Three Set Example Problems
Problem 1
There are a total of 103 foreign language students in a high school where they offer Spanish,
French, and German. There are 29 students who take at least 2 languages at once. If there are 40
Spanish students, 42 French students, and 46 German students, how many students take all three
languages at once?
We try to motivate this through the Venn diagram.
If you look at the counting:
Since all students are taking at least one language, nothing is outside of the
A or B or C set.
But we have an overlap, since the count for A with B with C is 40 + 42 +
46 = 128, but we have only 103 students!
So, the overlap is 128 - 103 = 25 only.
Now the statement that "29 students who take at least 2 languages at once"
includes the area shaded in ugly green. So the common intersection (red
crosshatch) is included in it redundantly.
It must be the source of disagreement between 25 and 29, and the overlap (those taking all three) is 29 - 25 = 4.
We could also use the formula below. The formula is developed by cutting apart the three-set Venn in a way
similar to the formula for two sets Venn diagrams in Lesson 3.1.
Then by substitutions:
© Arizona State University, Department of Mathematics and Statistics
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Three Set Example Problems
Problem 2:
In a survey of 279 college students, it is found that
29 like both Brussels sprouts and broccoli,
25 like both Brussels sprouts and cauliflower,
24 like both broccoli and cauliflower, and
14 of the students like all three vegetables.
68 like brussels sprouts,
90 like a broccoli,
57 like cauliflower,
In a problem like this one, a Venn Diagram is the best organizational tool.
This is what I call Mickey's Head in the videos and such.
Start with three interlocking circles. Each circle represents the count of a
characteristic, IE, A= "like brussels sprouts,” B = "like broccoli,” and C =
"like cauliflower." Label the circles in a way to remember what it stands for.
The next step is to find something you can fill in without any calculation.
Look for words like "all" or "none." In this problem, 14 of the students like
all. That means the number 14 goes into the area of Mickey's nose.
Now we look at the "pairwise" intersections. These are our "and" statements,
using only two characteristics. In the second diagram they are shaded gray.
Recall that these overlap the "all" region. The 14 we know about are already
counted! The three red results show how we adjust the counts so the "cat's
eye" shapes total properly for these statements:
29 like both Brussels sprouts and a broccoli,
25 like both Brussels sprouts and cauliflower,
24 like both broccoli and cauliflower
The rest of the diagram works in a similar way. Totals in each circle must add
to meet these statements:
68 like brussels sprouts,
90 like a broccoli,
57 like cauliflower
So, the "A-only" region must have a count of
68 - (15+14+11) = 28.
Then the "B-only" region must have a count of
90 - (15+14+10) = 51.
Finally, the "C-only" region must have a count of 57 - (11+14+10) = 22.
We filled them into the diagram to see the third result. We are almost done.
The possibility does exist that there is someone in the survey who doesn't
like any of these three taste delights! To fill in the "likes none of them" area,
add all the numbers in Mickey's head to get 151. Since the survey includes
279 students, the number counted outside of the head is 279 - 151 = 128. That is shown in green.
The diagram is complete.
On to Lesson 6
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© Arizona State University, Department of Mathematics and Statistics