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I ran across this statement I am trying to prove, but I am sure if I am in the right direction.

Statement: Show there is a natural correspondence between the set of sigma algebras on a finite set $X$ and the set of partitions of $X$.

Proof: To me, it seems that I must show two things. Given a set of sigma algebras, how do each of these sigma algebras relate to the set of partitions of $X$, and given a set of partitions of $X$, how do these relate to the sigma algebras.

(Partition $\rightarrow$ sigma algebra) For each partition, it generates a given sigma algebra by taking the power set of the cells in the given partition of $X$. The element in this set guarantees that I am closed under the operation of complements and union (finite). Also $X$ lie sin the given sigma-algebra.

(Sigma Algebra $\rightarrow$ Partition) For a given sigma algebra, I want to find its generating set. The set where I take its powerset will give me the sigma algebra. If we examine the set of all partitions of $X$ and superimpose each partition of $X$ on each other, the resulting partition is the desired partitition which will give us the sigma algebra.

At least, I believe, this is one way I see it.

Thanks in Advance!

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    @user178500: In fact, you need to do *four* things, not just two. You need to show how to assign a partition to every sigma algebra, how to assign a sigma algebra to every partition; but you also need to show that the sigma algebra of the partition of a sigma algebra is the original sigma algebra, and that the partition of the sigma algebra of a partition is the original partition, in order to show that you have a **correspondence** (and not merely maps from the set of partitions to the set of sigma algebras and vice versa).2012-02-10

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