What is the proof that the total number of subsets of a set is $2^n$? What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set).
Suppose you want to choose a subset. For each element, you have two choices: either you put it in your subset, or you don't; and these choices are all independent.

Remark: this works also for the empty set. An empty set has exactly one subset, namely the empty set. And the fact that $2^0=1$ reflects the fact that there is only one way to pick no elements at all!
Let us consider the following cases: 


Let a set be $\{a\}$, then its subsets are: $\{a\}, {\emptyset}$ . Therefore the number of subsets is $2^1$
Let another set be $\{a,b\}$, then its subsets are: $\{a\},\{b\},\{a,b\}, {\emptyset}$ Therefore the number of subsets is $2^2$
Let another set be $\{a,b,c\}$, then its subsets are: $\{a\},\{b\},\{c\},\{a,b,c\},\{a,b\},\{b,c\},\{a,c\}, {\emptyset}$. Therefore, the number of subsets is $2^3$


$\therefore$  (1), (2) &amp; (3) $\Rightarrow $ If there are $n$ sets, then the number of subsets is $2^n$