elementary set theory - Cardinality of the power set $mathcal Pleft(Sright),$ where $S$ is a set of $15$ elements?

What is the cardinality of the power set $\mathcal P\left(S\right)$ where $S$ is a set of $15$ elements?

I think the power set is a set of all the subsets of a given set or $2^n$. So would the cardinality of this set be $2^{15}$ or $32,768$?

Answer

Yes, the cardinality of the power set $\mathcal P(S)$ of a set $S$ is given by $2^n$, and so in your case, by $2^{15}$.

Note: There is a difference between a set, and its cardinality. The power set $\mathcal P(S)$ itself is the set of all subsets of $S$, whereas $2^n$ is the number of these sets (which are the elements) in $\mathcal P(S).$