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
a ˉ \bar{a} aˉ :非
PART 1 公理
交换律
A ⋅ B = B ⋅ A A\cdot B\ =B\cdot A A⋅B=B⋅A
结合律
A + ( B + C ) = ( A + B ) + C A+(B+C)=(A+B)+C A+(B+C)=(A+B)+C
A ⋅ ( B ⋅ C ) = ( A ⋅ B ) ⋅ C A\cdot(B\cdot C)=(A\cdot B)\cdot C A⋅(B⋅C)=(A⋅B)⋅C
分配律
A + ( B ⋅ C ) = ( A + B ) ⋅ ( A + C ) A+(B\cdot C)=(A+B)\cdot(A+C) A+(B⋅C)=(A+B)⋅(A+C)
A ⋅ ( B + C ) = ( A ⋅ B ) + ( A ⋅ C ) A\cdot(B+C)=(A\cdot B)+(A\cdot C) A⋅(B+C)=(A⋅B)+(A⋅C)
0-1律
A + 1 = 1 A ⋅ 1 = A A+1=1\ \ \ \ \ \ \ \ \ \ \ \ \ A\cdot 1=A A+1=1A⋅1=A
A + 0 = A A ⋅ 0 = 0 A+0=A\ \ \ \ \ \ \ \ \ \ \ \ \ A\cdot 0=0 A+0=AA⋅0=0
互补律
A + A ˉ = 1 A+\bar{A}=1 A+Aˉ=1
A ⋅ A ˉ = 0 A\cdot \bar{A}=0 A⋅Aˉ=0
PART 2 定理
1
0 + 0 = 0 0 + 1 = 1 1 + 1 = 1 0 ⋅ 0 = 0 0 ⋅ 1 = 0 1 ⋅ 1 = 1 0+0=0\\0+1=1\\1+1=1\\0\cdot0=0\\0\cdot1=0\\1\cdot1=1 0+0=00+1=11+1=10⋅0=00⋅1=01⋅1=1
2
A + A = A A ⋅ A = A A+A=A\\A\cdot A=A A+A=AA⋅A=A
3
A + A ⋅ B = A A ⋅ ( A + B ) = A A+A\cdot B=A\\A\cdot(A+B)=A A+A⋅B=AA⋅(A+B)=A
(KEY:= A ⋅ 1 + A ⋅ B A\cdot1+A\cdot B A⋅1+A⋅B)
4
A + A ˉ ⋅ B = A + B A ⋅ ( A ˉ + B ) = A ⋅ B A+\bar{A}\cdot B=A+B\\A\cdot(\bar{A}+B)=A\cdot B A+Aˉ⋅B=A+BA⋅(Aˉ+B)=A⋅B
(KEY: = ( ( A + A ˉ ) ⋅ ( A + B ) =((A+\bar{A})\cdot(A+B) =((A+Aˉ)⋅(A+B))
5
A ˉ ‾ = A \overline{\bar{A}}=A Aˉ=A
6
A + B ‾ = A ˉ ⋅ B ˉ A ⋅ B ‾ = A ˉ + B ˉ \overline{A+B}=\bar{A}\cdot\bar{B}\\ \overline{A\cdot B}=\bar{A}+\bar{B} A+B=Aˉ⋅BˉA⋅B=Aˉ+Bˉ
7
A ⋅ B + A ⋅ B ˉ = A ( A + B ) ⋅ ( A + B ˉ ) = A A\cdot B +A\cdot\bar{B}=A\\ (A+B)\cdot(A+\bar{B})=A A⋅B+A⋅Bˉ=A(A+B)⋅(A+Bˉ)=A
8
A ⋅ B + A ˉ ⋅ C + B ⋅ C = A ⋅ B + A ˉ ⋅ C ( A + B ) ⋅ ( A ˉ + C ) ⋅ ( B + C ) = ( A + B ) ⋅ ( A ˉ + C ) 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) A⋅B+Aˉ⋅C+B⋅C=A⋅B+Aˉ⋅C(A+B)⋅(Aˉ+C)⋅(B+C)=(A+B)⋅(Aˉ+C)
(KEY: = A ⋅ B + A ˉ ⋅ C + B ⋅ C ⋅ ( A + A ˉ ) =A\cdot B+\bar{A}\cdot C+B\cdot C\cdot(A+\bar{A}) =A⋅B+Aˉ⋅C+B⋅C⋅(A+Aˉ))
等有空了再详细解释.
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