I understand that the number of elements generated by sigma(A,B) is 16. But I'm not able to find all of them.
I have found: $A$, $B$, $A^c$, $B^c$, $A ∩ B$, $A^c ∩ B^c$, $A^c ∪ B^c$, the empty set, the sample space, $A^c ∩ B$, $A ∩ B^c$, $A^c ∪ B$, $A ∪B^c$, $A ∪ B$. What are the last two?
Let us be systematic: the smallest sigma-algebra $\mathcal F$ containing $A$ and $B$ is generated by the partition $\{C_1,C_2,C_3,C_4\}$ with $$ C_1=A\setminus B,\quad C_2=B\setminus A,\quad C_3=A\cap B,\quad C_4=\Omega\setminus(A\cup B). $$ Hence $\mathcal F=\{C_I\mid I\subseteq\{1,2,3,4\}\}$, where, for every $I\subseteq\{1,2,3,4\}$, $C_I=\bigcup\limits_{i\in I}C_i$ (for example, $A=C_{\{1,3\}}$ and $B=C_{\{2,3\}}$). Hence, in the general case (that is, when none of the $C_i$ is empty), $\mathcal F$ has size $2^4=16$.
Likewise, the smallest sigma-algebra containing $n$ subsets has, in the general case, size $2^{2^n}$.
Hint: Draw a Venn diagram where $A$ and $B$ intersect. There should be four distinct regions $(A\cup B)^c$, $A\cap B^c$, $B\cap A^c$ and $A\cap B$. Each element of the sigma algebra is a union of some of these sets. As there are four sets and each one is either part of the union or not, you will get $2^4 = 16$ sets, in particular, the two missing from your list.
The general methodology goes as follows: If $\mathcal{C}=\{A_1,A_2,\cdots\}$ is a class of subsets of a non-empty set $\Omega$, then construct the sets $$B_{\epsilon}=\bigcap_{n}A_n^{\epsilon_n}$$ where $\epsilon=(\epsilon_1,\epsilon_2)$, $\epsilon_i\in\{0,1\}$ and $$A_i^{\epsilon_i}=\left\{\begin{array}{cc} A_i&&\mbox{if}~\epsilon_i=0\\ A_i^c&&\mbox{if}~\epsilon_i=1 \end{array}\right.$$ Let $\mathcal{D}=\{B_{\epsilon}:B_{\epsilon}\neq\phi\}$. Then take all possible unions of the sets in $\mathcal{D}$. This will give you the field generated by $\mathcal{C}$. If $\mathcal{C}$ is finite then this is also the $\sigma$-field generated by $\mathcal{C}$.
