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### 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.

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
If such a problem appeared on a test, you would want to state clearly
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/
```
Associated Topics:
High School Discrete Mathematics
High School Logic
High School Puzzles

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