Question 8
Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
• $\mathcal{A} \cup \mathcal{B}$ $\small = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\},$
• $\mathcal{A} \cap \mathcal{B} = \emptyset,$
• the number of elements of $\mathcal{A}$ is not an element of $\mathcal{A},$
• the number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}.$
Find $N.$
• $\mathcal{A} \cup \mathcal{B}$ $\small = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\},$
• $\mathcal{A} \cap \mathcal{B} = \emptyset,$
• the number of elements of $\mathcal{A}$ is not an element of $\mathcal{A},$
• the number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}.$
Find $N.$