Example 9. Simplify the expression $(\bar{A}+B)(\bar{A}+\bar{B})$.
Solution. Using formulas (1.6.1), (1.6.4), (1.6.5) - (1.6.8), (1.6.11), we find that

$$
\begin{aligned}
& (\bar{A}+B)(\bar{A}+\bar{B})=\bar{A}(\bar{A}+\bar{B})+B(\bar{A}+\bar{B})=\overline{A A}+\bar{A} \bar{B}+B \bar{A}+B \bar{B}= \\
& =\overline{A A}+\bar{A}(\bar{B}+B)+B \bar{B}=\bar{A}+\bar{A} U+V=\bar{A}+\bar{A}+V=\bar{A}+V=\bar{A}
\end{aligned}
$$

Therefore, $(\bar{A}+B)(\bar{A}+\bar{B})=\bar{A}$.