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Таблица истинности для функции (X∨Y)∧(X∧¬Y∨X∧T∨¬Y∧¬X∨¬Y∨¬Y∧T∨¬T∧¬X∨¬T∧¬Y):
Промежуточные таблицы истинности:X∨Y: ¬Y: ¬X: ¬T: X∧(¬Y): X | Y | ¬Y | X∧(¬Y) | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 |
X∧T: (¬Y)∧(¬X): Y | X | ¬Y | ¬X | (¬Y)∧(¬X) | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
(¬Y)∧T: Y | T | ¬Y | (¬Y)∧T | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
(¬T)∧(¬X): T | X | ¬T | ¬X | (¬T)∧(¬X) | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
(¬T)∧(¬Y): T | Y | ¬T | ¬Y | (¬T)∧(¬Y) | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
(X∧(¬Y))∨(X∧T): X | Y | T | ¬Y | X∧(¬Y) | X∧T | (X∧(¬Y))∨(X∧T) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)): X | Y | T | ¬Y | X∧(¬Y) | X∧T | (X∧(¬Y))∨(X∧T) | ¬Y | ¬X | (¬Y)∧(¬X) | ((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
(((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y): X | Y | T | ¬Y | X∧(¬Y) | X∧T | (X∧(¬Y))∨(X∧T) | ¬Y | ¬X | (¬Y)∧(¬X) | ((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)) | ¬Y | (((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 |
((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T): X | Y | T | ¬Y | X∧(¬Y) | X∧T | (X∧(¬Y))∨(X∧T) | ¬Y | ¬X | (¬Y)∧(¬X) | ((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)) | ¬Y | (((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y) | ¬Y | (¬Y)∧T | ((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 |
(((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)): X | Y | T | ¬Y | X∧(¬Y) | X∧T | (X∧(¬Y))∨(X∧T) | ¬Y | ¬X | (¬Y)∧(¬X) | ((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)) | ¬Y | (((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y) | ¬Y | (¬Y)∧T | ((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T) | ¬T | ¬X | (¬T)∧(¬X) | (((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
((((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)))∨((¬T)∧(¬Y)): X | Y | T | ¬Y | X∧(¬Y) | X∧T | (X∧(¬Y))∨(X∧T) | ¬Y | ¬X | (¬Y)∧(¬X) | ((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)) | ¬Y | (((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y) | ¬Y | (¬Y)∧T | ((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T) | ¬T | ¬X | (¬T)∧(¬X) | (((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)) | ¬T | ¬Y | (¬T)∧(¬Y) | ((((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)))∨((¬T)∧(¬Y)) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
(X∨Y)∧(((((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)))∨((¬T)∧(¬Y))): X | Y | T | X∨Y | ¬Y | X∧(¬Y) | X∧T | (X∧(¬Y))∨(X∧T) | ¬Y | ¬X | (¬Y)∧(¬X) | ((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)) | ¬Y | (((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y) | ¬Y | (¬Y)∧T | ((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T) | ¬T | ¬X | (¬T)∧(¬X) | (((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)) | ¬T | ¬Y | (¬T)∧(¬Y) | ((((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)))∨((¬T)∧(¬Y)) | (X∨Y)∧(((((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)))∨((¬T)∧(¬Y))) | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
Общая таблица истинности:X | Y | T | X∨Y | ¬Y | ¬X | ¬T | X∧(¬Y) | X∧T | (¬Y)∧(¬X) | (¬Y)∧T | (¬T)∧(¬X) | (¬T)∧(¬Y) | (X∧(¬Y))∨(X∧T) | ((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)) | (((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y) | ((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T) | (((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)) | ((((((X∧(¬Y))∨(X∧T))∨((¬Y)∧(¬X)))∨(¬Y))∨((¬Y)∧T))∨((¬T)∧(¬X)))∨((¬T)∧(¬Y)) | (X∨Y)∧(X∧¬Y∨X∧T∨¬Y∧¬X∨¬Y∨¬Y∧T∨¬T∧¬X∨¬T∧¬Y) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Логическая схема:
Совершенная дизъюнктивная нормальная форма (СДНФ):
По таблице истинности: X | Y | T | F | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
F сднф = ¬X∧Y∧¬T ∨ X∧¬Y∧¬T ∨ X∧¬Y∧T ∨ X∧Y∧T Логическая cхема:
Совершенная конъюнктивная нормальная форма (СКНФ):
По таблице истинности: X | Y | T | F | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
F скнф = (X∨Y∨T) ∧ (X∨Y∨¬T) ∧ (X∨¬Y∨¬T) ∧ (¬X∨¬Y∨T) Логическая cхема:
Построение полинома Жегалкина:
По таблице истинности функции X | Y | T | Fж | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
Построим полином Жегалкина: F ж = C 000 ⊕ C 100∧X ⊕ C 010∧Y ⊕ C 001∧T ⊕ C 110∧X∧Y ⊕ C 101∧X∧T ⊕ C 011∧Y∧T ⊕ C 111∧X∧Y∧T 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 = 1 => С 100 = 0 ⊕ 1 = 1 F ж(010) = С 000 ⊕ С 010 = 1 => С 010 = 0 ⊕ 1 = 1 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 ⊕ 1 ⊕ 1 ⊕ 0 = 0 F ж(101) = С 000 ⊕ С 100 ⊕ С 001 ⊕ С 101 = 1 => С 101 = 0 ⊕ 1 ⊕ 0 ⊕ 1 = 0 F ж(011) = С 000 ⊕ С 010 ⊕ С 001 ⊕ С 011 = 0 => С 011 = 0 ⊕ 1 ⊕ 0 ⊕ 0 = 1 F ж(111) = С 000 ⊕ С 100 ⊕ С 010 ⊕ С 001 ⊕ С 110 ⊕ С 101 ⊕ С 011 ⊕ С 111 = 1 => С 111 = 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 = 0 Таким образом, полином Жегалкина будет равен: F ж = X ⊕ Y ⊕ Y∧T Логическая схема, соответствующая полиному Жегалкина:
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