Associated Topics || Dr. Math Home || Search Dr. Math

### 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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search