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