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