1.
a) ABC' + (ABC')' = 1
b) (AB +CD')(AB +D'E) = AB(AB + CD') + D'E(AB +CD') =
ABAB + ABCD' + ABD'E+ CD'D'E = AB + ABCD' ABD'E + C'D'E = AB(1 + CD' + D'E) + C'D'E = AB + CD'E
c) A + B'C + D'(A + B'C) = (A + B'C)(1 + D') = A + B'C
d) AB'(C +D) + (C + D)' = AB' + (C + D)'
e) [(EF)' + AB + C'D'](EF) = (EF)'(EF) + ABEF + C'D'EF = ABEF C'D'EF = EF(AB + C'D')
f) (AB + C) +(D + EF)(AB+C)' = [(AB + C) + (D + EF)][(AB +C) + (AB + C)'] =
AB + C + D+ EF
2.
a) (A + B)(A + C')(A + D)(BC'D + E) = (AA + AC' + BA + BC')(ABC'D + AE + DBC'D + DE) =[A(1 +C' + B) + BC'][BC'D(A +1) + AE + DE] = (A + BC')(BC'D + AE + DE) = A(BC'D + AE +DE) + BC'(BC'D + AE + DE) = ABC'D + AAE + ADE + BC'BC'D + AEBC' BC'DE = BC'D + AE
b) (A + B' + C)(B' + C + D)(A' + C) = (AB' + AC + AD + B'B' + B'C + B'D + B'C + CC +
CD)(A' + C) = (B' + C + AD)(A' + C) = A'B' + B'C + A'C + CC + AA'D + ACD = A'B' + C
3.
a) DE + F'G' = (DE + F')(DE + G') = (D + F')(E + F')(D + G')(E + G')
b) WX' + WY'Z' + WYZ = W[X' + (Y'Z' + Y)(Y'Z' +Z)] = W(X' + Y + Y'Z')(X' + Z + Y'Z') =W(X' + Y + Y')(X' + Y + Z')(X' + Y' + Z)(X' + Z + Z') = W(X' + Y + Z')(X' + Y' + Z)
c) A'CD + E'F + BCD = (A' + B)CD + E'F = (A' + B + E'F)(CD + E'F) = (A' + B + E')
(A' + B + F)(E' + CD)(F + CD) = (A' + B + F)(A' + B + F)(E' + C)(E' + D)(F + C)(F + D)
4.
NOR
A' = A NOR A
A + B = [(A + B)']' = (A NOR B)' = (A NOR B) NOR (A NOR B)
AB = [(AB)']' = (A' + B')' = [(A NOR A) + (B NOR B)]' = (A NOR A) NOR (B NOR B)
NAND
A' = A NAND A
AB = [(AB)']' = (A NAND B)' = (A NAND B) NAND (A NAND B)
A + B = [(A + B)']' = (A'B')' = [(A NOR A)(B NOR B)]' = (A NAND A) NAND (B NAND B)
5.
A) F(A,B,C,D) = ∑(1,3,7,10,11);
AB\CD |
00 |
01 |
11 |
10 |
00 |
|
1 |
1 |
|
01 |
|
|
1 |
|
11 |
|
|
|
|
10 |
|
|
1 |
1 |
AB'C+A'CD+A'B'D
B) G(A,B,C,D) = ∏(0,2,4,8,9,10,14) + d(1,3,13,15);
AB\CD |
00 |
01 |
11 |
10 |
00 |
0 |
- |
- |
0 |
01 |
0 |
|
|
|
11 |
|
- |
- |
0 |
10 |
0 |
0 |
|
0 |
(A+B)*(B+C)*(A'+C'+D)*(A+C+D)
C) X(A,B,C,D,E) = ∑(1,3,7,10,11,14,15,20,21,27,28,29) +d(17,19)
AB\CDE |
000 |
001 |
011 |
010 |
110 |
111 |
101 |
100 |
00 |
|
1 |
1 |
|
|
1 |
|
|
01 |
|
|
1 |
1 |
1 |
1 |
|
|
11 |
|
- |
1 |
|
|
|
1 |
1 |
10 |
|
|
- |
|
|
|
1 |
1 |
A'B'C'E+C'DE+ABC'E+A'BD+A'CDE+ACD'
D) Y(A,B,C,D,E) = ∑(1,2,3,4,6,9,10,11,14,18,19,20,22,27,28,29) + d(0,8,26,30);
AB\CDE |
000 |
001 |
011 |
010 |
110 |
111 |
101 |
100 |
00 |
- |
1 |
1 |
1 |
1 |
|
|
1 |
01 |
- |
1 |
1 |
1 |
1 |
|
|
|
11 |
|
|
1 |
- |
- |
|
1 |
1 |
10 |
|
|
1 |
1 |
1 |
|
|
1 |
A'B'+C'D+CDE'+CD'E'+ACD'E'
E) Z(A,B,C,D,E) = ∏(0,1,2,3,6,7,8,9,20,21,22,23) + d(28,29,30,31);
AB\CDE |
000 |
001 |
011 |
010 |
110 |
111 |
101 |
100 |
00 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
01 |
0 |
0 |
|
|
|
|
|
|
11 |
|
|
|
|
- |
- |
- |
- |
10 |
|
|
|
|
0 |
0 |
0 |
0 |
(A'+C'+D')*(A'+B'+D)*(A+C)
F) H(A,B,C,D) = ∑(1,3,5,7,9,11);
AB\CD |
00 |
01 |
11 |
10 |
00 |
|
1 |
1 |
|
01 |
|
1 |
1 |
|
11 |
|
|
|
|
10 |
|
1 |
1 |
|
B'D+A'D
G) I(A,B,C,D) = ∑(0,2,5,7,8,10,13,15);
AB\CD |
00 |
01 |
11 |
10 |
00 |
1 |
|
|
1 |
01 |
|
1 |
1 |
|
11 |
|
1 |
1 |
|
10 |
1 |
|
|
1 |
BD+B'D'
H) J(A,B,C,D) =∏(1,3,5,7,8,9,11,12,13);
AB\CD |
00 |
01 |
11 |
10 |
00 |
|
0 |
0 |
|
01 |
|
0 |
0 |
|
11 |
0 |
0 |
|
|
10 |
0 |
0 |
0 |
|
B+(D'A)+(D'A')+C
I) K(A,B,C,D,E)= ∑(0,2,4,6,9,11,13,15,16,18,20,22,25,27,29,31);
AB\CDE |
000 |
001 |
011 |
010 |
110 |
111 |
101 |
100 |
00 |
1 |
|
|
1 |
1 |
|
|
1 |
01 |
|
1 |
1 |
|
|
1 |
1 |
|
11 |
|
1 |
1 |
|
|
1 |
1 |
|
10 |
1 |
|
|
1 |
1 |
|
|
1 |
BE + B'E'
J) L(A,B,C,D,E)= ∏(1,2,5,6,8,11,12,15,16,19,20,23,25,26,29,30).
AB\CDE |
000 |
001 |
011 |
010 |
110 |
111 |
101 |
100 |
00 |
|
0 |
|
0 |
0 |
|
0 |
|
01 |
0 |
|
0 |
|
|
0 |
|
0 |
11 |
|
0 |
|
0 |
0 |
|
0 |
|
10 |
0 |
|
0 |
|
|
0 |
|
0 |
(A + B + D + E')(A + B + D' + E)(A + B' + D + E)(A + B' + D' + E)
(A' + B' + D + E')(A' + B' + D' + E)(A' + B + D + E)(A' + B + D' + E')