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Таблица истинности ONLINE
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Таблица истинности для функции (A1∨A2)|(¬A1→A3)∧A4:
Промежуточные таблицы истинности:A1∨A2: ¬A1: (¬A1)→A3: | A1 | A3 | ¬A1 | (¬A1)→A3 | | 0 | 0 | 1 | 0 | | 0 | 1 | 1 | 1 | | 1 | 0 | 0 | 1 | | 1 | 1 | 0 | 1 |
(A1∨A2)|((¬A1)→A3): | A1 | A2 | A3 | A1∨A2 | ¬A1 | (¬A1)→A3 | (A1∨A2)|((¬A1)→A3) | | 0 | 0 | 0 | 0 | 1 | 0 | 1 | | 0 | 0 | 1 | 0 | 1 | 1 | 1 | | 0 | 1 | 0 | 1 | 1 | 0 | 1 | | 0 | 1 | 1 | 1 | 1 | 1 | 0 | | 1 | 0 | 0 | 1 | 0 | 1 | 0 | | 1 | 0 | 1 | 1 | 0 | 1 | 0 | | 1 | 1 | 0 | 1 | 0 | 1 | 0 | | 1 | 1 | 1 | 1 | 0 | 1 | 0 |
((A1∨A2)|((¬A1)→A3))∧A4: | A1 | A2 | A3 | A4 | A1∨A2 | ¬A1 | (¬A1)→A3 | (A1∨A2)|((¬A1)→A3) | ((A1∨A2)|((¬A1)→A3))∧A4 | | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
Общая таблица истинности:| A1 | A2 | A3 | A4 | A1∨A2 | ¬A1 | (¬A1)→A3 | (A1∨A2)|((¬A1)→A3) | (A1∨A2)|(¬A1→A3)∧A4 | | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
Совершенная дизъюнктивная нормальная форма (СДНФ):
По таблице истинности: | A1 | A2 | A3 | A4 | 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 | 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 | 0 | | 1 | 1 | 0 | 1 | 0 | | 1 | 1 | 1 | 0 | 0 | | 1 | 1 | 1 | 1 | 0 |
F сднф = ¬A1∧¬A2∧¬A3∧A4 ∨ ¬A1∧¬A2∧A3∧A4 ∨ ¬A1∧A2∧¬A3∧A4 Логическая cхема:
Совершенная конъюнктивная нормальная форма (СКНФ):
По таблице истинности: | A1 | A2 | A3 | A4 | 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 | 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 | 0 | | 1 | 1 | 0 | 1 | 0 | | 1 | 1 | 1 | 0 | 0 | | 1 | 1 | 1 | 1 | 0 |
F скнф = (A1∨A2∨A3∨A4) ∧ (A1∨A2∨¬A3∨A4) ∧ (A1∨¬A2∨A3∨A4) ∧ (A1∨¬A2∨¬A3∨A4) ∧ (A1∨¬A2∨¬A3∨¬A4) ∧ (¬A1∨A2∨A3∨A4) ∧ (¬A1∨A2∨A3∨¬A4) ∧ (¬A1∨A2∨¬A3∨A4) ∧ (¬A1∨A2∨¬A3∨¬A4) ∧ (¬A1∨¬A2∨A3∨A4) ∧ (¬A1∨¬A2∨A3∨¬A4) ∧ (¬A1∨¬A2∨¬A3∨A4) ∧ (¬A1∨¬A2∨¬A3∨¬A4) Логическая cхема:
Построение полинома Жегалкина:
По таблице истинности функции | A1 | A2 | A3 | A4 | 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 | 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 | 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∧A1 ⊕ C 0100∧A2 ⊕ C 0010∧A3 ⊕ C 0001∧A4 ⊕ C 1100∧A1∧A2 ⊕ C 1010∧A1∧A3 ⊕ C 1001∧A1∧A4 ⊕ C 0110∧A2∧A3 ⊕ C 0101∧A2∧A4 ⊕ C 0011∧A3∧A4 ⊕ C 1110∧A1∧A2∧A3 ⊕ C 1101∧A1∧A2∧A4 ⊕ C 1011∧A1∧A3∧A4 ⊕ C 0111∧A2∧A3∧A4 ⊕ C 1111∧A1∧A2∧A3∧A4 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 = 0 => С 1000 = 0 ⊕ 0 = 0 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 ⊕ 0 ⊕ 0 ⊕ 0 = 0 F ж(1010) = С 0000 ⊕ С 1000 ⊕ С 0010 ⊕ С 1010 = 0 => С 1010 = 0 ⊕ 0 ⊕ 0 ⊕ 0 = 0 F ж(1001) = С 0000 ⊕ С 1000 ⊕ С 0001 ⊕ С 1001 = 0 => С 1001 = 0 ⊕ 0 ⊕ 1 ⊕ 0 = 1 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 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 = 0 F ж(1101) = С 0000 ⊕ С 1000 ⊕ С 0100 ⊕ С 0001 ⊕ С 1100 ⊕ С 1001 ⊕ С 0101 ⊕ С 1101 = 0 => С 1101 = 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 = 0 F ж(1011) = С 0000 ⊕ С 1000 ⊕ С 0010 ⊕ С 0001 ⊕ С 1010 ⊕ С 1001 ⊕ С 0011 ⊕ С 1011 = 0 => С 1011 = 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 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 = 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 = 1 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 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 = 1 Таким образом, полином Жегалкина будет равен: F ж = A4 ⊕ A1∧A4 ⊕ A2∧A3∧A4 ⊕ A1∧A2∧A3∧A4 Логическая схема, соответствующая полиному Жегалкина:
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