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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
sure about the "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/ 
Associated Topics:
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