derewolf4894 derewolf4894
  • 22-08-2019
  • Mathematics
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Give a combinatorial proof that the cardinality of the power set of a finite set A is 2^|A|

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LammettHash
LammettHash LammettHash
  • 22-08-2019

There are [tex]\dbinom{|A|}k[/tex] ways of building a subset of [tex]k[/tex] elements from [tex]A[/tex], so the total number of subsets you can build is

[tex]\displaystyle\sum_{k=0}^{|A|}\binom{|A|}k[/tex]

Recalling the binomial theorem, the above sum is equal to

[tex]\displaystyle\sum_{k=0}^{|A|}\binom{|A|}k1^k1^{|A|-k}=(1+1)^{|A|}=2^{|A|}[/tex]

as required.

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