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This question already has an answer here:

I was asked a high-school question today which was merely asking the number of sigma-algebras on a set.

Let $X$ be a set

Let $S\triangleq \{\Sigma\subset P(X): \Sigma \text{ is a sigma-algebra on }X\}$.

If $X$ is finite, what would be the cardinality of $S$?

The question i was asked was the case $|X|=3$. In this case $|S|=5$.

What would be the cardinality of $S$ in general? Is it possible to find this value via elementary function?

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marked as duplicate by Camilo Arosemena-Serrato, Thomas Andrews, user940, user127.0.0.1, Asaf Karagila Feb 11 '14 at 1:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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A sigma algebra on a finite set is uniquely determined by the minimal non-empty elements under inclusion. (Why?)

The minimal elements are pair-wise disjoint. (Why?)

The minimal elements must have the entire set as a union. (Why?)

So the minimal elements form a partition of the set.

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