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Definition of Set and How the Empty Set Fits within It

Date: 07/18/2004 at 09:34:25
From: Khalid
Subject: Definition of Set

The definition of a set is ďA collection of well-defined objects".
Whenever I teach this topic, my students become confused about the 
idea of the null set, because they think every collection must have 
some elements.  I say to them that a collection can be empty, but 
still they are not satisfied.  How can I get them to understand that
the definition allows for an empty collection?



Date: 07/18/2004 at 10:36:41
From: Doctor Tom
Subject: Re: Definition of Set

Hello Khalid,

You have actually asked an interesting (and difficult) question. 
Definitions in mathematics are not always easy to work with.

In formal mathematics, you begin with some "undefined objects", some
axioms or postulates about those objects, possibly some definitions,
and from those, you prove theorems.  In Euclidean geometry, for
example, the terms "point, line, plane" are undefined objects.

Euclid tried to define a point as "A point is that which has no part."
Of course as soon as he said that, we can ask, "What is a part?"  If
he had given the definition of "part", it would have had other
undefined terms, and the process would never end.

The only thing we officially know about these things is whatever the
postulates tell us.  For example, there is a postulate that tells us
that every two points lie on exactly one line.  From this, we can
prove the following very simple theorem: "If the system contains at
least two points, then it contains a line."

Because of this, more modern mathematicians realized that it is
impossible to define everything--you will always be forced into an 
infinite set of definitions--so the only solution is to begin with a 
set of undefined terms, and then to write down a series of properties 
of these terms that you call axioms or postulates and work from those.

Of course, if you're teaching beginners, you cannot use the formal
mathematical approach.  You have to help them by giving them a way to
think about what is going on.  For example, I can tell my beginning
geometry class that "a point is like the tiniest dot you can put on a
piece of paper", "a line is like what you get when you drag a pencil
along a ruler".  As the class goes on, you try to improve that idea
with the concept that a point is really smaller than any physical dot,
but it represents an "ideal, infinitely tiny dot", and so on.

In set theory, the word "set" is formally undefined.  All we know
about sets are what the axioms of set theory tell us.  You can look up
those formal definitions if you want.  I actually wrote a paper for
very bright high-school students that gives an introduction to formal
set theory and you can obtain it here:

    http://www.geometer.org/mathcircles/nothing.pdf 

The part that might interest you is called "Axiomatic Set Theory". 
You will see there that the existance of the empty set is simply one
of the axioms.  It cannot be proved.

But in the same way that you have to help beginning geometry students
with a more intuitive idea of what is meant by "point" and "line"
rather than just telling them that "those are undefined terms", you
need to help your beginning set theory students by giving them an idea
of something to think about when they are trying to think of what the
term "set" means.

I like to tell my students that a set is like a box that may or may
not contain objects.  So the set:  {1, 2, 3} is a box containing
those particular three numbers.  The empty set is simply an empty box.

This is particularly helpful when you think about something like "the
set containing the empty set".  Most beginners' first impulse is to
say that this must be the empty set, but it is not.  It is like a box
that contains an empty box inside it.  The outer box is NOT empty--
it has one thing inside it; namely, an empty box.

I think what you should do is understand yourself exactly what is
meant by formal set theory so that you REALLY know what is going on,
but realize that for beginning students, you have to start them out
with an informal idea.  Later, if they are interested and willing to
think harder about what is going on, they can refine their ideas to
the "mathematically precise" theory.

Good luck!

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

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