Given a set family $\mathscr{A}=(A_1,A_2,\cdots,A_n)$. An ordered sequence $(a_1,a_2,\cdots,a_n)$ is called an SDR if
- $a_i \in A_i$ for $i=1,\cdots,n$;
- $a_i \ne a_j$ for any $i \ne j$.
Given two set families $\mathscr{A}=(A_1,A_2,\cdots,A_n)$ and $\mathscr{B}=(B_1,B_2,\cdots,B_n)$. If for any $1 \le k \le n$ and $1 \le i_1<i_2<\cdots<i_k \le n$, we have $$\left|\bigcup_{j=1}^k A_{i_j} \right| \ge \left|\bigcup_{j=1}^k B_{i_j} \right|.$$
Can we prove that the number of SDR of $\mathscr A$ is not less than the number of SDR of $\mathscr B$?
I think it is true, but I don’t know how to prove it.