 Solution

A subset is determined uniquely by deciding, for each of the $n$ elements, whether to include it. Each decision has two possibilities. By the rule of product, there are $2\cdot 2\cdots 2=2^n$ decision strings. The correspondence between such binary strings and subsets is a bijection, so $|\mathcal{P}(\mathcal{U})|=2^n$. The all-zero string corresponds to the empty subset.