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### How Can a Set Be Empty?

Date: 09/29/2003 at 01:27:50
From: Hash
Subject: Why is empty set called a set ?

Why is the empty or null set called a set when it has no elements?  Is
there a mathematical proof that it's a set?

Date: 09/29/2003 at 08:33:38
From: Doctor Peterson
Subject: Re: Why is empty set called a set ?

Hi, Hash.

We try to make our definitions so that they are as useful as possible.
In this case, we would like all the operations we can do between sets
to yield sets, just as we want addition and multiplication of two
numbers to produce a number. Now, what happens when you take the
intersection of a pair of disjoint sets (sets with no elements in
common)? The result is an empty set, right? If we didn't call that a
set, then in this (rather common) case, the result of the intersection
operation would not be a set.

This is typical of the way math is done. We make some natural
definition (for example, thinking of a set as any collection of
objects), and then work with it; eventually we find that we have to
refine our definitions, or clarify the extreme cases, in order to make
our new branch of mathematics work neatly. We can't "prove" that the
empty set is a set, since we are defining it as such; but we do have
to demonstrate that it is a useful and consistent definition that
produces interesting mathematics. It does!

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

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