Solution
= Solution
The definition says
$$
\mathbf x\sim\mathbf y
\quad\Longleftrightarrow\quad
P(\mathbf x)=P(\mathbf y).
$$
Equality of sets is reflexive, symmetric, and transitive, so $\sim$ is an <equivalence relation>. Its classes are indexed by all subsets of $\{1,\ldots,n\}$, and every subset occurs as the support of its zero-one <indicator vector>. The number of classes is therefore
$$
\boxed{2^n}.
$$