逻辑代数几个重要定理

写作业要用的一些逻辑代数定理,写在这备查备忘.
符号声明:
+:或;
$\cdot$:与
$\bar{a}$ :非

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PART 1 公理

交换律

$A\cdot B=B\cdot A$

结合律

$A+(B+C)=(A+B)+C$
$A\cdot (B\cdot C)=(A\cdot B)\cdot C$

分配律

$A+(B\cdot C)=(A+B)\cdot (A+C)$
$A\cdot (B+C)=(A\cdot B)+(A\cdot C)$

0-1律

$A+1=1A\cdot 1=A$
$A+0=AA\cdot 0=0$

互补律

$A+\bar{A}=1$
$A\cdot \bar{A}=0$

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PART 2 定理

1

$$\begin{gathered} 0+0=0 \\ 0+1=1 \\ 1+1=1 \\ 0\cdot 0=0 \\ 0\cdot 1=0 \\ 1\cdot 1=1 \end{gathered}$$

2

$$\begin{gathered} A+A=A \\ A\cdot A=A \end{gathered}$$

3

$$\begin{gathered} A+A\cdot B=A \\ A\cdot (A+B)=A \end{gathered}$$
(KEY:=$A\cdot 1+A\cdot B$)

4

$$\begin{gathered} A+\bar{A}\cdot B=A+B \\ A\cdot (\bar{A}+B)=A\cdot B \end{gathered}$$
(KEY:$=((A+\bar{A})\cdot (A+B)$)

5

$\overline{\bar{A}}=A$

6

$$\begin{gathered} \overline{A+B}=\bar{A}\cdot \bar{B} \\ \overline{A\cdot B}=\bar{A}+\bar{B} \end{gathered}$$

7

$$\begin{gathered} A\cdot B+A\cdot \bar{B}=A \\ (A+B)\cdot (A+\bar{B})=A \end{gathered}$$

8

$$\begin{gathered} A\cdot B+\bar{A}\cdot C+B\cdot C=A\cdot B+\bar{A}\cdot C \\ (A+B)\cdot (\bar{A}+C)\cdot (B+C)=(A+B)\cdot (\bar{A}+C) \end{gathered}$$
(KEY:$=A\cdot B+\bar{A}\cdot C+B\cdot C\cdot (A+\bar{A})$)

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等有空了再详细解释.