逻辑代数与硬件描述语言基础.ppt

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1、2 .逻辑代数与硬件描述语言基础逻辑代数与硬件描述语言基础2.1 逻辑代数逻辑代数 2.2 逻辑函数的卡诺图化简法逻辑函数的卡诺图化简法 教学基本要求教学基本要求1 1、熟悉逻辑代数常用基本定律、恒等式熟悉逻辑代数常用基本定律、恒等式和规则。和规则。2 2、掌握逻辑代数的变换和卡诺图化简法；、掌握逻辑代数的变换和卡诺图化简法；Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Inform

2、ation Science Department of Electronics and Information Science http:/http:/ 逻 辑 代 数2.1.1 逻辑代数的基本定律与恒等式逻辑代数的基本定律与恒等式2.1.2 逻辑代数的基本规则逻辑代数的基本规则2.1.3 逻辑代数的代数变换与化简法逻辑代数的代数变换与化简法Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics an

3、d Information Science Department of Electronics and Information Science http:/http:/ 序号序号公式公式a公式公式b名称名称1A + 0=AA 0 = 00、1律律2A + 1 =1A 1 = A3A + A =AA A = A重叠律重叠律4 互互补律律5A + ( B + C)= (A + B) +CA (B C) = (A B) C结合律合律6A + B = B + AA B = B A交交换律律7A (B + C) = A B +A CA + B C= (A + B) (A + C)分配律分配律8反演律反演

4、律9还原律原律2.1.1 逻辑代数的基本定律和恒等式逻辑代数的基本定律和恒等式Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ (真值表证明真值表证明)例例 证明证明，按按A、B取值取值 A BA BA+BA+B0 01 1

5、0+0=1100 = 110 11 00+1=0001 = 111 00 11+0=0010 = 111 10 01+1=0011 = 00，情况列出真值表，从表中可以直接得出结果。情况列出真值表，从表中可以直接得出结果。2.1.1 逻辑代数的基本定律和恒等式逻辑代数的基本定律和恒等式Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Departm

6、ent of Electronics and Information Science http:/http:/ 2.1.2 逻辑代数的基本规则 1. 代入规则代入规则 2.2. 反演规则反演规则3.3. 对偶规则对偶规则1.代入规则代入规则代入规则代入规则： 在任何一个包含变量在任何一个包含变量A逻辑等式中，如果用另一个函数逻辑等式中，如果用另一个函数式代入式中式代入式中A的位置，的位置， 则等式仍然成立。这一规则称为代则等式仍然成立。这一规则称为代入规则。入规则。例：例：B (A + C) = BA+BC，用用A + D代替代替A，得得B (A +D) +C = B(A +D) + BC =

7、 BA + BD + BCDepartment of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 2. 反演规则反演规则反演规则反演规则：将逻辑表达式L中的与（ ）换成或（+），或（+）换成与（）；再将原变量换为非变量，非变量换为原变量；并将1换

8、成0，0换成1；那么，所得的函数式就是 。注意事项：注意事项： (1) (1) 保持原来的运算优先顺序保持原来的运算优先顺序. . (2) (2) 对于反变量以外的非号应保留不变。对于反变量以外的非号应保留不变。 2.1.2 逻辑代数的基本规则逻辑代数的基本规则 Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Elec

9、tronics and Information Science http:/http:/ 3. 对偶规则对偶规则对偶规则对偶规则：将逻辑表达式L中的与（ ）换成或（+），或（+）换成与（）；并将1换成0，0换成1；那么，所得的函数式就是L的对偶式，记作 。 例例 试证明试证明 A+BC=(A+B)(A+C)分别写出其对偶式：分别写出其对偶式：A(B+C) AB+AC由分配律知：由分配律知：A(B+C) = AB+AC 故故 A+BC=(A+B)(A+C) 2.1.2 逻辑代数的基本规则逻辑代数的基本规则 Department of Electronics and Information Sci

10、ence Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 2.1.3 逻辑函数的代数变换与化简法逻辑函数的代数变换与化简法“与或与或” “或与或与” “与非与非与非与非” “或非或非或非或非” “与或非与或非” “与非或非与非或非” “与或与或” 常见的几种逻辑函数表达式常见的几种逻辑函数表达式Department of

11、 Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ of Electronics and Information Science Department of Electronics and Information Science Department

12、of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 2.1.3 逻辑函数的代数变换与化简法逻辑函数的代数变换与化简法与非与非-与非式与非式或非或非-或非式或非式“与非或非与非或非” Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Inf

13、ormation Science Department of Electronics and Information Science http:/http:/ “与或与或”表达式：表达式： 相与项（即乘积项）的个数最少；相与项（即乘积项）的个数最少；（门的个数少）（门的个数少） 每个相与项中，所含的变量个数最少每个相与项中，所含的变量个数最少 （门的输入端少）。（门的输入端少）。 化简后电路简单、可靠性高化简后电路简单、可靠性高 2.1.3 逻辑函数的代数变换与化简法逻辑函数的代数变换与化简法Department of Electronics and Information Science D

14、epartment of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 运用逻辑代数的基本定律和恒等式进行化简的方法。运用逻辑代数的基本定律和恒等式进行化简的方法。 方法：方法：并项法并项法: : 吸收法：吸收法： A + AB = A 消去法：消去法： 配项法：配项法： A+AB=A+B 2.1.3 逻辑函数的代数化简与化简法逻辑函数的代数化简

15、与化简法Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 和和 。例如：例如：配项法配项法： 或或 。例如：例如： 2.1.3 逻辑函数的代数化简与化简法逻辑函数的代数化简与化简法Department of Electron

16、ics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 2.1.3 逻辑函数的代数化简与化简法逻辑函数的代数化简与化简法Department of Electronics and Information Science Department of Electronics a

17、nd Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics

18、 and Information Science http:/http:/ 逻辑函数的卡诺图化简法逻辑函数的卡诺图化简法2.2.1 最小项的定义及性质最小项的定义及性质2.2.2 逻辑函数的最小项表达式逻辑函数的最小项表达式2.2.3 用卡诺图表示逻辑函数用卡诺图表示逻辑函数2.2.4 用卡诺图化简逻辑函数用卡诺图化简逻辑函数Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Inform

19、ation Science Department of Electronics and Information Science http:/http:/ 逻辑函数的最小项的定义及其性质 n变量的最小项，是变量的最小项，是n个因子的乘积，每个变量都以个因子的乘积，每个变量都以它的原变量或非变量的形式它的原变量或非变量的形式在乘积中在乘积中出现，且只出出现，且只出现一次。现一次。1 1、最小项的定义：、最小项的定义：如三变量逻辑函数如三变量逻辑函数 f(A B C)A(B + C )A(B + C ) -不是最小项不是最小项-最小项最小项Department of Electronics and

20、Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 三个变量的所有最小项的真值表三个变量的所有最小项的真值表 m0m1m2m3m4m5m6m7最小项的表示：通常用最小项的表示：通常用mi表示最小项，表示最小项，m表示最小项表示最小项,下标下标 i为最为最小项编号。小项编号。 00010

21、000000001010000000100010000010000001000011000100001010000010011000000010111000000012.2.1 2.2.1 2.2.1 2.2.1 最小项的定义及其性质最小项的定义及其性质最小项的定义及其性质最小项的定义及其性质 Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science De

22、partment of Electronics and Information Science http:/http:/ B C0 0 0100000000 0 1010000000 1 0001000000 1 1000100001 0 0000010001 0 1000001001 1 0000000101 1 100000001l 对于任意一个最小项，对于任意一个最小项，只有一组变量取值使得它只有一组变量取值使得它的值为的值为1 1；l 不同的最小项，使它不同的最小项，使它的值为的值为1 1的那一组变量取的那一组变量取值也不同；值也不同；l 对于变量的任一组取对于变量的任一组取值，任意两

23、个最小项的乘值，任意两个最小项的乘积为积为0 0；l 对于变量的任一组取对于变量的任一组取值，全体最小项之和为值，全体最小项之和为1 1。2 2、最小项的性质、最小项的性质 2.2.1 2.2.1 最小项的定义及其性质最小项的定义及其性质 Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and

24、Information Science http:/http:/ 2.2.2 逻辑函数的最小项表达式逻辑函数的最小项表达式 逻辑函数的最小项表达式：逻辑函数的最小项表达式： l 为为“与或与或”逻辑表达式；逻辑表达式； l 在在“与或与或”式中的每个乘积项都是最小项。式中的每个乘积项都是最小项。例例1 将将化成最小项表达式化成最小项表达式= m7m6m3m1 唯一的唯一的唯一的唯一的Department of Electronics and Information Science Department of Electronics and Information Science Departm

25、ent of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 例例2 将将 化成最小项表达式化成最小项表达式 去掉非号去掉非号去括号去括号 将将AB乘以乘以 2.2.2 逻辑函数的最小项表达式 可见，任一逻辑函数都可以化成唯一的最小项表达式可见，任一逻辑函数都可以化成唯一的最小项表达式可见，任一逻辑函数都可以化成唯一的最小项表达式可见，任一逻辑函数都可以化成唯一的最小项表达式Department of Electronics and Inform

26、ation Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 2.2.3 用卡诺图表示逻辑函数用卡诺图表示逻辑函数 将一个逻辑函数最小项表达式中的各最小项相应地将一个逻辑函数最小项表达式中的各最小项相应地填入一个特定的方格图内，此方格图就称为卡诺图。填入一个特定的方格图内，此方格图就称为卡诺图。几何相邻几何

27、相邻某一方格和其它方格具有共同的边某一方格和其它方格具有共同的边 逻辑相邻逻辑相邻对于两个最小项，组成它们的变对于两个最小项，组成它们的变 量中，只有一个不同，其余都相同量中，只有一个不同，其余都相同.如如1、卡诺图：、卡诺图： 逻辑函数的图形表示法。逻辑函数的图形表示法。2、卡诺图的特点：、卡诺图的特点： 几何相邻对应着逻辑相邻几何相邻对应着逻辑相邻Department of Electronics and Information Science Department of Electronics and Information Science Department of Electroni

28、cs and Information Science Department of Electronics and Information Science http:/http:/ m0 m1 m2 m3 m4 m5 m6 m7 m12 m13 m14 m15 m8 m9 m10 m110001111000011110ABCD 2.2.3 用卡诺图表示逻辑函数用卡诺图表示逻辑函数 一变量卡诺图一变量卡诺图三变量卡诺图三变量卡诺图四变量卡诺图四变量卡诺图两变量卡诺图两变量卡诺图ABCDBCA m0 m1 m2 m3 m4 m5 m6 m7m0m1AAL=m0+m1=m0+m1+m2+m3m0m1m

29、2m3LABm2m314m104Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 将逻辑函数化为最小项表达式；将逻辑函数化为最小项表达式； 2. 填写卡诺图。填写卡诺图。 例例1 用卡诺图表示逻辑函数用卡诺图表示逻辑函数。

30、2.2.3 用卡诺图表示逻辑函数用卡诺图表示逻辑函数 Lm0m3m2m4m6m5m7m111111000解解1. 将逻辑函数化为最小项表达式；将逻辑函数化为最小项表达式；2. 填写卡诺图。填写卡诺图。 Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science

31、 http:/http:/ 用卡诺图表示逻辑函数用卡诺图表示逻辑函数 画出下式的卡诺图画出下式的卡诺图例例2解解1. 将逻辑函数化为最小项表达式；将逻辑函数化为最小项表达式；2. 填写卡诺图。填写卡诺图。Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Scienc

32、e http:/http:/ 2.2.4 用卡诺图化简逻辑函数 1、卡诺图化简的依据、卡诺图化简的依据 相邻项相加时，反复应用，相邻项相加时，反复应用， 公式，函数表达式的公式，函数表达式的项数和每项所含的因子数就会减小项数和每项所含的因子数就会减小.Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronic

33、s and Information Science http:/http:/ A.画出逻辑函数的卡诺图。画出逻辑函数的卡诺图。B. 合并最小项，即将相邻的为合并最小项，即将相邻的为1的方格圈成一组。的方格圈成一组。 C. 将所有包围圈对应的乘积项相加。将所有包围圈对应的乘积项相加。 2.2.4 用卡诺图化简逻辑函数用卡诺图化简逻辑函数 Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and I

34、nformation Science Department of Electronics and Information Science http:/http:/ 4. 一个包围圈的方格数要尽可一个包围圈的方格数要尽可能多能多, ,包围圈的数目要可能少。包围圈的数目要可能少。3.3.同一方格可以被不同的包同一方格可以被不同的包围圈重复包围多次，但新围圈重复包围多次，但新增的包围圈中一定要有原增的包围圈中一定要有原有包围圈未曾包围的方格。有包围圈未曾包围的方格。1.1.包围圈内的方格数一定是包围圈内的方格数一定是2 2n n个，且包围圈必须个，且包围圈必须呈矩形。呈矩形。2.2.循环相邻特性包括

35、上下底相邻，左右边相邻和循环相邻特性包括上下底相邻，左右边相邻和四角相邻。四角相邻。画包围圈时应遵循的原则：画包围圈时应遵循的原则： 2.2.4 用卡诺图化简逻辑函数用卡诺图化简逻辑函数 XDepartment of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:

36、/http:/ 例例1 用卡诺图化简逻辑函数用卡诺图化简逻辑函数1111111111Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 2 用卡诺图化简逻辑函数用卡诺图化简逻辑函数11111111111111111111Dep

37、artment of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 例例3 用卡诺图化简逻辑函数用卡诺图化简逻辑函数1111111111111100该例说明：画包围圈时，可包围1，也可包围0Department of Electronics and

38、 Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 含无关项的逻辑函数及其化简无关项：无关项：无关项：无关项：1、填、填卡诺图时，在对应的方格内填任意符号卡诺图时，在对应的方格内填任意符号“”。处理方法：处理方法：处理方法：处理方法：2、化化简简时时根根据据需需要要可可将将“”视视为

39、为“1”，也也可可视视为为“0”。 真真值值表表内内对对应应于于某某些些变变量量组组合合，函函数数值值可可以以是是任任意意的的。或或者者说说，这这些些变变量量组组合合根根本本不不会会出出现现，则则这这些些变变量量组组合合对对应应的的最最小小项项称称为为无无关关项项，也也称称任任意意项项。所所谓谓任任意意项项就就是是，其其取取值值是是任任意意的的，可可取取“1 1”，也可取，也可取“0 0”。Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science Department of Electronics and Information Science http:/http:/ 含无关项的逻辑函数及其化简含无关项的逻辑函数及其化简1 1、画出逻辑函数的、画出逻辑函数的卡诺图卡诺图BDBCA含无关项的逻辑函数化简举例含无关项的逻辑函数化简举例: :例例 试用卡诺图化简逻辑函数试用卡诺图化简逻辑函数化化简简时时可可根根据据需需要要视视为为“1 1”也也可可视视为为“0 0”，使使函函数数化化到到最最简简。2 2、化简逻辑函数、化简逻辑函数