Consider an abstract probability space $(,F,P).$ Answer the following questions.

$($a$)$ Suppose that $A_1$ and $A_2$ form a partition $,$ and that $B$ is another subset of $$ that non$-$trivially intersects $A_1$ and $A_2$ and contains neither. Write down the smallest $-$algebra $F$ containing $A_1,A_2,$ and $B.$ How many elements are in $F?$

$($b$)$ Extending $($a$),$ suppose now $A_1,A_2,$dots,$A_n$ form a partition of $$ for nin $\{3,4,5,$dots$\},$ and $B$ is another subset of $$ that non$-$trivially intersects $A_1,$dots,$A_n$ and contains none of them. What is the cardinality of the smallest $-$algebra, $F_n,$ containing $A_1,$dots,$A_n$ and $B?$