Промежуточные таблицы истинности:B∧C:
¬D:
A∧C:
(A∧C)∧(¬D):
A | C | D | A∧C | ¬D | (A∧C)∧(¬D) |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 |
1 | 1 | 1 | 1 | 0 | 0 |
(B∧C)∧D:
B | C | D | B∧C | (B∧C)∧D |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 0 |
1 | 1 | 0 | 1 | 0 |
1 | 1 | 1 | 1 | 1 |
¬((B∧C)∧D):
B | C | D | B∧C | (B∧C)∧D | ¬((B∧C)∧D) |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 1 |
0 | 1 | 1 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 | 0 |
A∧D:
(A∧D)∧(¬((B∧C)∧D)):
A | D | B | C | A∧D | B∧C | (B∧C)∧D | ¬((B∧C)∧D) | (A∧D)∧(¬((B∧C)∧D)) |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 |
1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
¬(B∧C):
B | C | B∧C | ¬(B∧C) |
0 | 0 | 0 | 1 |
0 | 1 | 0 | 1 |
1 | 0 | 0 | 1 |
1 | 1 | 1 | 0 |
¬((A∧C)∧(¬D)):
A | C | D | A∧C | ¬D | (A∧C)∧(¬D) | ¬((A∧C)∧(¬D)) |
0 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 1 | 0 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 | 1 | 0 |
1 | 1 | 1 | 1 | 0 | 0 | 1 |
¬((A∧D)∧(¬((B∧C)∧D))):
A | D | B | C | A∧D | B∧C | (B∧C)∧D | ¬((B∧C)∧D) | (A∧D)∧(¬((B∧C)∧D)) | ¬((A∧D)∧(¬((B∧C)∧D))) |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 |
(¬(B∧C))∨(¬((A∧C)∧(¬D))):
B | C | A | D | B∧C | ¬(B∧C) | A∧C | ¬D | (A∧C)∧(¬D) | ¬((A∧C)∧(¬D)) | (¬(B∧C))∨(¬((A∧C)∧(¬D))) |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 |
((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D)))):
B | C | A | D | B∧C | ¬(B∧C) | A∧C | ¬D | (A∧C)∧(¬D) | ¬((A∧C)∧(¬D)) | (¬(B∧C))∨(¬((A∧C)∧(¬D))) | A∧D | B∧C | (B∧C)∧D | ¬((B∧C)∧D) | (A∧D)∧(¬((B∧C)∧D)) | ¬((A∧D)∧(¬((B∧C)∧D))) | ((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D)))) |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
¬(((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D))))):
B | C | A | D | B∧C | ¬(B∧C) | A∧C | ¬D | (A∧C)∧(¬D) | ¬((A∧C)∧(¬D)) | (¬(B∧C))∨(¬((A∧C)∧(¬D))) | A∧D | B∧C | (B∧C)∧D | ¬((B∧C)∧D) | (A∧D)∧(¬((B∧C)∧D)) | ¬((A∧D)∧(¬((B∧C)∧D))) | ((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D)))) | ¬(((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D))))) |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 |
1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 |
1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
(¬(((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D))))))≡0:
B | C | A | D | B∧C | ¬(B∧C) | A∧C | ¬D | (A∧C)∧(¬D) | ¬((A∧C)∧(¬D)) | (¬(B∧C))∨(¬((A∧C)∧(¬D))) | A∧D | B∧C | (B∧C)∧D | ¬((B∧C)∧D) | (A∧D)∧(¬((B∧C)∧D)) | ¬((A∧D)∧(¬((B∧C)∧D))) | ((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D)))) | ¬(((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D))))) | (¬(((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D))))))≡0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
Общая таблица истинности:
B | C | A | D | B∧C | ¬D | A∧C | (A∧C)∧(¬D) | (B∧C)∧D | ¬((B∧C)∧D) | A∧D | (A∧D)∧(¬((B∧C)∧D)) | ¬(B∧C) | ¬((A∧C)∧(¬D)) | ¬((A∧D)∧(¬((B∧C)∧D))) | (¬(B∧C))∨(¬((A∧C)∧(¬D))) | ((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D)))) | ¬(((¬(B∧C))∨(¬((A∧C)∧(¬D))))∨(¬((A∧D)∧(¬((B∧C)∧D))))) | ¬(¬(B∧C)∨¬(A∧C∧¬D)∨¬(A∧D∧¬(B∧C∧D)))≡0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |
Логическая схема:
Совершенная дизъюнктивная нормальная форма (СДНФ):
По таблице истинности:
B | C | A | D | 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 | 1 |
0 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 |
F
сднф = ¬B∧¬C∧¬A∧¬D ∨ ¬B∧¬C∧¬A∧D ∨ ¬B∧¬C∧A∧¬D ∨ ¬B∧¬C∧A∧D ∨ ¬B∧C∧¬A∧¬D ∨ ¬B∧C∧¬A∧D ∨ ¬B∧C∧A∧¬D ∨ ¬B∧C∧A∧D ∨ B∧¬C∧¬A∧¬D ∨ B∧¬C∧¬A∧D ∨ B∧¬C∧A∧¬D ∨ B∧¬C∧A∧D ∨ B∧C∧¬A∧¬D ∨ B∧C∧¬A∧D ∨ B∧C∧A∧¬D ∨ B∧C∧A∧D
Логическая cхема:
Совершенная конъюнктивная нормальная форма (СКНФ):
По таблице истинности:
B | C | A | D | 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 | 1 |
0 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 |
В таблице истинности нет набора значений переменных при которых функция ложна!
Построение полинома Жегалкина:
По таблице истинности функции
B | C | A | D | 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 | 1 |
0 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 |
Построим полином Жегалкина:
F
ж = C
0000 ⊕ C
1000∧B ⊕ C
0100∧C ⊕ C
0010∧A ⊕ C
0001∧D ⊕ C
1100∧B∧C ⊕ C
1010∧B∧A ⊕ C
1001∧B∧D ⊕ C
0110∧C∧A ⊕ C
0101∧C∧D ⊕ C
0011∧A∧D ⊕ C
1110∧B∧C∧A ⊕ C
1101∧B∧C∧D ⊕ C
1011∧B∧A∧D ⊕ C
0111∧C∧A∧D ⊕ C
1111∧B∧C∧A∧D
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 = 1 => С
0100 = 1 ⊕ 1 = 0
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 = 1 => С
1100 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 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 ⊕ 0 ⊕ 0 ⊕ 1 = 0
F
ж(0101) = С
0000 ⊕ С
0100 ⊕ С
0001 ⊕ С
0101 = 1 => С
0101 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
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 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 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 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
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 = 1 => С
0111 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 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 = 1 => С
1111 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Таким образом, полином Жегалкина будет равен:
F
ж = 1