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Cardinal of P(S) (Adam's question at Yahoo! Answers)

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Here is the question:

Show for any n ϵ ℤ+ with |S| = n, then |P(S)| = 2n.?
Here is a link to the question:

Show for any n

I have posted a link there to this topic so the OP can find my response.
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello Adam,

The number of subsets of $S$ with $k$ elements is $\displaystyle\binom{n}{k}$. So, using the binomial theorem: $$\begin{aligned}\left|\mathcal{P}(S)\right|&= \displaystyle\binom{n}{0}+\displaystyle\binom{n}{1}+\displaystyle\binom{n}{2}+\ldots+ \displaystyle\binom{n}{n}\\&=\displaystyle\binom{n}{0}1^n\cdot 1^0+\displaystyle\binom{n}{1}1^{n-1}\cdot1+\displaystyle\binom{n}{2}1^{n-2}\cdot1^2+\ldots+\displaystyle\binom{n}{n}1^0 \cdot 1^n\\&=(1+1)^n\\&=2^n\end{aligned}$$