习题二
2.1:设A 、B 、C 为逻辑变量,试回答:
(1) 若已知A+B=A+C,则B=C,对吗?
(2) 若已知AB=AC,则B=C,对吗?
(3) 若已知A+B=A+C且AB=AC,则B=C,对吗?
解:(1)不对,当A=1时,B 可以不等于C 。
(2)不对,当A=0时,B 可以不等于C 。
(3)对。
2.2:指出使下列逻辑函数为1的变量取值组合 (1)F (A , B ) =A B +AB (2) F (A , B , C ) =AB +A C (3) F (A , B , C , D ) =(A +B C ) D +(A +B ) CD
解:(1) A、B=00和11时F=1,否则F=0 (2) F (A , B , C ) =AB +A C A,B 为11或A,C 为10时F=1,即A,B,C 的取值为100,110,111时F=1,否则F=0。 (3) F =A D +B C D +ACD +B CD
A,D 为10或B,C,D 为000, 或A,C,D 为111, 或B,C,D 为011时F=1,即A,B,C,D 的取值为0000, 0011,1000,1010, 1011,1100,1110, 1111时F=1,否则F=0。
2.3、用代数法证明下列等式
(1) AB +A C +BCD =AB +A C
(2) AB +A C +B C =AB +C
(3) A B +B D +A D +DC =A B +D
(4) BC +D +D (B +C )(DA +B ) =B +D
(5) (A+B)(A+B )(A +B)(A +B ) =0
(6) A B +B C +C A =A B +B C +C A
(7) (AB+A B )(BC +B C )(CD+C D ) =A B +B C +C D +D A (8)A ⊕B =A ⊕B (9)A +B +C +D +A +B +C +D =A B +AB +C D
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(10)B C D +B C D +ACD +A B C D +A B CD +B C D +BCD =B C +BD +B C 解: (1) AB +A C +BCD =AB +A C +BC +BCD =AB +A C +BC =AB +A C
(2) AB +A C +B C =AB +(A +B )C =AB +AB C =AB +C
(3) A B +B D +A D +DC =A B +B D +AD +A D +DC
=A B +B D +D +DC =A B +D
(4) BC +D +(+)(DA +B ) =BC +D +(+)(DA +B )
=BC +D +BC (DA +B ) =BC +D +DA +B =B +D
(5) (A+B)(A+B )(A +B)(A +B ) =A •A =0
(6) A B +B C +C A =A B C +A B C +A B C +A B C +A B C +A BC =A B C +A BC +A B C +A B C +A B C +A B C =A B +B C +C A
(7) (AB+)(BC +)(CD+) =(ABC+)(CD+)
=ABCD +A B C D
A B +B C +C D +D A =(A +B )(B +C )(C +D )(D +A )
=(A B +A C +BC )(C D +A C +AD) =A B C D +ABC D
∴ (AB+A B )(BC +B C )(CD+C D ) =A B +B C +C D +D A (8)A ⊕B =A B +A B =A B +A B =A ⊕B (9)A +B +C +D +A +B +C +D =A B +AB +C D (10)B C D +B C D +ACD +A B C D +A B CD +B C D +BCD =B C +BD +B C
2.4、用真值表验证下列等式
(1) A B +A B =(A +B )(A +B )
(2) A +A (B +C ) =A +B +C
(3) A B +B C +C A =A B C +ABC
解:略
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2.5、用反演规则和对偶规则求下列函数的反函数F 和对偶函数F ' 。 (1) F =AB +A C
(2) F =A B +B C +A (C +D )
(3) F =A [B +(C D +D E ) G ]
(4) F =A +B +C +D +E
(5) F =A B C D B D A D E
(6) F =A B C +C D (AC +B D )
解: (1) F =(A +B )(A +C ) , F ' =(A +B )(A +C )
(2) F =(A +B )(B +C )(A +C D ) , F ' =(A +B )(B +C )(A +C D )
(3) F =A +B [(C +D )(D +E ) +G ], F ' =A +B [(C +D )(D +E ) +G ]
(4) F =A •B C •D E , F ' =A •B C •D E
(5) F =A +B +C +D +B +D +A +D +E
F ' =A +B +C +D +B +D +A +D +E
(6) F =(A +B +C )(C +D ) +(A +C )(B +D ) ,
F ' =(A +B +C )(C +D ) +(A +C )(B +D )
2.6、试用逻辑代数的公理、定理和规则化简下列逻辑函数 (1)F =A B +A B C (2) F =A BC +A (A C +B C ) (3) F =B (C +A D ) +B (C +A D ) (4) F =A B C D +A B CD +A B C D +A B CD (5) F =AB +A B +BC +B C (6) F =AB +(A B +A B ) C (7) F =A B +A +DE +A +B +G +(A +D) (A +B +E) D
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(8) F =AC +A BC +B C +AB C (9) F =A (B +C ) +A (B +C ) +BCD +B C D
解:(1)F =A B +A B C =B (A+A C) =B (A +C ) =A B +B C (2) F =A BC +A (A C +B C ) =A BC +AB C (3) F =B (C +A D ) +B (C +A D ) =BC +A BD +B C +A B D =C +A D (4) F =A B C D +A B CD +A B C D +A B CD =A B C +A B C =B C (5) F =AB +A B +BC +B C =AB +A B C +A B C +A BC +ABC +B C
=AB +ABC ++++BC =AB ++ (6) F =AB +(A B +A B ) C =AB +A B C +A BC =A (B +C ) +B (A +C )
=AB +AC +A B +BC =AB +AC +A B (7) F =A +DE +B +(++D) =A +DE +B +D =A +D +(8) F =AC +A BC +B C +AB C =(AC +A BC )(B +C )(A +B +C )
=(ABC +A BC )(A +B +C ) =BC (A +B +C ) =A BC +BC =BC (9) F =A (B +C ) +A (B +C ) +BCD +B C D
=AB +A C +A B +A C +BCD +B C D
=AB C +ABC +A C +A B +A B C +A BC +BCD +B C D
=A C +A B +BC +BCD +B C D =A C +A B +B C +BC +BCD +B C D
=A C +A B +B C +BC =A C +A B +BC
2.7 将下列函数转换成最小项表达式和最大项表达式:
(1)F (A , B , C ) =A +B C +A B C
(2) F (A , B , C ) =(A +B )(A +B +C )
(3) F (A , B , C , D )) =A B C D +BCD +AC
(4) F (A , B , C , D ) =AB +BC (C +D )
解:(1)F (A , B , C ) =A +B C +A B C =A B +AB +B C +A B C
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=A +A +AB +ABC ++A +B =∑m (1, 2, 4, 5, 6, 7) =∏M (0, 3)
(2) F (A , B , C ) =(A +B )(A +B +C )
=(A +B +C )(A +B +C )(A +B +C ) =∏M (0, 1, 7) =∑m (2, 3, 4, 5, 6)
(3) F (A , B , C , D )) =A B C D +BCD +AC
=A B C D +A BCD +ABCD +A B C D +A B CD +ABC D +ABCD
=∑m (7, 9, 10, 11, 14, 15) =∏M (0, 1, 2, 3, 4, 5, 6, 8)
(4) F (A , B , C , D ) =AB +BC (C +D ) =AB +BC +CD
=AB C D +AB C D +ABC D +ABCD +A BC D +A BCD +ABC D +ABCD
+A B CD +A BCD +A B CD +ABCD ∑m (3, 6, 7, 11, 12, 111, 14, 15) =∏M (0, 1, 2, 4, 5, 8, 910)
2.8 列出下述问题的真值表,并写出逻辑最大项表达式和最小项表达式:
(1)有A,B,C 三个输入信号,如果三个输入信号均为0或其中一个为1时,输出信号F 为1,其余情况下,输出F 为0。
解:依题意得真值表:
F(A,B,C)=∑m(0,1,2,4)=ΠM(3,5,6,7)
(2)有三个温度探测器,当探测的温度超过600C 时,输出控制信号为1,如果探测的温度低于600C 输出控制信号为0,当有两个或两个以上的温度探测器输出1信号时,总控制器输出1信号,自动控制调控设备,使温度降到600C 以下,试写出总控制器的真值表和逻
辑表达式。
F 表示,得:
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F(A,B,C)=∑m(3,5,6,7)=Π2.9用卡诺图将下列函数化简成最简与或表达式和最简或与表达式
(1) F =A B +B C +A B +B C
F =A B +B C +A C 或F =B C +A B +A C
(2) F (A , B , C , D ) =∑m (0, 3, 5, 6, 7, 10, 11, 13, 15)
F =BD +CD +BC +A +
(3) F (A , B , C , D ) =∏M (3, 4, 6, 7, 11, 12, 13, 14, 15)
F =B D +A B +B C +AC D
(4) F (A , B , C , D ) =∑m (2, 3, 6, 7, 8, 10, 12)
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F =A C +A C D +B C D 或F =A C +A C D +A B D
2.10用卡诺图化简下列包含无关项逻辑函数
(1) F (A , B , C , D ) =∑m (0, 1, 5, 7, 8, 11, 14) +∑d (3, 9, 15)
F =CD +D ++ABC
(2) F (A , B , C , D ) =C D (A ⊕B ) +A B C +A C D , (约束条件AB +CD =0)
解:F (A , B , C , D ) =C D (A ⊕B ) +A B C +A C D =A BC D +A B C D +A B C +A C D =
F =B +AC +A C D +ABC
(3) F (A , B , C , D ) =∑m (3, 5, 6, 7, 10) , (约束条件m 0+m 1+m 2+m 4+m 8=0)
F =B D +A D
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2.11化简下列多输出逻辑函数
F CD +A B C +A C D 1(A , B , C , D ) =A B C +A
F 2(A , B , C , D ) =BCD +ACD +B C D +A C D F (A , B , C , D ) =AC +BCD +A C +B C D 3
F 1=A B +A BD F 2=A BD +AD F 3=A +A BD
2.12 试用列表法化简下列逻辑函数
(1) F (A , B , C , D ) =∑m (4, 5, 6, 13, 14, 15)
(2) F (A , B , C , D , E ) =∑m (0, 4, 8, 10, 20, 26, 28, 30) 解:(1) F =A B C +BC D +ABD
(2) F =A B D E +A B C E +ABD E +AC D E
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