Sets: Unions and Intersections

Date: 12/17/97 at 19:04:03
From: Alison Lawless
Subject: Unions and Intersections

I want to know about complements, union, intersection, and sets of 
numbers.  I am having  much trouble! Please send some advice.  I
would appreciate it very much.  Thank you so much!

Alison and her mother

Date: 12/22/97 at 19:20:36
From: Doctor Barney
Subject: Re: Unions and Intersections

Here are a few examples.  

A set is a group of elements that have some characteristic in common.  
For example, let our set be all of the whole numbers from 1 to 20 
inclusive. What do these elements have in common? They are all between 
0.5 and 20.5, among other things.  

A subset is a portion of the elements within a set that have some 
additional characteristic in common. For example, let's define subset 
A to be all of the numbers in my original set that are even, and 
subset B numbers in the original set that are divisible by three.

Now, the INTERSECTION these two subsets A and B is the set of all 
numbers that are in BOTH of these sets, just as a traffic intersection 
is that part of the ground that is in two streets. In our example, the 
intersection of A and B would be 6, 12, and 18, since these are the 
only whole numbers between 1 and 20 that are both even and divisible 
by 3.

The UNION of these two subsets is the set containing all elements that 
are in BOTH A and B: in our example, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 
16, 18, and 20. (Note that all elements contained in the intersection 
of two sets are also contained in the union.)

The COMPLEMENT of a set is everything that is not in the set - 
everything that would be needed to "complete" the area being 
considered (in our case, the numbers 1 to 20.) For example, for the 
sets I have defined above, the complement of A is all the odd numbers 
between 1 and 20 inclusive, and the complement of B is  1, 2, 4, 5, 7, 
8, 10, 11, 13, 14, 16, 17, 19, 20.

Here's an exercise. Remember that sets do not have to be numbers, they 
can be lots of things. Let's use the capital letters A through Z.

Let A be the set of capital letters that have straight lines in them.

Let B be the set of capital letters that have curved parts.

1) List set A.

2) List set B.

3) List the complement of A.

4) List the complement of B.

5) What is the union of A and B?

6) List the letters in the intersection of A and B.

7) What letters are in the complement of the intersection of A and B?
   Describe in words what makes these letters different from all the 
   others.


***ANSWERS  DO NOT READ THIS.  TRY THE PROBLEMS FIRST.   ANSWERS***

1) A,B,D,E,F,G,H,I,J,K,L,M,N,P,Q,R,T,U,V,W,X,Y,Z

2) B,C,D,G,J,O,P,Q,R,S,U

3) C,O,S

4) A,E,F,H,I,K,L,M,N,T,V,W,X,Y,Z

5) the whole alphabet

6) B,D,G,J,P,Q,R,U

7) A,C,E,F,H,I,K,L,M,N,O,S,T,V,W,X,Y,Z 

   None of these letters has both curved and straight lines in it. 

-Doctor Barney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/