Logic expressions

In China, everything becomes weird.

一、Basic thing

$A + B$ means $(A \text{ or } B)$

$A \cdot B$ means $(A \text{ and } B)$

$\overline{A}$ means $(\text{not } A)$

二、Simple rule

$AB + AC = A(B + C)$

$\overline{AB} = \overline{A} + \overline{B}$

$\overline{A + B} = \overline{A} \cdot \overline{B}$

三、Complex rule

$A + \overline{A}B = A + B$

$$\begin{align*} &AB + \overline{A}C + BC \\ \\ =& AB + \overline{A}C + BC \cdot (A + \overline{A}) \\ \\ =& AB + \overline{A}C + ABC + \overline{A}BC \\ \\ =& AB(C + 1) + \overline{A}C(B + 1) \\ \\ =& AB + \overline{A}C \end{align*}$$