
4. Let $X=\{1,2,3, \ldots, 10\}$. Find the the number of pairs $\{A, B\}$ such that $A \subseteq X$, $B \subseteq X, A \neq B$ and $A \cap B=\{2,3,5,7\}$.


Solution: Let $A \cup B=Y, B \backslash A=M, A \backslash B=N$ and $X \backslash Y=L$. Then $X$ is the disjoint union of $M, N, L$ and $A \cap B$. Now $A \cap B=\{2,3,5,7\}$ is fixed. The remaining six elements $1,4,6,8,9,10$ can be distributed in any of the remaining sets $M, N, L$. This can be done in $3^{6}$ ways. Of these if all the elements are in the set $L$, then $A=B=\{2,3,5,7\}$ and which this case has to be deleted. Hence the total number of pairs $\{A, B\}$ such that $A \subseteq X, B \subseteq X, A \neq B$ and $A \cap B=\{2,3,5,7\}$ is $3^{6}-1$.
