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Таблица истинности для функции X∧Y∧X∧V∧X∧Y∧Y∧V∧X∧Y:
Промежуточные таблицы истинности:X∧Y: (X∧Y)∧X: X | Y | X∧Y | (X∧Y)∧X | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
((X∧Y)∧X)∧V: X | Y | V | X∧Y | (X∧Y)∧X | ((X∧Y)∧X)∧V | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
(((X∧Y)∧X)∧V)∧X: X | Y | V | X∧Y | (X∧Y)∧X | ((X∧Y)∧X)∧V | (((X∧Y)∧X)∧V)∧X | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
((((X∧Y)∧X)∧V)∧X)∧Y: X | Y | V | X∧Y | (X∧Y)∧X | ((X∧Y)∧X)∧V | (((X∧Y)∧X)∧V)∧X | ((((X∧Y)∧X)∧V)∧X)∧Y | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
(((((X∧Y)∧X)∧V)∧X)∧Y)∧Y: X | Y | V | X∧Y | (X∧Y)∧X | ((X∧Y)∧X)∧V | (((X∧Y)∧X)∧V)∧X | ((((X∧Y)∧X)∧V)∧X)∧Y | (((((X∧Y)∧X)∧V)∧X)∧Y)∧Y | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V: X | Y | V | X∧Y | (X∧Y)∧X | ((X∧Y)∧X)∧V | (((X∧Y)∧X)∧V)∧X | ((((X∧Y)∧X)∧V)∧X)∧Y | (((((X∧Y)∧X)∧V)∧X)∧Y)∧Y | ((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
(((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V)∧X: X | Y | V | X∧Y | (X∧Y)∧X | ((X∧Y)∧X)∧V | (((X∧Y)∧X)∧V)∧X | ((((X∧Y)∧X)∧V)∧X)∧Y | (((((X∧Y)∧X)∧V)∧X)∧Y)∧Y | ((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V | (((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V)∧X | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
((((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V)∧X)∧Y: X | Y | V | X∧Y | (X∧Y)∧X | ((X∧Y)∧X)∧V | (((X∧Y)∧X)∧V)∧X | ((((X∧Y)∧X)∧V)∧X)∧Y | (((((X∧Y)∧X)∧V)∧X)∧Y)∧Y | ((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V | (((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V)∧X | ((((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V)∧X)∧Y | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Общая таблица истинности:X | Y | V | X∧Y | (X∧Y)∧X | ((X∧Y)∧X)∧V | (((X∧Y)∧X)∧V)∧X | ((((X∧Y)∧X)∧V)∧X)∧Y | (((((X∧Y)∧X)∧V)∧X)∧Y)∧Y | ((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V | (((((((X∧Y)∧X)∧V)∧X)∧Y)∧Y)∧V)∧X | X∧Y∧X∧V∧X∧Y∧Y∧V∧X∧Y | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Логическая схема:
Совершенная дизъюнктивная нормальная форма (СДНФ):
По таблице истинности: X | Y | V | F | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
F сднф = X∧Y∧V Логическая cхема:
Совершенная конъюнктивная нормальная форма (СКНФ):
По таблице истинности: X | Y | V | F | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
F скнф = (X∨Y∨V) ∧ (X∨Y∨¬V) ∧ (X∨¬Y∨V) ∧ (X∨¬Y∨¬V) ∧ (¬X∨Y∨V) ∧ (¬X∨Y∨¬V) ∧ (¬X∨¬Y∨V) Логическая cхема:
Построение полинома Жегалкина:
По таблице истинности функции X | Y | V | Fж | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
Построим полином Жегалкина: F ж = C 000 ⊕ C 100∧X ⊕ C 010∧Y ⊕ C 001∧V ⊕ C 110∧X∧Y ⊕ C 101∧X∧V ⊕ C 011∧Y∧V ⊕ C 111∧X∧Y∧V Так как F ж(000) = 0, то С 000 = 0. Далее подставляем все остальные наборы в порядке возрастания числа единиц, подставляя вновь полученные значения в следующие формулы: F ж(100) = С 000 ⊕ С 100 = 0 => С 100 = 0 ⊕ 0 = 0 F ж(010) = С 000 ⊕ С 010 = 0 => С 010 = 0 ⊕ 0 = 0 F ж(001) = С 000 ⊕ С 001 = 0 => С 001 = 0 ⊕ 0 = 0 F ж(110) = С 000 ⊕ С 100 ⊕ С 010 ⊕ С 110 = 0 => С 110 = 0 ⊕ 0 ⊕ 0 ⊕ 0 = 0 F ж(101) = С 000 ⊕ С 100 ⊕ С 001 ⊕ С 101 = 0 => С 101 = 0 ⊕ 0 ⊕ 0 ⊕ 0 = 0 F ж(011) = С 000 ⊕ С 010 ⊕ С 001 ⊕ С 011 = 0 => С 011 = 0 ⊕ 0 ⊕ 0 ⊕ 0 = 0 F ж(111) = С 000 ⊕ С 100 ⊕ С 010 ⊕ С 001 ⊕ С 110 ⊕ С 101 ⊕ С 011 ⊕ С 111 = 1 => С 111 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 1 Таким образом, полином Жегалкина будет равен: F ж = X∧Y∧V Логическая схема, соответствующая полиному Жегалкина:
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