Алгебра логики. Задача 4-21
Укажите, какое логическое выражение равносильно выражению ¬(A ∧ B) ∧ ¬(C ∨ ¬A) ∧ (B ∧ ¬C)
1) A ∧ ¬B ∨ ¬B ∧ ¬C
2) 0
3) ¬A ∧ B ∧ C
4) A ∨ ¬B ∨ ¬C
Преобразуем выражение: \( \overline{(A ⋅ B)} ⋅ \overline{(C ⋅ \overline{A})} ⋅ \overline{(B ⋅ \overline{C})} \) = \( (\overline{A} + \overline{B}) ⋅ (\overline{C} + A) ⋅ (\overline{B} + C) \) = \( (\overline{A} ⋅ \overline{C} + \overline{B} ⋅ \overline{C} + \overline{A} ⋅ A + \overline{B} ⋅ A) ⋅ (\overline{B} + C) \) = \( (\overline{A} ⋅ \overline{C} + \overline{B} ⋅ \overline{C} + \overline{B} ⋅ A) ⋅ (\overline{B} + C) \) = \( \overline{A} ⋅ \overline{C} ⋅ \overline{B} + \overline{B} ⋅ \overline{C} ⋅ \overline{B} + \overline{B} ⋅ A ⋅ \overline{B} + \overline{A} ⋅ \overline{C} ⋅ C + \overline{B} ⋅ \overline{C} ⋅ C + \overline{B} ⋅ A ⋅ C \) = \( \overline{A} ⋅ (\overline{B} ⋅ \overline{C}) + \overline{B} ⋅ \overline{C} + A ⋅ \overline{B} + (A ⋅\overline{B}) ⋅ C \) = \( \overline{B} ⋅ \overline{C} + A ⋅\overline{B} \) = \( A ⋅\overline{B} + \overline{B} ⋅ \overline{C} \)
Полученное выражение соответствует первому варианту ответа.