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Universal Sets

Date: 08/19/2002 at 12:53:41
From: Tim Melton
Subject: Universal Sets - explanation

I don't quite understand the concept of universal sets. For instance,
if I had to place the President of the United States in three 
universal sets, what are some possibilities? I need a complex yet 
simple definition with examples.


Date: 08/19/2002 at 15:31:06
From: Doctor Peterson
Subject: Re: Universal Sets - explanation

Hi, Tim.

The universal set in any particular context is the set of all objects 
under consideration. You might be relating the President to other 
American men, or to other current heads of state in the world, or to 
past presidents. Then the universal set would be the set of all 
American men, or the set of all current world heads of state, or the 
set of all U.S. presidents. Or it might be the set of current 
residents in the White House, or the set of children of George Bush, 
or of men named George, or of human beings throughout history. Or, 
you might not be thinking of the president as a specific person, but 
just as the holder of an office, and the universal set might be all 
federal offices.

What do these have in common? The fact that they contain the word 
"all" (or some equivalent), and that they include the element you were 
asked about. (If the president were a woman, my first example would 
not work.) A universal set must be big enough that everything you want 
to talk about (that is, to include in a set) will be within the 
universal set; but it need not contain every object anyone could ever 
want to talk about, or that might be mentioned in passing, such as the 
White House itself.

Here is one brief definition from Eric Weisstein's MathWorld:

   Universal Set
   http://mathworld.wolfram.com/UniversalSet.html 

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

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


Date: 08/19/2002 at 15:43:45
From: Tim Melton
Subject: Universal Sets - explanation

When I describe the universal set with an example you provided me, am 
I correct in doing as stated below?

U={American Men}
U={White House Resident}
U={Men Named George}

I just want to make sure I write it out as I should.


Date: 08/19/2002 at 16:08:06
From: Doctor Peterson
Subject: Re: Universal Sets - explanation

Hi, Tim.

It depends on the notation you are expected to use, which can vary 
from formal

    U = { x | x is an American man }

to informal

    U = all American men

I'm not quite comfortable with your notation, because when we use 
braces, what is inside them generally represents individual members 
of the set, rather than a description of the contents of the set. 
Perhaps you can show me an example of the notation your text is using 
to define sets, so we can find the right level for your notation. I 
personally would just say it in words unless I was told otherwise; 
but I haven't been a student in a long time!

Incidentally, the last example is likely to be objected to; it just 
doesn't sound broad enough to be a universal set. I included it as an 
extreme example, and there may conceivably be a situation in which it 
might be considered universal, but you should probably choose some 
really universal-sounding examples to be on the safe side. On the 
other hand, you might want to stick that in as a fourth, with an 
explanation, just to test the boundaries of the terminology as used 
in your class - and to see how good a sense of humor your instructor 
has.

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


Date: 08/19/2002 at 15:46:48
From: Doctor Mike
Subject: Re: Universal Sets - explanation

Hi Tim,   
  
Here are 5 examples of such a universal set. Note that BOTH of the 
presidents named Bush are in sets 1 through 4, but only the current
president Bush is in the final one. See? I hope this helps. 
  
1.  The set of all people in the Bush family.
2.  The set of all people who have been or
    are the President of the United States.
3.  The set of all living mammals.
4.  The set of all living males.
5.  The set of all people whose father was
    a president of the United States.  

- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Sets

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