Промежуточные таблицы истинности:¬B0:
A0∨(¬B0):
A0 | B0 | ¬B0 | A0∨(¬B0) |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |
¬B1:
A1∨(¬B1):
A1 | B1 | ¬B1 | A1∨(¬B1) |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |
¬A0:
B0∨(¬A0):
B0 | A0 | ¬A0 | B0∨(¬A0) |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |
¬A1:
B1∨(¬A1):
B1 | A1 | ¬A1 | B1∨(¬A1) |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |
(A0∨(¬B0))∧(A1∨(¬B1)):
A0 | B0 | A1 | B1 | ¬B0 | A0∨(¬B0) | ¬B1 | A1∨(¬B1) | (A0∨(¬B0))∧(A1∨(¬B1)) |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
0 | 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 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
((A0∨(¬B0))∧(A1∨(¬B1)))∧(B0∨(¬A0)):
A0 | B0 | A1 | B1 | ¬B0 | A0∨(¬B0) | ¬B1 | A1∨(¬B1) | (A0∨(¬B0))∧(A1∨(¬B1)) | ¬A0 | B0∨(¬A0) | ((A0∨(¬B0))∧(A1∨(¬B1)))∧(B0∨(¬A0)) |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 |
0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
(((A0∨(¬B0))∧(A1∨(¬B1)))∧(B0∨(¬A0)))∧(B1∨(¬A1)):
A0 | B0 | A1 | B1 | ¬B0 | A0∨(¬B0) | ¬B1 | A1∨(¬B1) | (A0∨(¬B0))∧(A1∨(¬B1)) | ¬A0 | B0∨(¬A0) | ((A0∨(¬B0))∧(A1∨(¬B1)))∧(B0∨(¬A0)) | ¬A1 | B1∨(¬A1) | (((A0∨(¬B0))∧(A1∨(¬B1)))∧(B0∨(¬A0)))∧(B1∨(¬A1)) |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 |
1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
Общая таблица истинности:
A0 | B0 | A1 | B1 | ¬B0 | A0∨(¬B0) | ¬B1 | A1∨(¬B1) | ¬A0 | B0∨(¬A0) | ¬A1 | B1∨(¬A1) | (A0∨(¬B0))∧(A1∨(¬B1)) | ((A0∨(¬B0))∧(A1∨(¬B1)))∧(B0∨(¬A0)) | (A0∨(¬B0))∧(A1∨(¬B1))∧(B0∨(¬A0))∧(B1∨(¬A1)) |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
Логическая схема:
Совершенная дизъюнктивная нормальная форма (СДНФ):
По таблице истинности:
A0 | B0 | A1 | B1 | F |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 1 | 1 |
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 | 0 |
1 | 1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 1 |
F
сднф = ¬A0∧¬B0∧¬A1∧¬B1 ∨ ¬A0∧¬B0∧A1∧B1 ∨ A0∧B0∧¬A1∧¬B1 ∨ A0∧B0∧A1∧B1
Логическая cхема:
Совершенная конъюнктивная нормальная форма (СКНФ):
По таблице истинности:
A0 | B0 | A1 | B1 | F |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 1 | 1 |
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 | 0 |
1 | 1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 1 |
F
скнф = (A0∨B0∨A1∨¬B1) ∧ (A0∨B0∨¬A1∨B1) ∧ (A0∨¬B0∨A1∨B1) ∧ (A0∨¬B0∨A1∨¬B1) ∧ (A0∨¬B0∨¬A1∨B1) ∧ (A0∨¬B0∨¬A1∨¬B1) ∧ (¬A0∨B0∨A1∨B1) ∧ (¬A0∨B0∨A1∨¬B1) ∧ (¬A0∨B0∨¬A1∨B1) ∧ (¬A0∨B0∨¬A1∨¬B1) ∧ (¬A0∨¬B0∨A1∨¬B1) ∧ (¬A0∨¬B0∨¬A1∨B1)
Логическая cхема:
Построение полинома Жегалкина:
По таблице истинности функции
A0 | B0 | A1 | B1 | Fж |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 1 | 1 |
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 | 0 |
1 | 1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 1 |
Построим полином Жегалкина:
F
ж = C
0000 ⊕ C
1000∧A0 ⊕ C
0100∧B0 ⊕ C
0010∧A1 ⊕ C
0001∧B1 ⊕ C
1100∧A0∧B0 ⊕ C
1010∧A0∧A1 ⊕ C
1001∧A0∧B1 ⊕ C
0110∧B0∧A1 ⊕ C
0101∧B0∧B1 ⊕ C
0011∧A1∧B1 ⊕ C
1110∧A0∧B0∧A1 ⊕ C
1101∧A0∧B0∧B1 ⊕ C
1011∧A0∧A1∧B1 ⊕ C
0111∧B0∧A1∧B1 ⊕ C
1111∧A0∧B0∧A1∧B1
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 = 0 => С
0010 = 1 ⊕ 0 = 1
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 ⊕ 1 ⊕ 0 = 1
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 ⊕ 1 ⊕ 0 = 1
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 = 1 => С
0011 = 1 ⊕ 1 ⊕ 1 ⊕ 1 = 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 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 = 0
F
ж(1101) = С
0000 ⊕ С
1000 ⊕ С
0100 ⊕ С
0001 ⊕ С
1100 ⊕ С
1001 ⊕ С
0101 ⊕ С
1101 = 0 => С
1101 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 = 0
F
ж(1011) = С
0000 ⊕ С
1000 ⊕ С
0010 ⊕ С
0001 ⊕ С
1010 ⊕ С
1001 ⊕ С
0011 ⊕ С
1011 = 0 => С
1011 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 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 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 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 = 1 => С
1111 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Таким образом, полином Жегалкина будет равен:
F
ж = 1 ⊕ A0 ⊕ B0 ⊕ A1 ⊕ B1 ⊕ A0∧A1 ⊕ A0∧B1 ⊕ B0∧A1 ⊕ B0∧B1
Логическая схема, соответствующая полиному Жегалкина: