Is Writing ##2^{\mathcal{A}}## an Abuse of Notation for the Power Set?

[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?

 

[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.

 

[SteveL27]

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?

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.

 

[Tenshou]

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.

Yes I know that. But the question is how do you raise a set to a power, that doesn't make sense.

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.

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?

 

[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.

 

[chiro]

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

 

[Tenshou]

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.

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?

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

Could you give an example to clear it up?

 

[pwsnafu]

Also, I thought the power set just listed the the elements in a set

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.

that is why the question arose, why not write it so that is easier to understand?

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.

Because this notation $2^{|\mathcal{A}|}$ isn't very difficult to understand that it has that many members,

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

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?

See matrix exponentiation

 

[Tenshou]

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.

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

I see, well thanks for clearing that up.