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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.

Thanks in advance!

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1 Answer

up vote 8 down vote accepted

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.

Alternatively, induct on the number of operations.

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Thank you! That was clever –  DoubleTrouble Nov 12 '12 at 8:02
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