The notation ##2^{\mathcal{A}}## represents the set of functions from set A to the set {0,1}, while ##\mathcal{P}(\mathcal{A})## denotes the power set of A, which contains all subsets of A. Both notations yield the same cardinality of ##2^{|\mathcal{A}|}## elements, but they are not interchangeable. The distinction lies in the fact that ##2^{\mathcal{A}}## is a set of functions, whereas ##\mathcal{P}(\mathcal{A})## explicitly refers to the collection of subsets. Understanding this difference is crucial for clarity in mathematical notation.
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Tenshou said: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?
Yes I know that. But the question is how do you raise a set to a power, that doesn't make sense.chiro said: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 said: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.
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?SteveL27 said: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.
Could you give an example to clear it up?chiro said:Also, one might want to look at a binomial expansion and its connections to the power set (in terms of [x+y]^n).
Tenshou said: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,
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?
pwsnafu said: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