If a set contains n elements, then its power set will contain

Let there is only& $1$~element in a set~$S$, and let~$S = \left\{a\right\}$.

So, power set of~$S\text{is}P\left(S\right) = \left\{ϕ, \left\{a\right\}\right\}$.~

Which shows that number of elements contained in the power set are~$2 = {2}^{1}$.~

Similarly,~if there are~$2$~elements in a set~$S$,~and let~$S = \left\{a, b\right\}$.

So, power set of~$S\text{is}P\left(S\right) = \left\{ϕ, \left\{a\right\}, \left\{b\right\}, \left\{a, b\right\}\right\}$.

Which shows that number of elements contained in the power set are~${2}^{2} .$~

Similarly,~if there are~$n$~elements in a~set~$S$,~then the number of elements in the power set of~$S$~will be~$= {2}^{n} .$