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Empty Sets


Date: 08/15/97 at 12:04:06
From: Dara  Hood
Subject: Algebra

I don't understand how my teachers can figure out a problem like empty 
sets and not explain it fully. Can you help me to understand it 
better? Thank you for your help.

Sincerely yours,
Dara Hood


Date: 08/15/97 at 16:10:08
From: Doctor Mike
Subject: Re: Algebra

Dear Dara,   
   
When you do your homework you probably write it out on paper
from a pad, or perhaps paper from a spiral notebook.  When you
buy it new there are a certain number of sheets of paper in it.
Just for an example, let's say yours has 100 sheets.
   
For an example of a set, I want to use the set of all sheets of
paper in the pad or notebook you are using (or will be using
when school starts in the fall).  Right when you bring it home
from the store the set of all sheets in it has 100 members, also
called elements. Right?  

Okay, time passes and you do a lot of homework and there are only 40 
sheets left, because you already used 60 for homework you turned in.  
The set of all sheets in the notebook is smaller now because it only 
has 40 members. 

More time passes and you use 39 more sheets, so there is only 1 sheet 
left. Now the set of all sheets in the notebook has only one thing in 
it.
   
You get scared because you have a big assignment coming due next
day so you write on that last sheet of paper  ......
                      ________________
                      |               |
                      |    M O M      |
                      |               |
                      |               |
                      | I gotta have  |
                      |               |
                      |  another      |
                      |               |
                      | notebook  by  |
                      |               |
                      | tonight !!!!  |
                      |_______________| 
    
.....  and you RIP IT OUT and stick it onto the refrigerator for Mom 
to see while you are in school today. How many sheets now are in the 
set of all sheets in the notebook?  Right, zero. 0. Zippo. Nada.  =The 
set is ***Empty*** because there is nothing in it. That's all there is 
to it. It's called the empty set because it is empty.   
    
Now there is one other sort of tricky little thing I should mention.
Your teacher might say to you things like "there is only ONE empty
set" or "the empty set is a subset of every other set."  Why would
such a thing be true? It is subtle, and you will have to think about
it a while before you are really comfortable, so for now I will just
give you an example:  
    
     Put 2 plates on the kitchen table. Put some jelly beans on 
     one plate, and put some quarters on the other plate. I'll
     say that set A is the set of jelly beans ON ONE plate, and 
     say that set B is the set of quarters ON THE OTHER plate. 
     Then your brother comes into the kitchen, eats all of the jelly
     beans and puts all the quarters in his pocket. Now the set
     of beans on the one plate is empty, and the set of quarters on
     the other plate is also empty, so both A and B are empty. Now
     invite your math teacher over to the house and see if he/she 
     can tell, JUST BY LOOKING AT THEM, which is the empty set of
     jelly beans, and which is the empty set of quarters. The math
     teacher might be able to guess correctly, but it would only be
     a guess, since there really is no difference between them. The
     Empty Set is just empty, that's all there is to it. So, there
     is only one empty set.  
   
This business of only one empty set can be made into a pretty good
excuse. You can say to your math teacher that you couldn't do your
homework because it was really really hard to concentrate at home.
Why? the teacher asks. Because there is a set of Elephants in my
bedroom. You don't have to mention that it's the Empty Set of 
elephants. If that doesn't work you can always fall back on that
old reliable excuse that my dog ate the homework.  Better yet, do
the homework.  I hope this helps.  

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra
Middle School Algebra

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