Prove that $|P(X)| = 2^{|X|}$
Basis
    $$ |X| = 0 $$
    $$|P(X)| = 2^{|X|} = 2^0 = 1$$
    True. Any set with zero elements takes the form $\{\}$, and thus its power set will be $\{\{\}\}$. One element. 

Inductive Step

We must show:
    $$|P(X+1)| = 2^{|X+1|}$$
    Thus:
    $$|P(X+1)| = 2^{|X+1|} = 2*2^{|X|} = 2*|P(X)|$$
    This is true because, if $X$ as a set grows by one, the power set, for every member in its set now has a new binary choice: include the new element or not. Thus the power set doubles.