《电子技术数字基础-Digital-Fundamentals》双语课件
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Associative laws (结合律） A+(B+C)=(A+B)+C A(BC)=(AB)C
Distributive Law (分配律） A(B+C)=AB+AC
6
4-2-2 Rules of Boolean Algebra (基本公式）
1. A+0= A A variable ORed with 0 is always equal to the variable.
2
4-1 Boolean Operations and Expressions
Boolean algebra (布尔代数） is the mathematics of digital systems.
Sum term （和的形式）: is a sum of literals (文字) ( a variable or the complement of a variable), produced by an OR gate.
X A B, X A B, X A B C D
A sum term is equal to 1 when one or more of the literals in the term are 1.
A sum term is equal to 0 only if each of the literals is 0.
19
4-4 Boolean Analysis of Logic Circuits
Ex. X=A(B+CD)
20
4-4 Boolean Analysis of Logic Circuits 4-7 Boolean Expressions and Truth Table
Determining logic expressions from a truth table (1) List the binary values of the input variables for
AB B A
ACDE
CDE B B CDE
13
4-3 Basic Theorems
2. Demorgan’s theorem （摩尔定理）
For a given expression X, if
01，10，+，+, AA, AA
then X (反函数)got.
7
4-2-2 Rules of Boolean Algebra
5. A+A= A A variable ORed with itself is always equal to the variable.
6. A·A= A A variable ANDed with itself is always equal to the variable.
4-4 Boolean Analysis of Logic Circuits
（逻辑电路分析）
1. Logic Circuit (逻辑电路图）
CD
B CD
A(B CD)
2. Boolean Expression （布尔表达式） X=A(B+CD)
17
4-4 Boolean Analysis of Logic Circuits 4-7 Boolean Expressions and Truth Table
3. Truth Table （真值表） Constructing the truth table from a logic expression.
(1) Determine the number of the input and output variables, and the number of the input variable possible combinations.
(2) Draw the truth table frame according to the input and output variables
18
4-4 Boolean Analysis of Logic Circuits
(3) List all of the input variable combinations of 1s and 0s in a binary sequence (按序）.
12
4-3 Basic Theorems
1. Replacement theorem (代入定理）
Any variable in laws or rules of Boolean expressions can also be replaced with a combination of other variables or combinations.
X X X
Procedure: Step 1: Replace OR with AND, AND with OR; Step 2: Complement each literal.
X ABC A CD BD
X (A B C )A(C D)(B D)
14
4-3 Basic Theorems
9
4-2-2 Rules of Boolean Algebra
9. A A The double complement of a variable is always equal to the variable.
10. A+AB= A
10
4-2-2 Rules of Boolean Algebra
11. (A+B)(A+C)=A+BC
12. A AB A B
A AB ( A A)( A B) A B
11
4-2-2 Rules of Boolean Algebra
13. AB AC BC AB BC AB AC BC AB AC ( A A)BC AB ABC AC ABC AB AC
2. A+1= 1 A variable ORed with 1 is always equal to 1.
3. A·0= 0 A variable ANDed with 0 is always equal to 0.
4. A·1= A A variable ANDed with 1 is always equal to the variable.
which the output is 1. (2) Convert each binary value to the corresponding
product term by replacing each 1 with the corresponding variable and each 0 with the complement. (3) Get the logic expression by summing all the combinations.
3. Duality theorem （对偶定理）
For a given expression X, if
X prime
01，10，+，+
then X’ (对偶式) got X X’
If X=Y, then X’=Y’
The dualistic expression of X
F ABC ABC ABC
5
4-2 Laws and Rules of Boolean Algebra 4-2-1 Laws of Boolean Algebra (布尔代数常用公式）
Commutative laws (交换律）for addition and multiplication A+B=B+A; AB=BA
XY X Y
The complement of two or more variables ANDed is equal to the OR of the complements of the individual variables.
X Y X Y
15
4-3 Basic Theorems
4 Boolean Algebra and Logic Simplification
1
Contents
Boolean Operations and Expressions Law and Rules of Boolean Algebra DeMorgan’s Theorems Boolean Analysis of Logic Circuits Simplification Using Boolean Algebra Standard Forms of Boolean Expressions Boolean Expressions and Truth Tables The Karnaugh Map Karnaugh Map SOP Minimization
21
4-7 Boolean Expressions and Truth Table
Ex. Truth Table
ABC
F
000 0 001 0 010 0 011 1
100 0 101 1
110 0 111 1
List the binary values where the outputs are “1” Convert each binary value to the corresponding product summing all the combinations.
8
4-2-2 Rules of Boolean Algebra
7. A A 1
A variable ORed with its complement is always equal to 1.
8. A A 0
A variable ANDed with its complement is always equal to 0.
Ex. X ACD 0 1 0 1 Only if A=0,C=1, and D=0, then X=1.
4
4-1 Boolean Operations and Expressions
ABC
F
000 0 001 0 010 0 011 1
100 0 101 1
110 0 111 1
3
4-1 Boolean Operations and Expressions
Product term （积的形式）: is the product of literals, produced by an AND gate. X AB, X AB, X ACD
A product term is equal to 1 only if each of the literals in the term is 1. A product term is equal to 0 when one or more of the literals are 0.
(4) Fill the truth table. If the input variable combinations make the output 1, then place a 1 in the corresponding output column, otherwise place a 0.
F ABC ABC ABC
22
4-4 Boolean Analysis of Logic Circuits
4. The Karnaugh Map （卡诺图） The Karnaugh map is similar to a truth table, it can make the simplification procedure of Boolean expression easy.
23
4-6 Standard Forms of Boolean Expressions
1. The SOP Form: (Sum-of-Products) (与或式） The expression is a sum of two or more products of literals.
Distributive Law (分配律） A(B+C)=AB+AC
6
4-2-2 Rules of Boolean Algebra (基本公式）
1. A+0= A A variable ORed with 0 is always equal to the variable.
2
4-1 Boolean Operations and Expressions
Boolean algebra (布尔代数） is the mathematics of digital systems.
Sum term （和的形式）: is a sum of literals (文字) ( a variable or the complement of a variable), produced by an OR gate.
X A B, X A B, X A B C D
A sum term is equal to 1 when one or more of the literals in the term are 1.
A sum term is equal to 0 only if each of the literals is 0.
19
4-4 Boolean Analysis of Logic Circuits
Ex. X=A(B+CD)
20
4-4 Boolean Analysis of Logic Circuits 4-7 Boolean Expressions and Truth Table
Determining logic expressions from a truth table (1) List the binary values of the input variables for
AB B A
ACDE
CDE B B CDE
13
4-3 Basic Theorems
2. Demorgan’s theorem （摩尔定理）
For a given expression X, if
01，10，+，+, AA, AA
then X (反函数)got.
7
4-2-2 Rules of Boolean Algebra
5. A+A= A A variable ORed with itself is always equal to the variable.
6. A·A= A A variable ANDed with itself is always equal to the variable.
4-4 Boolean Analysis of Logic Circuits
（逻辑电路分析）
1. Logic Circuit (逻辑电路图）
CD
B CD
A(B CD)
2. Boolean Expression （布尔表达式） X=A(B+CD)
17
4-4 Boolean Analysis of Logic Circuits 4-7 Boolean Expressions and Truth Table
3. Truth Table （真值表） Constructing the truth table from a logic expression.
(1) Determine the number of the input and output variables, and the number of the input variable possible combinations.
(2) Draw the truth table frame according to the input and output variables
18
4-4 Boolean Analysis of Logic Circuits
(3) List all of the input variable combinations of 1s and 0s in a binary sequence (按序）.
12
4-3 Basic Theorems
1. Replacement theorem (代入定理）
Any variable in laws or rules of Boolean expressions can also be replaced with a combination of other variables or combinations.
X X X
Procedure: Step 1: Replace OR with AND, AND with OR; Step 2: Complement each literal.
X ABC A CD BD
X (A B C )A(C D)(B D)
14
4-3 Basic Theorems
9
4-2-2 Rules of Boolean Algebra
9. A A The double complement of a variable is always equal to the variable.
10. A+AB= A
10
4-2-2 Rules of Boolean Algebra
11. (A+B)(A+C)=A+BC
12. A AB A B
A AB ( A A)( A B) A B
11
4-2-2 Rules of Boolean Algebra
13. AB AC BC AB BC AB AC BC AB AC ( A A)BC AB ABC AC ABC AB AC
2. A+1= 1 A variable ORed with 1 is always equal to 1.
3. A·0= 0 A variable ANDed with 0 is always equal to 0.
4. A·1= A A variable ANDed with 1 is always equal to the variable.
which the output is 1. (2) Convert each binary value to the corresponding
product term by replacing each 1 with the corresponding variable and each 0 with the complement. (3) Get the logic expression by summing all the combinations.
3. Duality theorem （对偶定理）
For a given expression X, if
X prime
01，10，+，+
then X’ (对偶式) got X X’
If X=Y, then X’=Y’
The dualistic expression of X
F ABC ABC ABC
5
4-2 Laws and Rules of Boolean Algebra 4-2-1 Laws of Boolean Algebra (布尔代数常用公式）
Commutative laws (交换律）for addition and multiplication A+B=B+A; AB=BA
XY X Y
The complement of two or more variables ANDed is equal to the OR of the complements of the individual variables.
X Y X Y
15
4-3 Basic Theorems
4 Boolean Algebra and Logic Simplification
1
Contents
Boolean Operations and Expressions Law and Rules of Boolean Algebra DeMorgan’s Theorems Boolean Analysis of Logic Circuits Simplification Using Boolean Algebra Standard Forms of Boolean Expressions Boolean Expressions and Truth Tables The Karnaugh Map Karnaugh Map SOP Minimization
21
4-7 Boolean Expressions and Truth Table
Ex. Truth Table
ABC
F
000 0 001 0 010 0 011 1
100 0 101 1
110 0 111 1
List the binary values where the outputs are “1” Convert each binary value to the corresponding product summing all the combinations.
8
4-2-2 Rules of Boolean Algebra
7. A A 1
A variable ORed with its complement is always equal to 1.
8. A A 0
A variable ANDed with its complement is always equal to 0.
Ex. X ACD 0 1 0 1 Only if A=0,C=1, and D=0, then X=1.
4
4-1 Boolean Operations and Expressions
ABC
F
000 0 001 0 010 0 011 1
100 0 101 1
110 0 111 1
3
4-1 Boolean Operations and Expressions
Product term （积的形式）: is the product of literals, produced by an AND gate. X AB, X AB, X ACD
A product term is equal to 1 only if each of the literals in the term is 1. A product term is equal to 0 when one or more of the literals are 0.
(4) Fill the truth table. If the input variable combinations make the output 1, then place a 1 in the corresponding output column, otherwise place a 0.
F ABC ABC ABC
22
4-4 Boolean Analysis of Logic Circuits
4. The Karnaugh Map （卡诺图） The Karnaugh map is similar to a truth table, it can make the simplification procedure of Boolean expression easy.
23
4-6 Standard Forms of Boolean Expressions
1. The SOP Form: (Sum-of-Products) (与或式） The expression is a sum of two or more products of literals.