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### How to Show 'Not' Statements on a Venn Diagram

```Date: 05/03/2004 at 22:40:24
From: Rachel
Subject: math tutoring

How can I use a Venn diagram to display the opposite of a union or
intersection of P & Q?  For instance, how do I show (not P) or (not Q)
versus not (P & Q)? I understand the positive aspect, I'm just not

```

```
Date: 05/04/2004 at 11:54:38
From: Doctor Ian
Subject: Re: math tutoring

Hi Rachel,

Let's make a simple Venn diagram.  (I'm using rectangles because
circles are hard to make with a keyboard, but it's the same idea.)
Note that I've included a universal set (U), which includes elements
that might not be in either of the sets P or Q.

+-----------U------------+
|                        |
|         +----Q----+    |
|         |         |    |
|  +------|----+    |    |
|  |      |    |    |    |
|  |      |    |    |    |
|  |      +---------+    |
|  |           |         |
|  |           |         |
|  +-----P-----+         |
|                        |
+------------------------+

Now, how can I represent "P and Q"?  That's the intersection of the
two sets, right?

+-----------U------------+
|                        |
|         +----Q----+    |
|         |         |    |
|  +------|----+    |    |
|  |      |....|    |    |        P and Q
|  |      |....|    |    |
|  |      +---------+    |
|  |           |         |
|  |           |         |
|  +-----P-----+         |
|                        |
+------------------------+

So what is "not (P and Q)"?  It's everything else:

+-----------U------------+
|........................|
|.........+----Q----+....|
|.........|.........|....|
|..+------|----+....|....|
|..|......|    |....|....|        not (P and Q)
|..|......|    |....|....|
|..|......+---------+....|
|..|...........|.........|
|..|...........|.........|
|..+-----P-----+.........|
|........................|
+------------------------+

Now, let's contrast this with '"not P) and (not Q)".  In this case, we
need to find everything that isn't in P,

+-----------U------------+
|........................|
|.........+----Q----+....|
|.........|.........|....|
|..+------|----+....|....|
|..|      |    |....|....|        not P
|..|      |    |....|....|
|..|      +---------+....|
|..|           |.........|
|..|           |.........|
|..+-----P-----+.........|
|........................|
+------------------------+

and then find everything that isn't in Q,

+-----------U------------+
|........................|
|.........+----Q----+....|
|.........|         |....|
|..+------|----+    |....|
|..|......|    |    |....|        not Q
|..|......|    |    |....|
|..|......+---------+....|
|..|...........|.........|
|..|...........|.........|
|..+-----P-----+.........|
|........................|
+------------------------+

and then intersect those two sets:

+-----------U------------+
|........................|
|.........+----Q----+....|
|.........|         |....|
|..+------|----+    |....|
|..|      |    |    |....|        (not P) and (not Q)
|..|      |    |    |....|
|..|      +---------+....|
|..|           |.........|
|..|           |.........|
|..+-----P-----+.........|
|........................|
+------------------------+

Now, if we look at this last diagram, we can see that what we have, in
fact, specified everything that is NOT in (P or Q):

+-----------U------------+
|........................|
|.........+----Q----+....|
|.........|         |....|
|..+------|----+    |....|
|..|      |    |    |....|        (not P) and (not Q)
|..|      |    |    |....|
|..|      +---------+....|
|..|           |.........|        not (P or Q)
|..|           |.........|
|..+-----P-----+.........|
|........................|
+------------------------+

That is to say, these expressions are equivalent:

(not P) and (not Q) <=> not (P or Q)

Try using the same technique to show that

not (P and Q) <=> (not P) or (not Q) ?

Does this help?

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