Joining sigma algebras I am having some problems joining sigma algebras. So I have:
$$\textit{K} =\left \{ A\cap B: A\in \sigma (D),B\in \sigma (E) \right \}$$
and I need to show
$$\sigma(K)= \sigma (D,E)$$
What I've done so far
I intend to do this by showing
$$\sigma(K)\subseteq  \sigma (D,E)$$ and $$\sigma(K)\supseteq\sigma (D,E)$$
The first part is easy enough, my problem is $\sigma(K)\supseteq\sigma (D,E)$ as I'm not sure where to begin.
I've started by saying by definition $\sigma(D,E)= \sigma(\sigma(D)\cup\sigma(E))$
and $\sigma(\sigma(D)\cup\sigma(E))=\sigma(\sigma(D)\cap\sigma(E))$ by De Morgan and $\sigma(D)\cap\sigma(E)=K$, but I think that is wrong as $\sigma(D)\cap\sigma(E)$ may be a sigma algebra and thus its smaller than $ \sigma (A,B)$.
I've also considered something like $\sigma(K^{c})$ to somehow argue that $\sigma(\sigma(D)\cup\sigma(E))$ is a subset of that but intuitively it feels like what I did with the De Morgan above.
Thanks
EDIT: sorry for the confusing notation everyone, everything has been changed.
 A: Notice that $K$ contains $A$ and $B$. The definition of $\sigma(A,B)$ would be $\sigma(A\cup B)$, I think. Not that it makes for much of a difference, but still...
I think you're overthinking it. I believe that the first part is actually harder (if only a little bit).
Also, you might want to use different symbols to denote the algebras and their elements (\mathcal might help you).
If you're still stuck, see the further hint:

If we have a set $G$ and a sigma-algebra $\Sigma$, then if $G\subseteq \Sigma$, then $\sigma(G)\subseteq \Sigma$. Thus, if for some $G_1,G_2$ we have that $\sigma(G_1)\supseteq G_2$ and $\sigma(G_2)\supseteq G_1$, then $\sigma(G_1)=\sigma(G_2)$.

A: Your definition of $K$ has no sense: you should say for instance $K=\sigma(X\cap Y, X \in\sigma(A), Y\in\sigma(B))$ (because $A \in \sigma(A)$ is puzzling).
The first inclusion $\sigma(K) \subset \sigma(A,B)$ is easy: it follows from the inclusions  $\sigma(A) \subset \sigma(A,B)$ and $\sigma(B) \subset \sigma(A,B)$ and from the stability of a $\sigma$-field under intersection.
The second inclusion $\sigma(K) \supset \sigma(A,B)$ is easy too: clearly $\sigma(K)$ is a $\sigma$-algebra containing $\sigma(A)$ and $\sigma(B)$, therefore it contains the smallest $\sigma$-algebra containing $\sigma(A)$ and $\sigma(B)$, which is nothing but $\sigma(A,B)$.
