Промежуточные таблицы истинности:¬B:
D∧A:
(D∧A)∧(¬B):
D | A | B | D∧A | ¬B | (D∧A)∧(¬B) |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 |
1 | 1 | 1 | 1 | 0 | 0 |
¬D:
A∨C:
(¬D)→(A∨C):
D | A | C | ¬D | A∨C | (¬D)→(A∨C) |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 1 | 1 |
0 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 1 | 0 | 1 | 1 |
1 | 1 | 0 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 | 1 |
B∧A:
¬((¬D)→(A∨C)):
D | A | C | ¬D | A∨C | (¬D)→(A∨C) | ¬((¬D)→(A∨C)) |
0 | 0 | 0 | 1 | 0 | 0 | 1 |
0 | 0 | 1 | 1 | 1 | 1 | 0 |
0 | 1 | 0 | 1 | 1 | 1 | 0 |
0 | 1 | 1 | 1 | 1 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 0 | 1 | 1 | 0 |
1 | 1 | 1 | 0 | 1 | 1 | 0 |
¬(B∧A):
B | A | B∧A | ¬(B∧A) |
0 | 0 | 0 | 1 |
0 | 1 | 0 | 1 |
1 | 0 | 0 | 1 |
1 | 1 | 1 | 0 |
(¬((¬D)→(A∨C)))∧(¬(B∧A)):
D | A | C | B | ¬D | A∨C | (¬D)→(A∨C) | ¬((¬D)→(A∨C)) | B∧A | ¬(B∧A) | (¬((¬D)→(A∨C)))∧(¬(B∧A)) |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
((D∧A)∧(¬B))∨((¬((¬D)→(A∨C)))∧(¬(B∧A))):
D | A | B | C | D∧A | ¬B | (D∧A)∧(¬B) | ¬D | A∨C | (¬D)→(A∨C) | ¬((¬D)→(A∨C)) | B∧A | ¬(B∧A) | (¬((¬D)→(A∨C)))∧(¬(B∧A)) | ((D∧A)∧(¬B))∨((¬((¬D)→(A∨C)))∧(¬(B∧A))) |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
Общая таблица истинности:
D | A | B | C | ¬B | D∧A | (D∧A)∧(¬B) | ¬D | A∨C | (¬D)→(A∨C) | B∧A | ¬((¬D)→(A∨C)) | ¬(B∧A) | (¬((¬D)→(A∨C)))∧(¬(B∧A)) | (D∧A∧¬B)∨¬(¬D→A∨C)∧¬(B∧A) |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
Логическая схема:
Совершенная дизъюнктивная нормальная форма (СДНФ):
По таблице истинности:
D | A | B | C | F |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 |
0 | 1 | 1 | 1 | 0 |
1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 0 |
F
сднф = ¬D∧¬A∧¬B∧¬C ∨ ¬D∧¬A∧B∧¬C ∨ D∧A∧¬B∧¬C ∨ D∧A∧¬B∧C
Логическая cхема:
Совершенная конъюнктивная нормальная форма (СКНФ):
По таблице истинности:
D | A | B | C | F |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 |
0 | 1 | 1 | 1 | 0 |
1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 0 |
F
скнф = (D∨A∨B∨¬C) ∧ (D∨A∨¬B∨¬C) ∧ (D∨¬A∨B∨C) ∧ (D∨¬A∨B∨¬C) ∧ (D∨¬A∨¬B∨C) ∧ (D∨¬A∨¬B∨¬C) ∧ (¬D∨A∨B∨C) ∧ (¬D∨A∨B∨¬C) ∧ (¬D∨A∨¬B∨C) ∧ (¬D∨A∨¬B∨¬C) ∧ (¬D∨¬A∨¬B∨C) ∧ (¬D∨¬A∨¬B∨¬C)
Логическая cхема:
Построение полинома Жегалкина:
По таблице истинности функции
D | A | B | C | Fж |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 |
0 | 1 | 1 | 1 | 0 |
1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 0 |
Построим полином Жегалкина:
F
ж = C
0000 ⊕ C
1000∧D ⊕ C
0100∧A ⊕ C
0010∧B ⊕ C
0001∧C ⊕ C
1100∧D∧A ⊕ C
1010∧D∧B ⊕ C
1001∧D∧C ⊕ C
0110∧A∧B ⊕ C
0101∧A∧C ⊕ C
0011∧B∧C ⊕ C
1110∧D∧A∧B ⊕ C
1101∧D∧A∧C ⊕ C
1011∧D∧B∧C ⊕ C
0111∧A∧B∧C ⊕ C
1111∧D∧A∧B∧C
Так как F
ж(0000) = 1, то С
0000 = 1.
Далее подставляем все остальные наборы в порядке возрастания числа единиц, подставляя вновь полученные значения в следующие формулы:
F
ж(1000) = С
0000 ⊕ С
1000 = 0 => С
1000 = 1 ⊕ 0 = 1
F
ж(0100) = С
0000 ⊕ С
0100 = 0 => С
0100 = 1 ⊕ 0 = 1
F
ж(0010) = С
0000 ⊕ С
0010 = 1 => С
0010 = 1 ⊕ 1 = 0
F
ж(0001) = С
0000 ⊕ С
0001 = 0 => С
0001 = 1 ⊕ 0 = 1
F
ж(1100) = С
0000 ⊕ С
1000 ⊕ С
0100 ⊕ С
1100 = 1 => С
1100 = 1 ⊕ 1 ⊕ 1 ⊕ 1 = 0
F
ж(1010) = С
0000 ⊕ С
1000 ⊕ С
0010 ⊕ С
1010 = 0 => С
1010 = 1 ⊕ 1 ⊕ 0 ⊕ 0 = 0
F
ж(1001) = С
0000 ⊕ С
1000 ⊕ С
0001 ⊕ С
1001 = 0 => С
1001 = 1 ⊕ 1 ⊕ 1 ⊕ 0 = 1
F
ж(0110) = С
0000 ⊕ С
0100 ⊕ С
0010 ⊕ С
0110 = 0 => С
0110 = 1 ⊕ 1 ⊕ 0 ⊕ 0 = 0
F
ж(0101) = С
0000 ⊕ С
0100 ⊕ С
0001 ⊕ С
0101 = 0 => С
0101 = 1 ⊕ 1 ⊕ 1 ⊕ 0 = 1
F
ж(0011) = С
0000 ⊕ С
0010 ⊕ С
0001 ⊕ С
0011 = 0 => С
0011 = 1 ⊕ 0 ⊕ 1 ⊕ 0 = 0
F
ж(1110) = С
0000 ⊕ С
1000 ⊕ С
0100 ⊕ С
0010 ⊕ С
1100 ⊕ С
1010 ⊕ С
0110 ⊕ С
1110 = 0 => С
1110 = 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 = 1
F
ж(1101) = С
0000 ⊕ С
1000 ⊕ С
0100 ⊕ С
0001 ⊕ С
1100 ⊕ С
1001 ⊕ С
0101 ⊕ С
1101 = 1 => С
1101 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 = 1
F
ж(1011) = С
0000 ⊕ С
1000 ⊕ С
0010 ⊕ С
0001 ⊕ С
1010 ⊕ С
1001 ⊕ С
0011 ⊕ С
1011 = 0 => С
1011 = 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 = 0
F
ж(0111) = С
0000 ⊕ С
0100 ⊕ С
0010 ⊕ С
0001 ⊕ С
0110 ⊕ С
0101 ⊕ С
0011 ⊕ С
0111 = 0 => С
0111 = 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 = 0
F
ж(1111) = С
0000 ⊕ С
1000 ⊕ С
0100 ⊕ С
0010 ⊕ С
0001 ⊕ С
1100 ⊕ С
1010 ⊕ С
1001 ⊕ С
0110 ⊕ С
0101 ⊕ С
0011 ⊕ С
1110 ⊕ С
1101 ⊕ С
1011 ⊕ С
0111 ⊕ С
1111 = 0 => С
1111 = 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 = 0
Таким образом, полином Жегалкина будет равен:
F
ж = 1 ⊕ D ⊕ A ⊕ C ⊕ D∧C ⊕ A∧C ⊕ D∧A∧B ⊕ D∧A∧C
Логическая схема, соответствующая полиному Жегалкина: