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What is a Venn Diagram?

Date: 02/26/98 at 09:59:57
From: Chris B.
Subject: Venn Diagrams

What is a Venn Diagram? What is its use, definition, and what does it 
look like?

Date: 02/26/98 at 14:38:32
From: Doctor Rob
Subject: Re: Venn Diagrams

A Venn Diagram is a picture that is used to illustrate intersections,
unions, and other operations on sets.

For one set A, it is represented by the points inside a circle in a 
plane. The complement of A is the points outside the circle, and in 
the plane. The Venn Diagram looks like a circle, with the letter A 
inside, near the edge.  It divides the plane into two regions.

For two sets A and B, each is represented by the points inside a 
circle in one plane.  The circles are made to overlap.  One is 
labeled A, the other B.  The points inside one circle represent A, 
and those outside are the complement of A.  The points inside the 
other circle represent B, and those outside are the complement of B.  
The overlap of the two circles is the intersection of A and B.  
All points inside either circle forms the union of A and B.  The two 
circles divide the plane into four regions.

To add a third set C, add a third circle which partially overlaps all
four regions of the preceding diagram. The three circles divide the 
plane into eight regions.

You cannot add a fourth circle to represent a fourth set D, but you 
can use a more complicated simple, closed, smooth curve for the same 
purpose. The interior of this curve must partially overlap all eight 
regions mentioned in the preceding paragraph.  The four curves divide 
the plane into 16 regions.

This can continue, but the diagram becomes quite unwieldy.

A Venn Diagram is often used to illustrate the truth or falsity of 
statements in symbolic logic.  The sets are the values of the 
variables. Set A represents variable a being true, and its complement 
represents a being false.  Likewise for set B and variable b, set C 
and variable c, and so on.  "a and b" is represented by the 
intersection of A and B, for example.

-Doctor Rob,  The Math Forum
Check out our web site http://mathforum.org/dr.math/   
Associated Topics:
High School Discrete Mathematics
High School Logic
High School Sets

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