As stated in the title, I want to show that the sigma algebra generated by a finite number of subsets of P(X) is finite. I feel like this should be easy but I am stuck. I tried showing that it is contained in a finite sigma algebra but I must not be choosing the right one (something like the set of all unions of the generating elements and their complements but I don't think that is right).
Thanks for the help!
Let $X_1,X_2,\ldots,X_n$ be subsets of $X$. For each $x\in X$ consider the following function $f:X\to\{0,1\}^n$:
$$f(x)_k=\begin{cases}1&\text{ if }x\in X_k\\0&\text{ otherwise}\end{cases}$$
Let $Y_1,\ldots,Y_{2^n}$ be the preimages of each element in $\{0,1\}^n$.
Note that every set of the $\sigma$-algebra $\Sigma$ generated by $X_1,\ldots,X_n$ is the union of some of the $Y_k$'s. So $\Sigma$ has at most $2^{2^n}$ sets (they can be less than that, because some $Y_k$'s may be empty).