2 problems

For each $n$, let $\Sigma_n=(\sigma_{n,1},\ldots,\sigma_{n,n})$, where each $\sigma_{n,j}$ is a subset of $[n]=\{1,2,\ldots,n\}$. Let $X_n\in\operatorname{M}_n(\mathbb{Z}_p)$ be Ha…

Let $X_m:=\{x_1,\ldots,x_m\}$ be a finite ordered subset of $\mathbb{Z}_p$ whose elements have distinct reductions modulo $p$, and let…