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Samuelb88
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Homework Statement
Let [n] denote the set consisting of the first n natural numbers 1, 2, 3, ..., n. Use induction to prove that the power set P([n]) has exactly 2^n elements.
The Attempt at a Solution
1) Base case: P([1]) = {{1}, null}. 2^1 = 2 elements.
2) Inductive step: Assume P([n]) has 2^n elements and prove P([n+1]) has 2^(n+1) elements.
I am stuck on the inductive step. I am not sure how to prove P([n+1]) has twice as many elements as P([n]). Through numerical experimentation for some small values of n, I've found that the number of new subset elements for each value of n is can be "represented" by pascal's triangle so I tried expressing the numbers of elements for P([n]) as a sum of the binomial coefficients:
[tex]2^n = \sum_{j=1}^{n} \frac{n(n-1)...(n-j+1)}{j!}\right)[/tex]
And by multiplying both sides by 2 would yield 2^(n+1) on the LHS, however, I can't seem to equate the RHS to:
[tex] \sum_{j=1}^{n+1} \frac{(n+1)(n)(n-1)...(n-j+2)}{j!}\right)[/tex]
Am I on the right track? Does this idea work? Any hints would be great! :)
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