka number of elements in that set.)
But, here drawing Venn diagrams is a lot more complicated. I did it anyway, and god an answer of B* = 6, I would be really grateful if someone could check it, because it's really complicated. Apart from that, I want to know if there is a simpler way of doing this instead of drawing 4 set Venn diagrams (which are really complicated), is there a way to use the Inclusion Exclusion principle on this particular problem?
Thanks in advance.
 
I find that the notation that you use makes the question unreadable.
Some common notation would help. A\cap B gives \(A\cap B\) or A intersect B.
If you use the reply tab you can see the codes for \(A\cup B\), union,
\(\#(A)\) number of elements in A, \(B^c \text{ or }\overline{B}\) for the complement of B.
What is the space? What are all of the sets. give \(\#(X)\) for each set X in S
 
Yes, I was worried about that... The actual question is: there are some students living in a dormitory. 12 of them are taking biology classes (hence B=12, or as I should have put it [MATH]\lvert B \rvert = 12[/MATH], 20 of them are taking math etc etc. B&M = 5 means that there are 5 students taking both math and biology ([MATH]\lvert B \cap M \rvert = 5[/MATH]. Hope this clears it up. Also, there are 71 students in the dorm that aren’t taking any of these classes. The first question was how many students are in the dorm in total (i got 100) and the second problem (which is why I posted the question) is how many students are JUST taking biology? Can I solve this without using elliptical Venn Diagrams? Thanks in advance
 
Zermelo said:
The actual question is: there are some students living in a dormitory. 12 of them are taking biology classes (hence B=12, or as I should have put it [MATH]\lvert B \rvert = 12[/MATH], 20 of them are taking math etc etc. B&M = 5 means that there are 5 students taking both math and biology ([MATH]\lvert B \cap M \rvert = 5[/MATH]. Hope this clears it up. Also, there are 71 students in the dorm that aren’t taking any of these classes. The first question was how many students are in the dorm in total (i got 100) and the second problem (which is why I posted the question) is how many students are JUST taking biology? Can I solve this without using elliptical Venn Diagrams? Thanks in advance
Click to expand...
What in the world does "[MATH]\lvert B \rvert = 12[/MATH], 20 of them are taking math etc etc." mean?
 
[MATH]\text{ }\\ \text{ }\\ n(B) \hspace2ex -\hspace2ex [n(B \cap M)+n(B \cap F)+n(B \cap H)] \hspace2ex + \hspace2ex [n(B \cap M \cap F) + n(B \cap M \cap H) + n(B \cap F \cap H)]\hspace2ex - \hspace2ex n(B \cap M \cap F \cap H)[/MATH]
Answer should be 2

(The above finds the number of elements of B not in any of the other sets.)
 
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