anonymous
  • anonymous
I was enumerating the elements of the power set of this set S:= {1,2,3,4,5} and I thought that the number of these elements could be obtained with this: \[\#\wp S = 1 + \sum_{k=1}^n {n\choose k} \] where \(n=\#S\) I saw that it holds for this set. But I'm not sure what if it could be applied to a different kind of set.
Mathematics
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
That formula is right, but there is a much shorter formula for the number of elements in the power set of an n-element set. If write out the power sets for sets with 2, 3 and 4 elements, it should become clear what the shorter formula is.
anonymous
  • anonymous
Yeah I know. The formula is \(2^n\)
anonymous
  • anonymous
I'm ashamed this is actually the binomial theorem

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sirm3d
  • sirm3d
the binomial expansion of \((x+y)^n\) is \[\Large (x+y)^n=\sum_{k=0}^n \left(\begin{matrix}n \\ k\end{matrix}\right)x^ky^{n-k}\] put \(x=y=1\) and you'll get the answer to your question.
anonymous
  • anonymous
\[\#\wp S=2^{\#S}= \sum_{k=0}^{n} {n\choose k}\]
anonymous
  • anonymous
thank you @sirm3d
sirm3d
  • sirm3d
YW

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