布尔代数与逻辑简化（PPT讲稿）Boolean Algebra and Logic Simplification

4 Boolean Algebra and Logic Simplification (重点

1 4 Boolean Algebra and Logic Simplification 重点

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

2 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

4-4 Boolean Anal ysis of Logic Circuits (逻辑电路分析) 1.Logic Circuit(逻辑电路图) D CDB (3+CD To derive the boolean expression for a given combinational logic circuit, begin at the left-most inputs and work toward the final output, writing the expression for each gate 2. Boolean Expression ⅹ=A(B+CD

3 4-4 Boolean Analysis of Logic Circuits （逻辑电路分析） 1.Logic Circuit (逻辑电路图） To derive the Boolean expression for a given combinational logic circuit , begin at the left-most inputs and work toward the final output , writing the expression for each gate. 2.Boolean Expression A ( B + C D ) C D B + C D X=A(B+CD)

4-4 Boolean Analysis of Logic Circuits 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 ble combinations 2)Draw the truth table frame according to the input and output variables

4 4-4 Boolean Analysis of Logic Circuits 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

4-4 Boolean Analysis of Logic Circuits (3)List all of the input variable combinations of Is and0 s in a binary sequence(按序) (4)Fill the truth table. If the input variable combinations make the output 1, then place a 1 in the corresponding output column, otherwise lace a o

5 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）. (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

4-4 Boolean Analysis of Logic Circuits Ex XFA(B+CD) Inputs Output A C0 A (B C 0) D 00000000 B0000 +00000000000 0000 0011001100 D010101010101010 6

6 4-4 Boolean Analysis of Logic Circuits Ex. X=A(B+CD)

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 which the output is I (2)Convert each binary value to the corresponding product term(积的形式) by replacing each I with the corresponding variable and each o with the complement B3) Get the logic expression by summing all the combinations

7 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 which the output is 1. (2) Convert each binary value to the corresponding product term （积的形式） by replacing each 1 with the corresponding variable and each 0 with the complement. (3) Get the logic expression by summing all the combinations

4-7 Boolean Expressions and Truth Table Ex. Truth table AB c F List the binary values where the 000 outputs are“1” 00001111 010 100. Convert each binary value to the 111 corresponding product 000 011. summing all the combinations 100 F:(+了 总结:逻辑图→逻辑表达式→真值表

8 4-7 Boolean Expressions and Truth Table A B C F 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 1 • List the binary values where the outputs are “1” • Convert each binary value to the corresponding product • summing all the combinations. Ex. Truth Table F = A B C + A B C + A B C 总结：逻辑图 逻辑表达式 真值表

引出本章的核心问题 AB+A(B+C)+B(B+C) B B+AC B A These iwo circuits are equivalent ◆通过与出两个逻辑图的表达式和具值表,逻辑 功能一样,但结构复杂程度不同,引出化简的 好处

引出本章的核心问题  通过写出两个逻辑图的表达式和真值表，逻辑 功能一样，但结构复杂程度不同，引出化简的 好处。 9

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+btc+D a sum term is equal to I when one or more of the literals in the term are 1 a sum term is equal to o only if each of the literals is 0

10 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.  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. X = A+ B, X = A+ B, X = A + B +C + D