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