# The cardinality of a finite sigma algebra is a power of 2?

Let $$X$$ be a finite set. Let $$\Sigma \subseteq P(X)$$ be a finite sigma algebra on $$X$$. I have been able to prove that the cardinality of $$\Sigma$$ is always an even number: the collection $$\left\{\left\{Y{,}\ X\setminus Y\right\}:Y\in \Sigma\right\}$$ is a partition of $$\Sigma$$. Thus, $$\left|\Sigma\right|=\left|\left\{\left\{Y{,}\ X\setminus Y\right\}:Y\in \Sigma\right\}\right|\cdot\left|\left\{Y{,}\ X\setminus Y\right\}\right|=2\left|\left\{\left\{Y{,}\ X\setminus Y\right\}:Y\in \Sigma\right\}\right|{,}$$ and the cardinality of $$\Sigma$$ is necessarily an even number.

However, there is more: the cardinality of $$\Sigma$$ is a power of $$2$$. This I am unable to prove. What would be an elementary way of showing that fact? I guess that one should construct some collection so that $$\Sigma$$ is its power set, thus having a cardinality of some power of $$2$$.

Hints: For each $$x \in X$$ there is a smallest member $$A_x$$ of the sigma algebra containing $$x$$. $$A_x$$ and $$A_y$$ are either equal or disjoint. Any set in the sigma algebra is a union of $$A_x$$'s. This gives a one-to-one function from the sigma algebra onto a power set.
Define on $$X$$ the equivalence relation $$x \sim y$$ if and only if for each $$A\in\Sigma$$, $$\mathbf{1}_A(x)=\mathbf{1}_A(y)$$. For any $$x\in X$$, the equivalence class of $$x$$ is $$C_x:=\bigcap_{A\in \Sigma\,:\,x\in A}A$$ and since $$\Sigma$$ is finite, we know that $$C_x$$ belongs to $$\Sigma$$. The collection of equivalence classes of elements of $$X$$ forms a finite partition, says $$A_i, 1 \leqslant i\leqslant N$$, of elements of $$\Sigma$$.
For $$I\subset\{1,\dots,N\}$$, let $$B_I=\bigcup_{i\in I}A_i$$ if $$I$$ is not empty and $$B_\emptyset=\emptyset$$. Then the map $$\tau\colon \mathcal P(\{1,\dots,N\})\mapsto \Sigma$$ given by $$\tau(I)=B_I$$ is a bijection.