Complements, Unions, and Intersections of Sets

Date: 10/13/2004 at 18:45:57
From: Turtle
Subject: Universal sets

How do you shade in the complement, union, and intersection?  I tried
to look it everywhere up but I got so frustrated.

Date: 10/13/2004 at 22:23:28
From: Doctor Peterson
Subject: Re: Universal sets

Hi, Turtle.

I suppose you are asking about problems where you are given a Venn
diagram, perhaps of two sets, or perhaps more, and are told to shade
various sets such as the complement of one set, the union of two sets,
or the intersection of two sets.

Think about the meaning of each term.

The complement of a set is everything EXCEPT that set. So if you have
a set shown inside a universal set (a larger set that is thought of as
containing all the objects under consideration), just shade in the
OUTSIDE of the given set--everything in the universal set that is NOT 
in that set:

   +---------------------+   +---------------------+
   |                     |   |:::::::::::::::::::::|
   |     +---------+     |   |:::::+---------+:::::|
   |    /           \    |   |::::/           \::::|
   |   /             \   |   |:::/             \:::|
   |  +               +  |   |::+               +::|
   |  |               |  |   |::|               |::|
   |  |       A       |  |   |::|               |::|
   |  |               |  |   |::|               |::|
   |  +               +  |   |::+               +::|
   |   \             /   |   |:::\             /:::|
   |    \           /    |   |::::\           /::::|
   |     +---------+     |   |:::::+---------+:::::|
   |                     |   |:::::::::::::::::::::|
   +---------------------+   +---------------------+
            set A                complement of A
    inside universal set       in the universal set

The union of two sets is the set of all objects that are members of
EITHER set; that is, it is the result of combining the sets, as if you
poured the contents of two bags into a new one.  To shade it in, shade
in all of both sets:

   +---------------------+   +---------------------+
   |                     |   |                     |
   |    +-----+          |   |    +-----+          |
   |   /       \         |   |   /:::::::\         |
   |  +         +        |   |  +:::::::::+        |
   |  |    A    |        |   |  |:::::::::|        |
   |  +      +--+--+     |   |  +:::::::::+--+     |
   |   \    /  /    \    |   |   \::::::::::::\    |
   |    +--+--+      +   |   |    +--+:::::::::+   |
   |       |    B    |   |   |       |:::::::::|   |
   |       +         +   |   |       +:::::::::+   |
   |        \       /    |   |        \:::::::/    |
   |         +-----+     |   |         +-----+     |
   |                     |   |                     |
   +---------------------+   +---------------------+
                                 union of A and B

The intersection of two sets is the set of all objects that are 
members of BOTH sets; it's the overlap between them.  To shade that 
in, just shade in only the part that is inside both sets:

   +---------------------+   +---------------------+
   |                     |   |                     |
   |    +-----+          |   |    +-----+          |
   |   /       \         |   |   /       \         |
   |  +         +        |   |  +         +        |
   |  |    A    |        |   |  |         |        |
   |  +      +--+--+     |   |  +      +--+--+     |
   |   \    /  /    \    |   |   \    /::/    \    |
   |    +--+--+      +   |   |    +--+--+      +   |
   |       |    B    |   |   |       |         |   |
   |       +         +   |   |       +         +   |
   |        \       /    |   |        \       /    |
   |         +-----+     |   |         +-----+     |
   |                     |   |                     |
   +---------------------+   +---------------------+
                             intersection of A and B

Here are some pages that discuss this:

  What is a Venn Diagram?
    http://mathforum.org/library/drmath/view/52420.html 

  Sets: Unions and Intersections
    http://mathforum.org/library/drmath/view/52395.html 

If you have any further questions, feel free to write back.

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