What is a Set?

Date: 04/04/97 at 18:15:03
From: Luis Cifuentes
Subject: Group of Objects

What is the correct term for a group of objects, e.g. 3 cars, 
7 pencils, or 5 apples? Would it be correct for us to say "a set of 
cars" or what would be the proper way to do it? 

Date: 04/05/97 at 00:24:12
From: Doctor Toby
Subject: Re: Group of Objects

I think the word you want is "set". 

A set is determined precisely by what is in it. If one set has three 
cars in it (and nothing else) and another set has three cars in it 
(and nothing else), they are the same set if they have the same three 
cars in them. Otherwise, they are different sets.

A set is a very abstract concept. It's not the same as, for example, 
a box. The same thing can't be in two different boxes at the same 
time, but it could easily be in two different sets. 

A set is more like a club whose members are defined abstractly, not by 
their physical location. But a set isn't quite the same as a club 
either. There could be two different clubs that just happened to have 
exactly the same members. One might be the math club, while the other 
was the chess club. Or one might be headquartered in New York and the 
other in Tokyo. But sets have no properties other than their members. 
If a set has exactly the same members as another, they are the exact 
same set.

If a set is small enough, you can identify it just by listing its 
elements (the things that are in it). For example, {Doctor Toby, 
Doctor Ken, Doctor Wilkinson} is a set of Math Doctors. Doctor Ken, 
Doctor Wilkinson, and I are in the set, and everything else in the 
universe is not. The order in which you list the elements doesn't 
matter. {Doctor Wilkinson, Doctor Ken, Doctor Toby} is a different 
list, but it names the same set, because it has the same elements. But 
{Doctor Toby, Doctor Ken} is a different set, because Doctor Wilkinson 
is not in it.

In some ways, a set is a very simple idea. Clubs can have subjects and 
headquarters, while boxes have physical locations. A set has nothing 
but its elements. But the very simplicity of the idea introduces new 
complications. For example, consider the set whose elements are 
exactly those sets that aren't in themselves. (Every set is either in 
itself or not in itself; we want those sets which aren't.) Is this set 
an element of itself or not? Let me know when you think you have the 
answer :-).

-Doctor Toby,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/