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