Bit Problem - Query Analysis

Let's decode what each query asks as subsets:

$1.$ $x \mid y = x$ means every bit of $y$ is also in $x$. So $y$ is a subset of $x$. Count subsets of $x$ in the array.

$2.$ $x \land y = x$ means every bit of $x$ is also in $y$. So $x$ is a subset of $y$, meaning $y$ is a superset of $x$. Count supersets of $x$.

$3.$ $x \land y \neq 0$ means $x$ and $y$ share at least one bit. The complement: $x \land y = 0$ means $y$ is a subset of $\sim x$ (the bitwise complement). So count $n$ minus subsets of $\sim x$.