Noting Nothing: Braces around Zero vs. Empty Braces

Date: 08/02/2010 at 09:36:22
From: Ien
Subject: difference between null set,phi and 0 in braces 

I'm confused: what is the difference between the null set, the "phi," and
a zero enclosed in braces?

I don't know which is which.

Date: 08/02/2010 at 10:12:55
From: Doctor Ian
Subject: Re: difference between null set,phi and 0 in braces

Hi Ien,

The null set is a set that contains no elements. We can represent that
with empty brackets, like so, ...

   {}

... or with a phi character. Those two representations are equivalent, and
either can be used in place of the other. 

But a zero enclosed in braces, like this, ...

   {0}
   
... is a set with one element -- namely, the number zero. It's similar to
the set {5}, except it has 0 as an element, instead of 5.

Note, too, that the set {phi} is a set that contains a set. For example,
we can have a set like this:

   A = { {a,b}, {c,d,e}, {a,c} }

Here, A is a set that contains some other sets as elements. Since this
set can contain sets, it can contain the empty set, too:

   A = { {a,b}, {c,d,e}, {a,c}, {} }

And we can remove elements from the set, one at a time:

   A = { {c,d,e}, {a,c}, {} }

   A = { {a,c}, {} }

   A = { {} }

That last set has a single element. That element is the empty set. And 
since {} and phi are two ways of saying the same thing, we can write 
that as

  A = { phi }

But this is very different than ...

   B = phi = {}

... which is the empty set itself -- not a set containing the empty set. 

Does this make sense? 

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

Date: 08/04/2010 at 05:51:36
From: Ien
Subject: Thank you (difference between null set,phi and 0 in braces )

Thanks for the answer. It helped me a lot.