Example 8. Simplify the expression $(A+B)(A+\bar{B})$.
Solution. Denoting the certain event by $U$, the impossible event by $V$, and using formulas (1.6.1), (1.6.4), (1.6.5) - (1.6.8) and $(1.6 .11)$, we get

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

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