# Thread: Finding the smallest sigma algebra containing two sets

1. ## Finding the smallest sigma algebra containing two sets

Hello,

the problem is the following: Let $\displaystyle A, B \subseteq \Omega$. Find the smallest $\displaystyle \sigma$-algebra ($\displaystyle \mathcal{F}$) on $\displaystyle \Omega$ containing the sets $\displaystyle A$ and $\displaystyle B$.

I have tried to answer this, and would appreciate if you could tell me if this the correct answer, or if I have left something out:

$\displaystyle \{ \emptyset, \Omega, A, B, A^c, B^c, A \cup B, A \cap B, A \Delta B , A \backslash B, B \backslash A,$$\displaystyle (A \cup B)^c, (A \cap B)^c, (A \Delta B)^c , (A \backslash B)^c, (B \backslash A)^c \}$

Thanks!

2. Hello,

It looks correct.
Of course, if A=B or $\displaystyle A\subset B$ or... there will be redundance.

Anyway, if you want to be 'clearer', you can change the complements (using de Moivre's formula)