A new thought and a problem |P(A)| = 1?

[flyingpig]

Homework Statement

P(A) = power set of A (in my book it is funky looking P, someone tell me how to read it and what it is)

What is |A| and |P(A)|?

a) A = \o

Book say that

|A| = 0, and |P(A)| = 1

Why?

P(A) = {\o} I know that this is one thing (or one element), but the symbol represents empty set meaning nothing so

{\o} = {{no elements}} = {}

SO Why isn't |P(A)| = 0?

EDIT: \o is supposed to be that symbol that looks like the greek letter phi, but I don't know why it isn't showing

 

[Dick]

You can also write phi as {}. {} contains no elements. It's the empty set. |{}| means the number of elements in {}. That's 0. P({}) is the set of all subsets of {}. There's only one, {}. So P({})={{}}. It contains one element {}. So |{{}}|=1.

 

[SammyS]

Do you mean this \mathcal{P} ? use \mathcal{P}

For \emptyset\,, use \emptyset, although, \o does work for some implementations of LaTeX.

 

[flyingpig]

Do you mean this \mathcal{P} ? use \mathcal{P}

For \emptyset\,, use \emptyset, although, \o does work for some implementations of LaTeX.

Yes thank you lol, now I am wondering the heck is \o lol

 

[flyingpig]

You can also write phi as {}. {} contains no elements. It's the empty set. |{}| means the number of elements in {}. That's 0. P({}) is the set of all subsets of {}. There's only one, {}. So P({})={{}}. It contains one element {}. So |{{}}|=1.

But {} is nothing

 

[flyingpig]

Actually why is the empty set is a subset of A anyways?

 

[vela]

But {} is nothing

It's not nothing. It's a set containing no elements. You can't just erase the curly brackets willy-nilly.

Actually why is the empty set is a subset of A anyways?

It's because it's true that every element of the empty set is an element of A. Or to put it in a way that may be a little clearer, {} doesn't contain an element that is not also in A.

 

[Dick]

Actually why is the empty set is a subset of A anyways?

Try it this way. The empty set is like a bookcase containing no books. It's not nothing, it's still a bookcase.