Subsets of a set generating a sigma algebra I have a notation difficulty while writing correctly a sigma algebra generated not by a set of elements but rather by a set of subsets of a set.
Let $E=\{ 1,2,3,4\}$. I need to determine the $\sigma$-algebra generated by the family of subsets $X=\{\{1\},\{2\}\}$ of $E$.
I first thought about this solution: $\sigma(\{1\}, \{2\})=\{\emptyset ,\{1,2\}, \{3,4\}, \{1,2,3,4\}\}$ which seems not to hold. 
I am wondering, whats the complement of $X$ in respect to $E$ ? Also, can one say that $\{\{1\},\{2\}\}$ is a subset of $E$
?
Thanks for your comment.

By the definition of $\sigma$-algebra generated by a set (i.e. smallest $\sigma$-algebra containing given set),
$\sigma(X) = \{ \emptyset, \{1\},\{2\},\{1,2\},\{3,4\},\{2,3,4\},\{1,3,4\},E\}$ 
You can check that above collection satisfies all properties required for $\sigma$-algebra. It contains $X$ and it’s smallest $\sigma$-algebra containing $X$.
