Cardinality of Power Set of Finite Set/Proof 2

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Theorem

Let $S$ be a set such that:

$\card S = n$

where $\card S$ denotes the cardinality of $S$,


Then:

$\card {\powerset S} = 2^n$

where $\powerset S$ denotes the power set of $S$.


Proof

Enumerating the subsets of $S$ is equivalent to counting all of the ways of selecting $k$ out of the $n$ elements of $S$ with $k = 0, 1, \ldots, n$.

So, from Cardinality of Set of Subsets, the number we are looking for is:

$\displaystyle \card {\powerset S} = \sum_{k \mathop = 0}^n \binom n k$

But from the binomial theorem:

$\displaystyle \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$

It follows that:

$2^n = \displaystyle \paren {1 + 1}^n = \sum_{k \mathop = 0}^n \binom n k \paren 1^{n - k} \paren 1^k = \sum_{k \mathop = 0}^n \binom n k = \card {\powerset S}$

$\blacksquare$


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