Set Theory: that squirrelly 'empty set'

[hain]

Okay, I'm new to set theory and having a little trouble wrapping my head around how empty set works.

(I'll represent empty set with the unicode character "∅")

As I understand it, ∅ is a set containing no elements.
Then I think it makes sense that |∅|=0. (Cardinality of empty set is zero).
So... this is where things get hairy.

1.
What's the cardinality of {∅}? I think according to set theory that |{∅}|=1.
This is because {∅} would contain itself right?

2.
If 1, then I'd assume that |{{∅}}|=2?

Further, what would the cardinality of the following sets be:
a. {∅,{∅}}
b. {∅} ∩ {{∅}} (and what does this set contain? quite confusing)
c. {{∅},∅} (this set is the same as a. I believe)

Actually if anyone could also let me know which of these sets is equal I think it might help me figure it out. Anyway I'm just trying to undertand empty set a little because my textbook just glazes over it. Thanks~!

 

[pka]

Okay, I'm new to set theory and having a little trouble wrapping my head around how empty set works.
As I understand it, ∅ is a set containing no elements.
Then I think it makes sense that |∅|=0. (Cardinality of empty set is zero).
Further, what would the cardinality of the following sets be:
a. {∅,{∅}}
b. {∅}∩{{∅}}=∅ The two sets have nothing in common.
c. {{∅},∅} (this set is the same as a. I believe) Correct!

There is quite a bit of history here.
One of the most important mathematicians and teachers of the twentieth century, R.L. Moore, did not believe that a set could be empty. He would not allow anyone to use the concept in any of his classes. Many of his students and their students have carried on the tradition. However, those are clearly in the minority. Many of Moore’s own students gave in to the idea.

If a set theory class, one normally defines it this way: ∅={x:x≠x}.
The given any proposition, “not equal to itself”, there is a set of elements for which it is true. In this case that set must be empty.
Because the set, ∅, is equal to itself so {∅}≠∅.
Thus: |∅|=0, |{∅}|=1, |{∅{∅}}|=2.
For any set A, \(\displaystyle A \cap \emptyset = \emptyset\).

 

[soroban]

Hello, hain!

Okay, I'm new to set theory and having a little trouble wrapping my head around how empty set works.

As I understand it, \(\displaystyle \not{O}\) is a set containing no elements.
Then I think it makes sense that \(\displaystyle \left|\not{O}\right|\,=\,0\). (Cardinality of empty set is zero).
So... this is where things get hairy.

1. What's the cardinality of \(\displaystyle \{\not{O}}\)?
I think according to set theory that \(\displaystyle \left|\{\not{O}\}\right|\,=\,1\;\) . . . right!
This is because \(\displaystyle \{\not{O}\}\) would contain itself, right?
It has cardinality 1 because it contains one thing.

Think of Sets as shopping bags.
Some have a few items in them, others have many items.
And one of them has no items; it is empty.

It can be written: \(\displaystyle \,\{\;\}\) . . . We can see that it contains nothing.

The set \(\displaystyle \{0\}\) is not an empty set; it contains "0".

The set \(\displaystyle \{\not{O}\}\) is not an empty set; it contains \(\displaystyle \not{O}\).

The set \(\displaystyle \{\text{nothing}\}\) is not an empty set; it contains the word "nothing".

So, the set \(\displaystyle \,\(\not{O}\}\:=\:\{\{\;\}\}\) is not empty.
\(\displaystyle \;\;\)It is a shopping bag with another bag inside.


[quote:2t96xdrd]2. If 1, then I'd assume that \(\displaystyle \left|\{\not{O},\,\{\not{O}\}\}\right|\,=\,2\;\;\) . . . right!

This is a bag with two things in it: an empty bag and a bag with another bag inside.

Further, what would the cardinality of the following sets be:

(a) and (c) are the same as #2 above.


\(\displaystyle \;\;b.\;\; \{\{\not{O}\}\,\cap\,\{\not{O}\}\}\) (and what does this set contain? quite confusing)

[/quote:2t96xdrd]
\(\displaystyle \{\not{O}\}\,\cap\,\{\not{O}\}\:=\:\{\not{O}\}\)
\(\displaystyle \;\;\)We have a bag with a bag inside ... and another bag with a bag inside.
\(\displaystyle \;\;\)What do the two bags have in common? . . . a bag.
The set has cardinality 1.