Venn Diagram: Two Possibilities

Date: 01/14/2003 at 10:39:23
From: Tracey Sangermano
Subject: Venn Diagrams/Problem Solving

Here is the specific question: The book says the answer is 32, but 26 
seems to be correct.

The science club advisor asked club members what science courses they 
liked. Eighteen members said they liked physics, 17 liked chemistry, 
and 10 liked biology. However, of these, 9 liked physics and 
chemistry, 4 liked biology and chemistry, 2 liked physics and biology, 
and 2 liked all three. How many science club members were interviewed?

I don't think the problem is stated clearly. Should we include the 
people who liked 2 and 3 subjects in the people who liked 1? If so, 
how did the answer become 32?

Date: 01/14/2003 at 12:49:18
From: Doctor Peterson
Subject: Re: Venn Diagrams/Problem Solving

Hi, Tracey.

These problems are rarely stated clearly enough to satisfy me. I 
think it is very clear in this case that those who like two or three 
subjects are included in the numbers for those who like one, since 
it says "of these." It is not entirely clear whether the 2 who like 
everything are included in the numbers who like two; those should 
have said "9 like ONLY physics and biology," or "2 OF THOSE like all 
three," depending on the intent.

If we take the 3 to be included in the twos, then the total number in 
the union of the sets is the alternating sum

    Like at least one - like at least two + like all three
      (18 + 17 + 10)  -    (9 + 4 + 2)    +     2
            45        -         15        +     2

which gives 32. I assume you are familiar with this "inclusion-
exclusion rule," which recognizes that in adding the three "at least 
one" numbers, we are counting twice all the "exactly two" numbers, so 
we subtract them; but then we are subtracting the "all three" 
people three times, rather than twice, so we have to add them again.

Of course, that interpretation is a little odd, since the 2 who like 
physics and biology ALL like chemistry as well. But it agrees with 
the fact that they clearly signalled at the start that they were 
speaking inclusively.

The Venn diagram of this interpretation looks like this:

     chem: 17
   +---------------+
   |               |
   |   6   +-------+-------+
   |       |   2   |       |bio: 10
   |   +---+-------+---+   |
   |   |   |       |   |   |
   |   | 7 |   2   |   | 6 |
   |   |   |       | 0 |   |
   +---+---+-------+   |   |
       |   |           |   |
       |   +-----------+---+
       |       9       |
       +---------------+
         phys: 18

If, instead, you take those who like all three to be distinct from 
those who like two, but still include them among those who like (at 
least) one, then you have to add those who like all three to those 
who like only two before doing that step:

    Like at least one - like at least two + like all three
      (18 + 17 + 10)  - (9+2 + 4+2 + 2+2) +     2
            45        -         21        +     2

and you get 26.

The Venn diagram this time looks like this, with 6 fewer people:

     chem: 17
   +---------------+
   |               |
   |   2   +-------+-------+
   |       |   4   |       |bio: 10
   |   +---+-------+---+   |
   |   |   |       |   |   |
   |   | 9 |   2   |   | 2 |
   |   |   |       | 2 |   |
   +---+---+-------+   |   |
       |   |           |   |
       |   +-----------+---+
       |       5       |
       +---------------+
         phys: 18


So the book's answer is the correct answer to one interpretation of 
the problem, and your answer is correct for another interpretation. 
If such a problem appeared on a test, you would want to state clearly 
your interpretation of the problem before giving the answer, in order 
to make it clear what you are answering.

But there is a bigger difficulty in the problem. I don't see anything 
that says everyone interviewed said they liked one of those three 
subjects; maybe one is in the club only because she likes boys. 
(Please ignore any stereotype suggested by that possibility!)

The point is, mathematical problems have to be stated clearly; part 
of the job of anyone using mathematical reasoning is to determine the 
precise meaning of the problem before trying to solve it. In my work 
as a programmer, I have to ask the clients exactly what they want to 
do, and then express it in a clear form for them to approve, before 
saying that I am solving their problem. You have to do that, too, even 
when the client is a teacher. When it's a textbook, it's a little 
harder to question it.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/