Advanced Math/Empty Sets

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Expert: - 5/19/2009


Question
Hey there, Scott. I'm a college student who has never really grasped the concept of an empty set. I have a test dealing with it this evening, but hopefully you can help me understand one thing for the final. :)

I understand why an empty set doesn't equal a set with the empty set symbol inside it, and most of the other confusing true/false questions about an empty set. My only difficulty is getting an answer and explanation for this one:

Does 0 equal the empty set? I know that 0 is the number of elements IN the empty set, but I don't know if it equals zero, or the explanation for why it does or doesn't.

Thanks for any help!

Answer
The number 0 is a number.  

It can be treated like any ohter element.

If a set has an element, its not an empty set.

An empty set is defined as one with no elements in it at all.
The number 0, even though it means having nothing, is a number.

The empty sets mean nothing is known.
This means the set doesn't even know what having 0 elements means.

Take, for example the set A that has one number in it called x.
You can see that A is not an empty set, since it has one number.
It can also be seen that A is not anywhere near complete, since x+x gives a number that is not in A.  Unless, of course x=0.
Then I believe A is a complete set since any sum of the elements in A and any product of the elements in A is in A, since there are no elements in A - there is only 1 element.

Summary
------------
If the set is empty, there is nothing to do operations with.

If the set has a 0, you can add the element to itself or
multiply the element by itself, so operations can be done
on that set.

Yes, it counts as a set, even though the element is nothing,
for it still has 1 element.

Nothing to it, right?

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