[Math] Finding the intersection of three sets

discrete mathematicselementary-set-theory

80 students were asked if they like math, science or humanities. 24
students did not like either of the subjects, 9 liked math only, 16
liked science only, 9 liked humanities only, 12 liked math and
humanities, 7 liked math and science and 9 liked humanities and
science.

a) How many students like all three subjects?

b) How many students like math or science?

c) How many students don't like humanities?

Here's a venn diagram displaying the given information:

enter image description here

a) finding the intersection of sets M, S and H

|M∩S∩H|=|M∪S∪H|−(|M|+|S|+|H|)+|M∩S|+|M∩H|+|S∩H|

-2 |M∩S∩H|= (80 – 24) – (9 + 16 + 9) – (12 + 7 + 9)

-2 |M∩S∩H|= 56 – 34 – 28

-2 |M∩S∩H|= 22 – 28

-2 |M∩S∩H|= -6

|M∩S∩H|= 3

I can't do b) or c) because when I say the intersection is 3, then all the other numbers in the venn diagram change (obviously). For example, if the intersection is 3, then the number of people who like math and science = 4 (7 – 3) and the number of people who like math and humanities = 9 (12 – 3). But when I add up the newfound numbers (3 + 9 + 4), I get 16 and I can't do 9 – 16 (which is -5, A NEGATIVE NUMBER!!!) Could someone please let me know what I have done wrong and how the heck I'm supposed to figure out the intersection of three sets?! Any help would be greatly appreciated.

Best Answer

So your first issue is that your Venn diagram does not display the given information, as you note at the end. This is confusing you, despite your solution being essentially correct, if somewhat oddly calculated (though I have no idea why you're trying to calculate $9 - 16$: you don't need to adjust the outer values, because those are already in your Venn diagram correctly; indeed, they're given in the question).

Let's call the number of students who like all three subjects $x$. Then your actual Venn diagram looks like this:

Venn diagram

Now, summing all of those values, we see that $80 = 24 + 9 + 16 + 9 + 7 - x + 9 - x + 12 - x + x = 86 - 2x$, and so $2x = 6$, and $x = 3$. Thus, the full Venn diagram looks like this:

Venn diagram with correct values

We can now solve the questions by just reading off the diagram.

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