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Set, Subset, Element


Date: 03/10/97 at 20:48:08
From: FRANCIS LAU
Subject: Sets

I was trying to explain to my daughter what are:

UNION, ELEMENT, SET, SUBSET, INTERSECTION, and MEMBER.

Since I did not learn this in my day I am hoping you will lend me a 
helping hand. 

Your attention and help are deeply appreciated.
                           
Sincerely,
FRANCIS LAU


Date: 03/11/97 at 02:05:24
From: Doctor Mike
Subject: Re: Sets

Dear Mr. Lau,  
  
Glad to help.  This is a pretty big part of modern math, so I'll have 
to cut a few corners, but I can give you the main idea.
  
A SET is like a "bunch" or "collection" or "group" of things. An 
example is the set of girls in your daughter's school class. Another 
example is the set of all 2-digit perfect square numbers greater than 
your age.  That is written { 49, 64, 81 }.  This is a finite set so 
you can list the things in it.  The 3 things in it are its ELEMENTs or 
its MEMBERs. 

The order in which you list the elements makes no difference. For 
instance, {2, 3, 5, 7} is considered to be exactly the same set as 
{5, 7, 3, 2}.  Whether you list them in numerical order or 
alphabetical order, this is still the set of all one-digit prime 
numbers.  

Some sets are infinite, like the set of all even numbers greater than 
your age, which can be written 

  { 2*N | N is a whole number and N > 23 }.  

You say this "The set of all numbers of the form 2*N where N is a
whole number and N is greater than 23". You can also write this set as 
{ 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, ..... } .
  
The INTERSECTION of 2 sets is another set. The members of the 
INTERSECTION set have to be in both of those 2 sets. Think of the sets 
of numbers I mentioned in the previous paragraph. The only number that 
is in both sets is 64, so the intersection is { 64 } which is a set 
with exactly one element.  
  
What about the UNION? Whereas the INTERSECTION of 2 sets contains the 
elements that are in BOTH of the 2 sets, the UNION of 2 sets contains 
the elements in EITHER one of the 2 sets. Here is an example: If the 
first set is { 2, 3, 4, 5, 6 } and the second set is { 4, 5, 6, 7 } 
then the intersection of the 2 is { 4, 5, 6 } and the union of the 2 
is { 2, 3, 4, 5, 6, 7 }.  
  
Let's see, what's left? SUBSET. It's sort of what it sounds like.  
Let's do this with another example. I will specify 2 sets, called set 
A and set B, as follows:
  
   A is the set of all girls in your daughter's class at school.
  
   B is the set of all girls in your daughter's class at school
     whose first name begins with a vowel.
  
It is clear that any member of B is also a member of A, just by the 
way these 2 sets are defined. This is what we mean by saying that B is 
a SUBSET of the set A. I don't know your daughter's name, so I'm not 
going to be very accurate here, but let's see how this could turn out. 
I'll present two possibilities for the set A.  I'll assume your 
daughter is Francesca.  
  
    A = { Francesca , Maria , Anita , Jean , Irene }
    
    A = { Francesca , Maria , Donna , Jean , Kendra , Hillary }
  
If the top version is the true one, then B is { Anita , Irene }. If 
the bottom version is the true one, then B is ..... wait a minute 
here! .... there are **NO NAMES** that begin with a vowel. Precisely, 
so B is still a perfectly good set which just happens not to have ANY 
members. This is called the EMPTY SET.  The empty set is a subset of 
all sets. Strange but true.  
  
This is a start for you. At your Public Library in the math section
they usually have books at many levels of learning.  Often the 
reference librarian can steer you to something useful if you describe 
what you are looking for. Good luck and have fun.  

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

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