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