Venn Diagram of Natural Numbers

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/