It is called the "power set" of $A$ because for any set $A$ with $n$ elements $\mathcal{P}(A)$ has $2^{n}$ elements. By induction. If $A$ has 1 element, then there are two subsets of $A$ : the empty set and the set itself. Suppose that for any set with $n-1$ elements its power set has $2^{n-1}$ elements. Take a set $A$ with $n$ elements, and let $x$ be an element in $A$ . Each of its subset either contains $x$ or it does not. We can construct all subsets of $A$ by taking each subset of $A-\{x\}$ (which has $n-1$ elements) and by optionally adding $x$ to it. Therefore, the number of subsets of $A$ is twice the number of subsets of $A-\{x\}$ , which, by the inductive hypothesis, is $2^{n-1}$ . So, we get $2^{n}$ subsets of $A$ . (Since we based the induction on the case $n=1$ , there is one more case to consider, namely, $n=0$ , but obviously $\mathcal{P}(\emptyset)=1$ as there is only one subset of the empty set — the empty set itself.)