I know that a sigma algebra generated by a subset a, $\sigma(a) = \{\emptyset,a,a^c,E\}$.
But what about $\sigma({a,b})$? Would it be $\{\emptyset,a,a^c, b, b^c,E\}$?
I know that a sigma algebra generated by a subset a, $\sigma(a) = \{\emptyset,a,a^c,E\}$.
But what about $\sigma({a,b})$? Would it be $\{\emptyset,a,a^c, b, b^c,E\}$?
It's quite easy to describe the $\sigma$-algebra generated by a finite partition $\{A_1,\dots,A_N\}$ of $\Omega$: it consists of the sets $\bigcup_{j\in J}A_j$, where $j\subset [N]$.
In this particular case, consider $A_1=A\cap B$, $A_2=A^c\cap B$, $A_3:=A\cap B^c$, $A_4:=A^c\cap B^c$. These sets form a partition of $\Omega$.
Of course, we can describe in the same way the $\sigma$-algebra generated by a finite collection $S_1,\dots,S_n$.