Finding the intersection of three sets

Question

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:

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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

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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.

Answer 1

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:

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:

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

Answer 2

You are overlooking that the $7$ who like math and science, and the $12$ who like math and humanities and the $9$ who like humanities and science, OVERLAP and each of those groups include those who like all three.

You assumed those were all separate and each group excluded the $?$ that liked all three.

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Use this image instead

When you say $12$ like math and humanities it is ambiguous as to whether is is meant there are $12$ who like math, humanities and dislike science. Or if there are $12$ who like math and humanities and may or may not like science.

If you interpret it the first way, there are $12$ who like math and humanities but do not like science and use the drawing you drew you will get a negative number for those who like all three.

But if you interpret it the second way (which is the logical and literal and mathematical way to interpret it; if you are told they like math and humanities that means everyone who likes math and humanities regardless of what else they may or may not like) then if there are $?$ who like all three then there are $12$ who like math and humanities, and may or may not like science; and there are $12 -?$ who like math and humanities and don't like science.

Another way to view this is:

But here is should be clear then regions of $7,12,9$ overlap.

Answer 3

The Venn diagram doesn't seem quite correct. If you add up the numbers, you get 86, which is greater than the universe of 80 students. I think that the 12 who like math and humanities includes those who like math, humanities, and science (and likewise for the 7 who like math and science, and the 9 who like science and humanities).

So, when we add up the numbers, we get 86 students, but we have counted the students who like all three subjects three times each - two times more than we should have.

We know that we should have 80 students so our overcount is 86-80=6. Students who like math, science, and humanities have each been counted two times too many. So there must be 6/2=3 such students.

Once we know this, we can conclude that there are:

• 12-3=9 students who like math and humanities but not science
• 7-3=4 students who like math and science but not humanities
• 9-3=6 students who like science and humanities but not math.

Fill these numbers into the Venn diagram and you should be able to complete the rest of the problems.