数电第三章
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8
3）
AB AB A;
(A B)(A B) A 合并律
Pr oof : AB+AB A(B B) A
4) AB+AC+BC = AB+AC; Proof:
(A+B)(A+C)(B+C) = (A+B)(A+C)
冗余定理
AB+AC+BC =AB+AC+(A+A)BC =AB+AC+ABC+ABC
如果 F 成立，F’ 也成立
6
Note: 1. Keep the order of the operations (运算顺序不变)
2. Keep the bar which is on a function
(不是一个变量上的反号保持不变，函数)
Example 1:
Given F=A(B+C) CD Find F' and F, respectively
5
3) Duality 对偶规则
设F为逻辑函数，如果将函数表达式中所有的与(•) 换成或(+),或(+) 换成与(•);0换成1，1换成0；则得到的逻辑函数是F的对偶式F’ The dual expression is found by replacing all + operation with • , all • operation with +, all ones with zeros, all zeros with ones.
Function F
+
1
•
0
New function F'
New function F’ is called the dual of the function F. If an expression F is valid in logic algebra, the dual of the expression, F’, is also valid.
4) Theorem 4
5) Theorem 5
A+0=A, A+1=1;
A+A=1
A•0 =0 A•A=0
A•1=A
(01律/互补)
6) Theorem 6
7) Theorem 7
A+A=A;
A•A=A
(重叠律)
(还原律)
AA
A•B = A+B
8) DeMorgan’s theorems A+B = A•B Deductions (推论）A+B+C=A•B•C
1
§3.1 Operations of Logic Algebra
逻辑代数运算法则
AND operation 与 Logic multiplication OR operation或 Logic addition “+” 0+0=0 0+1=1 1+0=1 1+1=1 The complement 非 of A is A
14
Example:
A B C: 010
Example: 2 variables A, B:
ABC 1
010 = 2
so minterm ABC is numbered m2
m1= AB,
m3 = AB
4 variables A, B, C, D:
m1 = A B C D m5 = ABCD m13 = ABCD
Right: A BC A B C
We have used it in the “Deduction” of Morgan’s Theorems. 多个变量摩根定理
4
2) Complementary Theorem 反演规则
设F为逻辑函数，如果将函数表达式中所有的与(•) 换成或(+),或(+)换成与 (•);0换成1，1换成0；原变量换成反变量，反变量换成原变量，则得到的逻 辑函数是F的反函数F The complement expression is formed by replacing all + operations with • , all • operations with +, all ones with zeros, all zeros with ones, all variables with inversed variables, and all inversed variables with variables.
F
1 + uncomplemented variable 原变量
0 • complemented variable反变量
New function
F
Function F is called the complement of the function F. if the function F is valid, the complement of the function, F , is also valid.若原函数成立，反函数也成立
Chapter 3
Logic Algebra 逻辑代数基础
Logic Algebra is presented by George Boole in 1849 and is also called Boolean Algebra. Logic algebra constitute a mathematical tool for describing the input-output behavior of the logical gates. 逻辑代数描述二值变量(0,1)的运算规律。由英国数学家布尔在19 世纪中提出，也称为布尔代数。
A variable is a symbol used to represent a logical quantity. Any single variable can have a 1 or a 0 value.
逻辑代数中变量是表示逻辑量，只能取0或1.
Logic algebra and arithmetic are different. 逻辑代数与算术运算不同。
证明(Proof): A+AB = A (1+B) = A
Absorption 吸收律
2)
A+AB A B;
A(A+B)=AB
吸收律
Proof: Using distributive law (分配律 A+BC=(A+B)(A+C) )
A+AB (A A)(A B) A B
A C D B; B C D A;
10
Causality (因果关系）
§3.2 Standard Forms of Logic Function
逻辑函数的标准形式 逻辑函数有两种标准形式：标准与或式（最小项和） 标准或与式（最大项积）
3.2.1 Minterms and Standard Sum of Products
other.
(每条定理有两种表达形式：逻辑加及逻辑乘。对偶式)
2
基本定律
1) Theorem 1 2) Theorem 2 3) Theorem 3
逻辑加
A+B=B+A; A+(B+C)=(A+B)+C;
逻辑乘
AB=BA A(BC)=(AB)C (交换律) (结合律)
A +BC= (A+B)(A+C); A(B+C)=AB+AC; (分配律)
ABC ABC ABC ABC ABC
ABC
ABC ABC
12
2. Truth Table of Minterms 最小项真值表
Minterm Number
Variables
m0
m1
m2
m3
m4
m5
m6
m7
A B C 0 0 0
Minterm
ABC ABC ABC ABC ABC ABC ABC ABC
标准与项(最小项)：n 变量函数, n 变量组成的与项中, 每个变 量都以原变量或反变量形式出现一次 , 且只出现一次。若变量数 为n，则有2n个最小项。
n variables 2n minterms
For example: 3 variables A, B, C, there are 23 8 minterms:
与项：多个变量(原变量或反变量)的乘积形式
AB
BCD
AE
11
Minterms (Standard Product Terms): For a function with n variables if a product term contains each of the n variables exactly one time in complemented or uncomplemented form, the product term is called a Minterm or a Standard Product Term.
Solution: F' = (A+BC)(C+D) F = (A+BC)(C+D)
Example 2:
G WX Y Z X
G' (W X)Y Z X
G (W X)Y Z X
7
3. Formula of Boolean Algebra
常用公式
1) A+AB=A; A(A+B)=A
“•”
0•0 = 0 0•1 = 0 1•0 = 0 1•1 = 1
0 1 1 0
1. Fundamental Theorems of Logic Algebra 基本定律
Every theorem is given in two forms: one for addition(加) and another for multiplication(乘). Two forms are equivalent and are called “Dual” each
(摩根定理)
ABC A B C
3
2. Basic Rules 基本规则
1) Substitution 代入规则
等式两侧某一变量都用一个逻辑函数代入，等式仍成立。
For example: If Left: So
AX A X
AX ABC ABC A B C
X BC
1 0 0 0 0 0
0
0 1
0 0 0 1 0
0 0 0 0
0
0
0 0
0
0 0
0 0 1
0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
1
0 0
0
0 0 0 1 0 0
0
0 0
0
0 0
0
0 0 0 1
13
0
0 0 0
1
0 0
0
0 0
0
0 0
0
0
1
0
0
We can see that from the truth table, when variables A B C are given a set of numbers, only one minterm equals 1, and other minterms all equal 0. (当 A B C 取某一组值时, 只有一个最小项值为 1 , 其他 都等于 0 ) Minterm Number is defined as mi where i is the decimal integer equal to the corresponding binary code which makes the minterm 1. (使某一最小项为1时, 变量取值的二进制数对应的十 进制数为此最小项的编号)
A A 1,
A 0 A,
A 1 A
多变量异或，运算结果只与变量为1的个数有关，与变量为0的个 数无关。变量为 1 的个数为奇数，异或结果为 1；变量为 1 的个数为 偶数，结果为 0 。(变量与1异或一次，相当于取反一次)
6) If
A BC D
then
Hale Waihona Puke A B D C;=AB+AC
Deduction : AB AC BCDE AB AC
9
5) Exclusive-OR (XOR) Formula (异或公式) A⊕B = A⊙B Proof: AB AB (A B)(A B) AB AB A B
A A 0,
1: variable
0: complemented variable
1对应于原变量
最小项及标准与或式
1. Minterms (Standard Product Form) 最小项：也称为标准与项 A product term was defined as a term consisting of the product (logic multiplication) of literals (variables or their complements).
3）
AB AB A;
(A B)(A B) A 合并律
Pr oof : AB+AB A(B B) A
4) AB+AC+BC = AB+AC; Proof:
(A+B)(A+C)(B+C) = (A+B)(A+C)
冗余定理
AB+AC+BC =AB+AC+(A+A)BC =AB+AC+ABC+ABC
如果 F 成立，F’ 也成立
6
Note: 1. Keep the order of the operations (运算顺序不变)
2. Keep the bar which is on a function
(不是一个变量上的反号保持不变，函数)
Example 1:
Given F=A(B+C) CD Find F' and F, respectively
5
3) Duality 对偶规则
设F为逻辑函数，如果将函数表达式中所有的与(•) 换成或(+),或(+) 换成与(•);0换成1，1换成0；则得到的逻辑函数是F的对偶式F’ The dual expression is found by replacing all + operation with • , all • operation with +, all ones with zeros, all zeros with ones.
Function F
+
1
•
0
New function F'
New function F’ is called the dual of the function F. If an expression F is valid in logic algebra, the dual of the expression, F’, is also valid.
4) Theorem 4
5) Theorem 5
A+0=A, A+1=1;
A+A=1
A•0 =0 A•A=0
A•1=A
(01律/互补)
6) Theorem 6
7) Theorem 7
A+A=A;
A•A=A
(重叠律)
(还原律)
AA
A•B = A+B
8) DeMorgan’s theorems A+B = A•B Deductions (推论）A+B+C=A•B•C
1
§3.1 Operations of Logic Algebra
逻辑代数运算法则
AND operation 与 Logic multiplication OR operation或 Logic addition “+” 0+0=0 0+1=1 1+0=1 1+1=1 The complement 非 of A is A
14
Example:
A B C: 010
Example: 2 variables A, B:
ABC 1
010 = 2
so minterm ABC is numbered m2
m1= AB,
m3 = AB
4 variables A, B, C, D:
m1 = A B C D m5 = ABCD m13 = ABCD
Right: A BC A B C
We have used it in the “Deduction” of Morgan’s Theorems. 多个变量摩根定理
4
2) Complementary Theorem 反演规则
设F为逻辑函数，如果将函数表达式中所有的与(•) 换成或(+),或(+)换成与 (•);0换成1，1换成0；原变量换成反变量，反变量换成原变量，则得到的逻 辑函数是F的反函数F The complement expression is formed by replacing all + operations with • , all • operations with +, all ones with zeros, all zeros with ones, all variables with inversed variables, and all inversed variables with variables.
F
1 + uncomplemented variable 原变量
0 • complemented variable反变量
New function
F
Function F is called the complement of the function F. if the function F is valid, the complement of the function, F , is also valid.若原函数成立，反函数也成立
Chapter 3
Logic Algebra 逻辑代数基础
Logic Algebra is presented by George Boole in 1849 and is also called Boolean Algebra. Logic algebra constitute a mathematical tool for describing the input-output behavior of the logical gates. 逻辑代数描述二值变量(0,1)的运算规律。由英国数学家布尔在19 世纪中提出，也称为布尔代数。
A variable is a symbol used to represent a logical quantity. Any single variable can have a 1 or a 0 value.
逻辑代数中变量是表示逻辑量，只能取0或1.
Logic algebra and arithmetic are different. 逻辑代数与算术运算不同。
证明(Proof): A+AB = A (1+B) = A
Absorption 吸收律
2)
A+AB A B;
A(A+B)=AB
吸收律
Proof: Using distributive law (分配律 A+BC=(A+B)(A+C) )
A+AB (A A)(A B) A B
A C D B; B C D A;
10
Causality (因果关系）
§3.2 Standard Forms of Logic Function
逻辑函数的标准形式 逻辑函数有两种标准形式：标准与或式（最小项和） 标准或与式（最大项积）
3.2.1 Minterms and Standard Sum of Products
other.
(每条定理有两种表达形式：逻辑加及逻辑乘。对偶式)
2
基本定律
1) Theorem 1 2) Theorem 2 3) Theorem 3
逻辑加
A+B=B+A; A+(B+C)=(A+B)+C;
逻辑乘
AB=BA A(BC)=(AB)C (交换律) (结合律)
A +BC= (A+B)(A+C); A(B+C)=AB+AC; (分配律)
ABC ABC ABC ABC ABC
ABC
ABC ABC
12
2. Truth Table of Minterms 最小项真值表
Minterm Number
Variables
m0
m1
m2
m3
m4
m5
m6
m7
A B C 0 0 0
Minterm
ABC ABC ABC ABC ABC ABC ABC ABC
标准与项(最小项)：n 变量函数, n 变量组成的与项中, 每个变 量都以原变量或反变量形式出现一次 , 且只出现一次。若变量数 为n，则有2n个最小项。
n variables 2n minterms
For example: 3 variables A, B, C, there are 23 8 minterms:
与项：多个变量(原变量或反变量)的乘积形式
AB
BCD
AE
11
Minterms (Standard Product Terms): For a function with n variables if a product term contains each of the n variables exactly one time in complemented or uncomplemented form, the product term is called a Minterm or a Standard Product Term.
Solution: F' = (A+BC)(C+D) F = (A+BC)(C+D)
Example 2:
G WX Y Z X
G' (W X)Y Z X
G (W X)Y Z X
7
3. Formula of Boolean Algebra
常用公式
1) A+AB=A; A(A+B)=A
“•”
0•0 = 0 0•1 = 0 1•0 = 0 1•1 = 1
0 1 1 0
1. Fundamental Theorems of Logic Algebra 基本定律
Every theorem is given in two forms: one for addition(加) and another for multiplication(乘). Two forms are equivalent and are called “Dual” each
(摩根定理)
ABC A B C
3
2. Basic Rules 基本规则
1) Substitution 代入规则
等式两侧某一变量都用一个逻辑函数代入，等式仍成立。
For example: If Left: So
AX A X
AX ABC ABC A B C
X BC
1 0 0 0 0 0
0
0 1
0 0 0 1 0
0 0 0 0
0
0
0 0
0
0 0
0 0 1
0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
1
0 0
0
0 0 0 1 0 0
0
0 0
0
0 0
0
0 0 0 1
13
0
0 0 0
1
0 0
0
0 0
0
0 0
0
0
1
0
0
We can see that from the truth table, when variables A B C are given a set of numbers, only one minterm equals 1, and other minterms all equal 0. (当 A B C 取某一组值时, 只有一个最小项值为 1 , 其他 都等于 0 ) Minterm Number is defined as mi where i is the decimal integer equal to the corresponding binary code which makes the minterm 1. (使某一最小项为1时, 变量取值的二进制数对应的十 进制数为此最小项的编号)
A A 1,
A 0 A,
A 1 A
多变量异或，运算结果只与变量为1的个数有关，与变量为0的个 数无关。变量为 1 的个数为奇数，异或结果为 1；变量为 1 的个数为 偶数，结果为 0 。(变量与1异或一次，相当于取反一次)
6) If
A BC D
then
Hale Waihona Puke A B D C;=AB+AC
Deduction : AB AC BCDE AB AC
9
5) Exclusive-OR (XOR) Formula (异或公式) A⊕B = A⊙B Proof: AB AB (A B)(A B) AB AB A B
A A 0,
1: variable
0: complemented variable
1对应于原变量
最小项及标准与或式
1. Minterms (Standard Product Form) 最小项：也称为标准与项 A product term was defined as a term consisting of the product (logic multiplication) of literals (variables or their complements).