11. (16 points) Let $A$ and $B$ be two different subsets of the set $\left\{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right\}$, such that $A$ is not a subset of $B$, and $B$ is not a subset of $A$. Find the number of different ordered pairs $(A, B)$.
11. The set $\left\{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right\}$ has $2^{5}$ subsets, and the number of different ordered pairs $(A, B)$ is $2^{5}\left(2^{5}-1\right)$.

If $A \subset B$, and suppose $B$ contains $k(1 \leqslant k \leqslant 5)$ elements. Then the number of ordered pairs $(A, B)$ satisfying $A \subset B$ is
$$
\begin{array}{l}
\sum_{k=1}^{5} \mathrm{C}_{5}^{k}\left(2^{k}-1\right)=\sum_{k=0}^{5} \mathrm{C}_{5}^{k}\left(2^{k}-1\right) \\
=\sum_{k=0}^{5} \mathrm{C}_{5}^{k} 2^{k}-\sum_{k=0}^{5} \mathrm{C}_{5}^{k}=3^{5}-2^{5} .
\end{array}
$$

Similarly, the number of ordered pairs $(A, B)$ satisfying $B \subset A$ is also $3^{5}-2^{5}$.

Therefore, the number of ordered pairs $(A, B)$ satisfying the conditions is
$$
2^{5}\left(2^{5}-1\right)-2\left(3^{5}-2^{5}\right)=570 \text {. }
$$