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Question about the power set

  1. Jul 4, 2013 #1
    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?
     
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  3. Jul 4, 2013 #2

    chiro

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    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.
     
  4. Jul 4, 2013 #3
    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
  5. Jul 4, 2013 #4
    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?
     
  6. Jul 5, 2013 #5
    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.
     
  7. Jul 5, 2013 #6

    chiro

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    Also, one might want to look at a binomial expansion and its connections to the power set (in terms of [x+y]^n).
     
  8. Jul 5, 2013 #7
    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?
     
  9. Jul 5, 2013 #8

    pwsnafu

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    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
  10. Jul 5, 2013 #9
    I see, well thanks for clearing that up.
     
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