基本定律
求反规则: 1 ‾ = 0 \overline{1} = 0 1=0; 0 ‾ = 1 \overline{0} = 1 0=1
常变量规则: 0 ⋅ A = 0 0 \cdot A = 0 0⋅A=0, 1 ⋅ A = A 1 \cdot A = A 1⋅A=A; 1 + A = 1 1 + A = 1 1+A=1, 0 + A = A 0 + A = A 0+A=A
重叠律: A ⋅ A = A A \cdot A = A A⋅A=A; A + A = A A + A = A A+A=A
互补律: A ⋅ A ‾ = 0 A \cdot \overline{A} = 0 A⋅A=0; A + A ‾ = 1 A + \overline{A} = 1 A+A=1
交换律: A ⋅ B = B ⋅ A A \cdot B = B \cdot A A⋅B=B⋅A; A + B = B + A A + B = B + A A+B=B+A
结合律: A ⋅ ( B ⋅ C ) = ( A ⋅ B ) ⋅ C A \cdot (B \cdot C) = (A \cdot B) \cdot C A⋅(B⋅C)=(A⋅B)⋅C; A + ( B + C ) = ( A + B ) + C A + (B + C) =(A+ B) + C A+(B+C)=(A+B)+C
分配律: A ⋅ ( B + C ) = A ⋅ B + A ⋅ C A \cdot (B + C) = A \cdot B + A \cdot C A⋅(B+C)=A⋅B+A⋅C; A + ( B ⋅ C ) = ( A + B ) ⋅ ( A + C ) A + (B \cdot C) =(A+ B) \cdot (A +C) A+(B⋅C)=(A+B)⋅(A+C)
狄摩根定律: A ⋅ B ‾ = A ‾ + B ‾ \overline{A \cdot B} = \overline{A} + \overline{B} A⋅B=A+B; A + B ‾ = A ‾ ⋅ B ‾ \overline{A+B} = \overline{A} \cdot \overline{B} A+B=A⋅B
还原律: A ‾ = A \overline{A} = A A=A
三大规则
带入规则
任何一个逻辑等式,如果将等式两边所出现的某一变量都代之以同一逻辑函数,则等式仍然成立,这个规则称为代入规则。代入规则可以扩大基本定律的运用范围。
例如,已知 A + B ‾ = A ‾ ⋅ B ‾ \overline{A+B} = \overline{A} \cdot \overline{B} A+B=A⋅B (反演律),若用 F = B + C F = B + C F=B+C 代替等式中的B,则可以得到适用于多变量的反演律,即 A + B + C ‾ = A ‾ ⋅ B ‾ ⋅ C ‾ \overline{A+B + C} = \overline{A} \cdot \overline{B} \cdot \overline{C} A+B+C=A⋅B⋅C
反演规则
任一逻辑式 F F F,如果将所有的“ ⋅ \cdot ⋅”换成“ + + +”、“ + + +”换成“ ⋅ \cdot ⋅”、0换成1、1换成0、原变量换成反变量、反变量变成原变量,则结果就是 F ‾ \overline{F} F。
- 优先次序:"()" > “.” > “+”
- 不属于单个变量上的反号应保留。
F = A ⋅ B ‾ ‾ + D ‾ + C F = \overline{\overline{A \cdot \overline{B}} + D} + C F=A⋅B+D+C 则
F ‾ = ( A ‾ + B ) ⋅ C ‾ ‾ ⋅ D ‾ ‾ ⋅ C ‾ \overline{F} = \overline{\overline{(\overline{A} + B) \cdot \overline{C} } \cdot \overline{D}} \cdot \overline{C} F=(A+B)⋅C⋅D⋅C
对偶规则
对偶式的定义:任一逻辑式F,如果将所有的“ ⋅ \cdot ⋅”换成“+”、“+”换成“ ⋅ \cdot ⋅”、0换成1、1换成0,而变量保持不变,得出的就是F的对偶式F’。
比如分配律:
分配律: A ⋅ ( B + C ) = A ⋅ B + A ⋅ C A \cdot (B + C) = A \cdot B + A \cdot C A⋅(B+C)=A⋅B+A⋅C; A + ( B ⋅ C ) = ( A + B ) ⋅ ( A + C ) A + (B \cdot C) =(A+ B) \cdot (A +C) A+(B⋅C)=(A+B)⋅(A+C)
常用公式
A + A ⋅ B = A A + A \cdot B = A A+A⋅B=A
A + A ‾ ⋅ B = A + B A + \overline{A} \cdot B = A + B A+A⋅B=A+B
A ⋅ B + A ⋅ B ‾ = A A \cdot B + A \cdot \overline{B} = A A⋅B+A⋅B=A
A ⋅ ( A + B ) = A A \cdot (A + B) = A A⋅(A+B)=A
A ⋅ B + A ‾ ⋅ C + B C D = A ⋅ B + A ‾ ⋅ C A \cdot B + \overline{A} \cdot C + BCD= A \cdot B + \overline{A} \cdot C A⋅B+A⋅C+BCD=A⋅B+A⋅C
A ⋅ A ⋅ B ‾ = A ⋅ B ‾ A \cdot \overline{A \cdot B} = A \cdot \overline{B} A⋅A⋅B=A⋅B
A ‾ ⋅ A ⋅ B ‾ = A ‾ \overline{A} \cdot \overline{A \cdot B} = \overline{A} A⋅A⋅B=A