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