# Question about the power set

1. Jul 4, 2013

### Tenshou

I am confused I understand that the power set has $2^{|\mathcal{A}|}$ members, but they write it as $2^{\mathcal{A}}$ I don't understand why they just don't write it as $\mathcal{P}(\mathcal{A})$ to refer to the power set which has $2^{|\mathcal{A}|}$ elements, isn't that an abuse of notation?

2. Jul 4, 2013

### chiro

Hey Tenshou.

The easy answer is that A is a set and |A| usually refers to the cardinality of the set (which is a number).

2^|A| would be a number while 2^A would be a set.

3. Jul 4, 2013

### SteveL27

The notation $2^{\mathcal{A}}$ means by definition the set of functions from the set A to the set 2. Now what is "the set 2?" It's the set {0,1}.

We can identify the elements of $2^{\mathcal{A}}$ with the elements of $\mathcal{P}(\mathcal{A})$ by corresponding a function f:A->2 with the subset of A that is mapped by f to 1.

In other words any function f:A->{0,1} defines a particular subset of A, namely the preimage of 1; and given a subset of A, its characteristic function is a function f:A->{0,1}. So $2^{\mathcal{A}}$ and $\mathcal{P}(\mathcal{A})$ are conceptually the same set. [Even though if you wrote out their elements, they would not literally be the same set!]

That's actually what's meant by the exponent notation for sets.

It's true that $2^{\mathcal{A}}$ has $2^{|\mathcal{A}|}$ elements; but the two notations are not interchangeable. The first is the set of functions from A to 2; the second is the cardinality of that set of functions.

Last edited: Jul 4, 2013
4. Jul 4, 2013

### Tenshou

Yes I know that. But the question is how do you raise a set to a power, that doesn't make sense.
Well couldn't some one define "2" by a set of ordinals? meaning couldn't "the set 2" be $\{ ,\{ \}\}$ so then it would be "2" with "2"elements, or the power set of a set with 1 element??? You have me some what lost. I don't understand how they are conceptually they same, yet aren't. It seems like bad notation to me, it seems like it would be better if books and people in general use a much more clearer notation. to denote the power set by $\mathcal{P}(\mathcal{A})$ which has $2^{| \mathcal{A}|}$ members it seems a lot more clearer don't you think?

5. Jul 5, 2013

### SteveL27

I suppose which notation is more clear is a matter of individual preference. There are 8 functions from {a,b,c} to {0,1} and there are 8 subsets of {a,b,c}; each subset picked out by the preimage of 1 for some function f:{a,b,c}->{0,1}. So it really is the same concept. And the exponentiation notation generalizes; for example the set of sequences of reals is the same as the set of functions from $\mathcal{N} \rightarrow \mathcal{R}$, or $R^{ \mathcal{N}}$. The $\mathcal{P}(\mathcal{A})$ notation is for the special case where the range of a function is {0,1}. Don't know if that helps ... this might be one of those things you just get used to.

6. Jul 5, 2013

### chiro

Also, one might want to look at a binomial expansion and its connections to the power set (in terms of [x+y]^n).

7. Jul 5, 2013

### Tenshou

I thought the only special case was reserved for the indicator function? you know $\mathbb{1}_{\mathcal{A}} : \mathcal{A} \to \{1,0\}$, because this function has that range space, and what this does is "project" all the elements giving it a value of 1 or 0, then when you integrate this you get the vol of the set, this is all so strange to me... Also, I thought the power set just listed the the elements in a set, that is why the question arose, why not write it so that is easier to understand? Because this notation $2^{|\mathcal{A}|}$ isn't very difficult to understand that it has that many members, and is the $\mathcal{P}(\mathcal{A})$ is the set which contains all those members instead of writing $2^{\mathcal{A}}$ Like how do you take a scalar and give it the power of a matrix, that is kind of what it looks like it is doing, you know?

Could you give an example to clear it up?

8. Jul 5, 2013

### pwsnafu

The power set is the set that is obtained through the axiom of power set. "List" is not really a good term. It just exists. There are algorithms which can compute the power set of a finite set though.

Mathematics isn't concerned with being "easy to understand". Notational conventions are just that, conventions. Somebody thought it was a good notation, and everybody else just follows suit.

With cardinal arithmetic, we wanted definitions such that $\left|{\mathcal A}^{\mathcal B} \right|= |\mathcal A|^{|\mathcal B|}$ works. That's all.

See matrix exponentiation

Last edited: Jul 5, 2013
9. Jul 5, 2013

### Tenshou

I see, well thanks for clearing that up.

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