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### Venn Diagram of Natural Numbers

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Date: 09/22/1999 at 15:08:13
From: Mollie Coons
Subject: Venn diagrams

I have been asked to construct a Venn diagram comparing the numbers 1
to 100 in these 4 areas: odd, even, composite and prime. I have tried
to make the circles work and I can't seem to get the overlap I need. I
read that you can't do this with 4 circles. There is no commonality
within all four groups that I can see, but in 3 groups (prime, even,
and composite) the number 2 should have a separate space. Is there any
reason why I can't make a separate circle off to the side to show this
one number? Any help would be appreciated.

Thanks,
Mollie
```

```
Date: 09/24/1999 at 13:18:08
From: Doctor Lilla
Subject: Re: Venn diagrams

Dear Molly,

Thanks for writing to Dr. Math. You have an interesting problem.
Actually, we will have a little problem with number 1, not with
number 2, Let's see why.

I would draw a rectangle or ellipse, which would represent the
universe for natural numbers 1-100. A horizontal line divides this
rectangle into an upper and a lower part - let's say these correspond
to composite and prime numbers, respectively. A vertical line divides
it now into a left and a right part - let's say these are the odd and
the even numbers, respectively. This will work for us, since both the
odd-even and the composite-prime pairs are mutually exclusive (have no
common elements).

We have now four parts of the universe:

odd - composite,
even - composite,
odd - prime,
even - prime.

The number 2 belongs to sector "even-prime," and actually this is the
only number here. Note that 2 is NOT a composite number. All of your
other numbers will have one and only one place in which they belong,
except the 1, which has no place here.

One is an odd number, but neither prime nor composite. Therefore I
would enlarge the set of odd numbers below the primes, and would
include a new place for 1.

A set in a Venn diagram doesn't have to be rectangle, a circle, or any
specific shape. The only criterion is that it has to be a closed part
of the plane. So you can include a new part for one or more elements
any time, as long as you take care of all necessary overlaps and set
relations.

The Dr. Math FAQ provides a page about prime numbers to check where 2
belongs, or to learn how to find them:

http://mathforum.org/dr.math/faq/faq.prime.num.html

I hope it makes sense to you. Please write back if you have more
questions.

- Doctor Lilla, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Discrete Mathematics
High School Logic
High School Sets

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