Subsets and Proper Subsets

Date: 07/10/98 at 20:47:49
From: Melinda Dwelly
Subject: Subsets

I don't understand the difference between subsets and proper subsets. 
Please explain and give examples. Thank you.

Melinda Dwelly

Date: 07/11/98 at 02:06:00
From: Doctor Mike
Subject: Re: Subsets

Dear Melinda,
   
If every member of one set is also a member of a second set, then the 
first set is said to be a subset of the second set. Usually, it turns 
out that the first set is smaller than the second, but not always. The 
definition of "subset" allows the possibility that the first set is 
the same as (equal to) the second set. But a "proper subset" must be 
smaller than the second set.  
  
Example 1: 

   The set {2,3,5,7} is a subset of {2,3,5,7}.
   The set {2,3,5,7} is NOT a proper subset of {2,3,5,7}.
   The set {2,3,5} is a proper subset of {2,3,5,7}.
     
Example 2: 

   It is possible to specify sets in different ways. 
   A = {0,1,2,3,4,5,6,7,8,9} means the set of all digits.
   A = {0,2,4,6,8,1,3,5,7,9} means exactly the same thing.
   B = {D | D is a digit in Bill Clinton's phone number, including 
        the area code} 
   has exactly the digits of the President's phone number. You know 
   that B must be a subset of A, because everything in B is a digit, 
   and every digit is in A. But is it a proper subset? The answer is 
   yes because the area code in Washington D.C. is 202, so his number 
   is of the form 202/abc-defg, and if you put all of the digits of 
   that phone number in a set you have at most nine members, which is 
   less than the size of set A. But if C is the set of digits in Bill 
   Clinton's phone number back in Arkansas when he was the governor, 
   then it gets more complicated. The area code for all of the state 
   of Arkansas is 501. So, C could be a proper subset of A or not.  
   Let's look at 2 possibilities. If his phone number was 
   501/555-1212, then the set C would be {0,1,2,5}. However, if his   
   phone number was 501/234-6789, then set C would be
   {0,1,2,3,4,5,6,7,8,9} which is not a proper set of A. 
    
I hope this helps.
       
- Doctor Mike, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 09/18/2001 at 15:03:55
From: Dan
Subject: Sets

What is a proper subset? 

I think when two sets are identical, then you can say one is the 
proper subset of the other, but I'm not sure if I'm right.

Date: 09/18/2001 at 17:35:27
From: Doctor Jodi
Subject: Re: Sets

Hi Dan,

Do you know what a subset is?

A PROPER subset is any subset of a set EXCEPT ITSELF.

So for instance, if you start with the set {Green Eggs, Ham, Cheese},
{Ham, Cheese} is a proper subset, but {Green Eggs, Ham, Cheese} is NOT 
a proper subset.

Hope this helps!

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