Set Theory: that squirrelly 'empty set'

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~!