Промежуточные таблицы истинности:¬A:
¬C:
(¬A)∧B:
A | B | ¬A | (¬A)∧B |
0 | 0 | 1 | 0 |
0 | 1 | 1 | 1 |
1 | 0 | 0 | 0 |
1 | 1 | 0 | 0 |
((¬A)∧B)∨(¬C):
A | B | C | ¬A | (¬A)∧B | ¬C | ((¬A)∧B)∨(¬C) |
0 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 0 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 0 | 0 | 0 |
(((¬A)∧B)∨(¬C))∨A:
A | B | C | ¬A | (¬A)∧B | ¬C | ((¬A)∧B)∨(¬C) | (((¬A)∧B)∨(¬C))∨A |
0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 |
1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
¬D:
¬B:
¬((((¬A)∧B)∨(¬C))∨A):
A | B | C | ¬A | (¬A)∧B | ¬C | ((¬A)∧B)∨(¬C) | (((¬A)∧B)∨(¬C))∨A | ¬((((¬A)∧B)∨(¬C))∨A) |
0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
A∧B:
C∧(¬D):
C | D | ¬D | C∧(¬D) |
0 | 0 | 1 | 0 |
0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 0 |
(¬C)∨(¬((((¬A)∧B)∨(¬C))∨A)):
C | A | B | ¬C | ¬A | (¬A)∧B | ¬C | ((¬A)∧B)∨(¬C) | (((¬A)∧B)∨(¬C))∨A | ¬((((¬A)∧B)∨(¬C))∨A) | (¬C)∨(¬((((¬A)∧B)∨(¬C))∨A)) |
0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
(A∧B)→(C∧(¬D)):
A | B | C | D | A∧B | ¬D | C∧(¬D) | (A∧B)→(C∧(¬D)) |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 |
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
((A∧B)→(C∧(¬D)))→(¬B):
A | B | C | D | A∧B | ¬D | C∧(¬D) | (A∧B)→(C∧(¬D)) | ¬B | ((A∧B)→(C∧(¬D)))→(¬B) |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
(((A∧B)→(C∧(¬D)))→(¬B))≡((¬C)∨(¬((((¬A)∧B)∨(¬C))∨A))):
A | B | C | D | A∧B | ¬D | C∧(¬D) | (A∧B)→(C∧(¬D)) | ¬B | ((A∧B)→(C∧(¬D)))→(¬B) | ¬C | ¬A | (¬A)∧B | ¬C | ((¬A)∧B)∨(¬C) | (((¬A)∧B)∨(¬C))∨A | ¬((((¬A)∧B)∨(¬C))∨A) | (¬C)∨(¬((((¬A)∧B)∨(¬C))∨A)) | (((A∧B)→(C∧(¬D)))→(¬B))≡((¬C)∨(¬((((¬A)∧B)∨(¬C))∨A))) |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
Общая таблица истинности:
A | B | C | D | ¬A | ¬C | (¬A)∧B | ((¬A)∧B)∨(¬C) | (((¬A)∧B)∨(¬C))∨A | ¬D | ¬B | ¬((((¬A)∧B)∨(¬C))∨A) | A∧B | C∧(¬D) | (¬C)∨(¬((((¬A)∧B)∨(¬C))∨A)) | (A∧B)→(C∧(¬D)) | ((A∧B)→(C∧(¬D)))→(¬B) | A∧B→C∧¬D→¬B≡¬C∨¬(¬A∧B∨¬C∨A) |
0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
Логическая схема:
Совершенная дизъюнктивная нормальная форма (СДНФ):
По таблице истинности:
A | B | C | D | F |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 0 |
F
сднф = ¬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
Логическая cхема:
Совершенная конъюнктивная нормальная форма (СКНФ):
По таблице истинности:
A | B | C | D | F |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 0 |
F
скнф = (A∨¬B∨C∨D) ∧ (A∨¬B∨C∨¬D) ∧ (¬A∨B∨¬C∨D) ∧ (¬A∨B∨¬C∨¬D) ∧ (¬A∨¬B∨¬C∨¬D)
Логическая cхема:
Построение полинома Жегалкина:
По таблице истинности функции
A | B | C | D | Fж |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 0 |
Построим полином Жегалкина:
F
ж = C
0000 ⊕ C
1000∧A ⊕ C
0100∧B ⊕ C
0010∧C ⊕ C
0001∧D ⊕ C
1100∧A∧B ⊕ C
1010∧A∧C ⊕ C
1001∧A∧D ⊕ C
0110∧B∧C ⊕ C
0101∧B∧D ⊕ C
0011∧C∧D ⊕ C
1110∧A∧B∧C ⊕ C
1101∧A∧B∧D ⊕ C
1011∧A∧C∧D ⊕ C
0111∧B∧C∧D ⊕ C
1111∧A∧B∧C∧D
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 = 1 => С
1000 = 1 ⊕ 1 = 0
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 = 1 => С
0001 = 1 ⊕ 1 = 0
F
ж(1100) = С
0000 ⊕ С
1000 ⊕ С
0100 ⊕ С
1100 = 1 => С
1100 = 1 ⊕ 0 ⊕ 1 ⊕ 1 = 1
F
ж(1010) = С
0000 ⊕ С
1000 ⊕ С
0010 ⊕ С
1010 = 0 => С
1010 = 1 ⊕ 0 ⊕ 0 ⊕ 0 = 1
F
ж(1001) = С
0000 ⊕ С
1000 ⊕ С
0001 ⊕ С
1001 = 1 => С
1001 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
F
ж(0110) = С
0000 ⊕ С
0100 ⊕ С
0010 ⊕ С
0110 = 1 => С
0110 = 1 ⊕ 1 ⊕ 0 ⊕ 1 = 1
F
ж(0101) = С
0000 ⊕ С
0100 ⊕ С
0001 ⊕ С
0101 = 0 => С
0101 = 1 ⊕ 1 ⊕ 0 ⊕ 0 = 0
F
ж(0011) = С
0000 ⊕ С
0010 ⊕ С
0001 ⊕ С
0011 = 1 => С
0011 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
F
ж(1110) = С
0000 ⊕ С
1000 ⊕ С
0100 ⊕ С
0010 ⊕ С
1100 ⊕ С
1010 ⊕ С
0110 ⊕ С
1110 = 1 => С
1110 = 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 = 0
F
ж(1101) = С
0000 ⊕ С
1000 ⊕ С
0100 ⊕ С
0001 ⊕ С
1100 ⊕ С
1001 ⊕ С
0101 ⊕ С
1101 = 1 => С
1101 = 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
F
ж(1011) = С
0000 ⊕ С
1000 ⊕ С
0010 ⊕ С
0001 ⊕ С
1010 ⊕ С
1001 ⊕ С
0011 ⊕ С
1011 = 0 => С
1011 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 = 0
F
ж(0111) = С
0000 ⊕ С
0100 ⊕ С
0010 ⊕ С
0001 ⊕ С
0110 ⊕ С
0101 ⊕ С
0011 ⊕ С
0111 = 1 => С
0111 = 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 = 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 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 = 1
Таким образом, полином Жегалкина будет равен:
F
ж = 1 ⊕ B ⊕ A∧B ⊕ A∧C ⊕ B∧C ⊕ A∧B∧C∧D
Логическая схема, соответствующая полиному Жегалкина: