# Algebra generated by countable family of sets is countable?

If I have a countable family of sets $\mathcal{A}=\{A_1,A_2,...\}$ and construct the Algebra generated by $\mathcal{A}$. Will it also be countable?

My intuition screams YES, but I cannot seem to construct the Algebra in any clever way. Hints are very much appreciated.

A member of the algebra can be written as an expression using the $A_j$ and the operations of intersection, union and complement, and thus encoded as a finite string over a finite alphabet. There are only countably many such strings.