Для функции (¬XYP∨Z)⊕¬P∧¬Y→Z≡¬Z∨X:


Промежуточные таблицы истинности:
¬XYP:
XYP¬XYP
01
10

(¬XYP)∨Z:
XYPZ¬XYP(¬XYP)∨Z
0011
0111
1000
1101

¬P:
P¬P
01
10

¬Y:
Y¬Y
01
10

¬Z:
Z¬Z
01
10

(¬P)∧(¬Y):
PY¬P¬Y(¬P)∧(¬Y)
00111
01100
10010
11000

(¬Z)∨X:
ZX¬Z(¬Z)∨X
0011
0111
1000
1101

((¬XYP)∨Z)⊕((¬P)∧(¬Y)):
XYPZPY¬XYP(¬XYP)∨Z¬P¬Y(¬P)∧(¬Y)((¬XYP)∨Z)⊕((¬P)∧(¬Y))
0000111110
0001111001
0010110101
0011110001
0100111110
0101111001
0110110101
0111110001
1000001111
1001001000
1010000100
1011000000
1100011110
1101011001
1110010101
1111010001

(((¬XYP)∨Z)⊕((¬P)∧(¬Y)))→Z:
XYPZPY¬XYP(¬XYP)∨Z¬P¬Y(¬P)∧(¬Y)((¬XYP)∨Z)⊕((¬P)∧(¬Y))(((¬XYP)∨Z)⊕((¬P)∧(¬Y)))→Z
00001111101
00011110010
00101101010
00111100010
01001111101
01011110011
01101101011
01111100011
10000011110
10010010001
10100001001
10110000001
11000111101
11010110011
11100101011
11110100011

((((¬XYP)∨Z)⊕((¬P)∧(¬Y)))→Z)≡((¬Z)∨X):
XYPZPYX¬XYP(¬XYP)∨Z¬P¬Y(¬P)∧(¬Y)((¬XYP)∨Z)⊕((¬P)∧(¬Y))(((¬XYP)∨Z)⊕((¬P)∧(¬Y)))→Z¬Z(¬Z)∨X((((¬XYP)∨Z)⊕((¬P)∧(¬Y)))→Z)≡((¬Z)∨X)
000001111101111
000011111101111
000101110010110
000111110010110
001001101010110
001011101010110
001101100010110
001111100010110
010001111101000
010011111101011
010101110011000
010111110011011
011001101011000
011011101011011
011101100011000
011111100011011
100000011110110
100010011110110
100100010001111
100110010001111
101000001001111
101010001001111
101100000001111
101110000001111
110000111101000
110010111101011
110100110011000
110110110011011
111000101011000
111010101011011
111100100011000
111110100011011

Общая таблица истинности:

XYPZPYX¬XYP(¬XYP)∨Z¬P¬Y¬Z(¬P)∧(¬Y)(¬Z)∨X((¬XYP)∨Z)⊕((¬P)∧(¬Y))(((¬XYP)∨Z)⊕((¬P)∧(¬Y)))→Z(¬XYP∨Z)⊕¬P∧¬Y→Z≡¬Z∨X
000001111111011
000011111111011
000101110101100
000111110101100
001001101101100
001011101101100
001101100101100
001111100101100
010001111010010
010011111011011
010101110000110
010111110001111
011001101000110
011011101001111
011101100000110
011111100001111
100000011111100
100010011111100
100100010101011
100110010101011
101000001101011
101010001101011
101100000101011
101110000101011
110000111010010
110010111011011
110100110000110
110110110001111
111000101000110
111010101001111
111100100000110
111110100001111

Логическая схема:

Совершенная дизъюнктивная нормальная форма (СДНФ):

По таблице истинности:
XYPZPYXF
000001
000011
000100
000110
001000
001010
001100
001110
010000
010011
010100
010111
011000
011011
011100
011111
100000
100010
100101
100111
101001
101011
101101
101111
110000
110011
110100
110111
111000
111011
111100
111111
Fсднф = ¬XYP∧¬Z∧¬P∧¬Y∧¬X ∨ ¬XYP∧¬Z∧¬P∧¬Y∧X ∨ ¬XYP∧Z∧¬P∧¬Y∧X ∨ ¬XYP∧Z∧¬P∧Y∧X ∨ ¬XYP∧Z∧P∧¬Y∧X ∨ ¬XYP∧Z∧P∧Y∧X ∨ XYP∧¬Z∧¬P∧Y∧¬X ∨ XYP∧¬Z∧¬P∧Y∧X ∨ XYP∧¬Z∧P∧¬Y∧¬X ∨ XYP∧¬Z∧P∧¬Y∧X ∨ XYP∧¬Z∧P∧Y∧¬X ∨ XYP∧¬Z∧P∧Y∧X ∨ XYP∧Z∧¬P∧¬Y∧X ∨ XYP∧Z∧¬P∧Y∧X ∨ XYP∧Z∧P∧¬Y∧X ∨ XYP∧Z∧P∧Y∧X
Логическая cхема:

Совершенная конъюнктивная нормальная форма (СКНФ):

По таблице истинности:
XYPZPYXF
000001
000011
000100
000110
001000
001010
001100
001110
010000
010011
010100
010111
011000
011011
011100
011111
100000
100010
100101
100111
101001
101011
101101
101111
110000
110011
110100
110111
111000
111011
111100
111111
Fскнф = (XYP∨Z∨P∨¬Y∨X) ∧ (XYP∨Z∨P∨¬Y∨¬X) ∧ (XYP∨Z∨¬P∨Y∨X) ∧ (XYP∨Z∨¬P∨Y∨¬X) ∧ (XYP∨Z∨¬P∨¬Y∨X) ∧ (XYP∨Z∨¬P∨¬Y∨¬X) ∧ (XYP∨¬Z∨P∨Y∨X) ∧ (XYP∨¬Z∨P∨¬Y∨X) ∧ (XYP∨¬Z∨¬P∨Y∨X) ∧ (XYP∨¬Z∨¬P∨¬Y∨X) ∧ (¬XYP∨Z∨P∨Y∨X) ∧ (¬XYP∨Z∨P∨Y∨¬X) ∧ (¬XYP∨¬Z∨P∨Y∨X) ∧ (¬XYP∨¬Z∨P∨¬Y∨X) ∧ (¬XYP∨¬Z∨¬P∨Y∨X) ∧ (¬XYP∨¬Z∨¬P∨¬Y∨X)
Логическая cхема:

Построение полинома Жегалкина:

По таблице истинности функции
XYPZPYXFж
000001
000011
000100
000110
001000
001010
001100
001110
010000
010011
010100
010111
011000
011011
011100
011111
100000
100010
100101
100111
101001
101011
101101
101111
110000
110011
110100
110111
111000
111011
111100
111111

Построим полином Жегалкина:
Fж = C00000 ⊕ C10000∧XYP ⊕ C01000∧Z ⊕ C00100∧P ⊕ C00010∧Y ⊕ C00001∧X ⊕ C11000∧XYP∧Z ⊕ C10100∧XYP∧P ⊕ C10010∧XYP∧Y ⊕ C10001∧XYP∧X ⊕ C01100∧Z∧P ⊕ C01010∧Z∧Y ⊕ C01001∧Z∧X ⊕ C00110∧P∧Y ⊕ C00101∧P∧X ⊕ C00011∧Y∧X ⊕ C11100∧XYP∧Z∧P ⊕ C11010∧XYP∧Z∧Y ⊕ C11001∧XYP∧Z∧X ⊕ C10110∧XYP∧P∧Y ⊕ C10101∧XYP∧P∧X ⊕ C10011∧XYP∧Y∧X ⊕ C01110∧Z∧P∧Y ⊕ C01101∧Z∧P∧X ⊕ C01011∧Z∧Y∧X ⊕ C00111∧P∧Y∧X ⊕ C11110∧XYP∧Z∧P∧Y ⊕ C11101∧XYP∧Z∧P∧X ⊕ C11011∧XYP∧Z∧Y∧X ⊕ C10111∧XYP∧P∧Y∧X ⊕ C01111∧Z∧P∧Y∧X ⊕ C11111∧XYP∧Z∧P∧Y∧X

Так как Fж(00000) = 1, то С00000 = 1.

Далее подставляем все остальные наборы в порядке возрастания числа единиц, подставляя вновь полученные значения в следующие формулы:
Fж(10000) = С00000 ⊕ С10000 = 0 => С10000 = 1 ⊕ 0 = 1
Fж(01000) = С00000 ⊕ С01000 = 0 => С01000 = 1 ⊕ 0 = 1
Fж(00100) = С00000 ⊕ С00100 = 0 => С00100 = 1 ⊕ 0 = 1
Fж(00010) = С00000 ⊕ С00010 = 0 => С00010 = 1 ⊕ 0 = 1
Fж(00001) = С00000 ⊕ С00001 = 1 => С00001 = 1 ⊕ 1 = 0
Fж(11000) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С11000 = 0 => С11000 = 1 ⊕ 1 ⊕ 1 ⊕ 0 = 1
Fж(10100) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С10100 = 1 => С10100 = 1 ⊕ 1 ⊕ 1 ⊕ 1 = 0
Fж(10010) = С00000 ⊕ С10000 ⊕ С00010 ⊕ С10010 = 1 => С10010 = 1 ⊕ 1 ⊕ 1 ⊕ 1 = 0
Fж(10001) = С00000 ⊕ С10000 ⊕ С00001 ⊕ С10001 = 0 => С10001 = 1 ⊕ 1 ⊕ 0 ⊕ 0 = 0
Fж(01100) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С01100 = 0 => С01100 = 1 ⊕ 1 ⊕ 1 ⊕ 0 = 1
Fж(01010) = С00000 ⊕ С01000 ⊕ С00010 ⊕ С01010 = 0 => С01010 = 1 ⊕ 1 ⊕ 1 ⊕ 0 = 1
Fж(01001) = С00000 ⊕ С01000 ⊕ С00001 ⊕ С01001 = 1 => С01001 = 1 ⊕ 1 ⊕ 0 ⊕ 1 = 1
Fж(00110) = С00000 ⊕ С00100 ⊕ С00010 ⊕ С00110 = 0 => С00110 = 1 ⊕ 1 ⊕ 1 ⊕ 0 = 1
Fж(00101) = С00000 ⊕ С00100 ⊕ С00001 ⊕ С00101 = 0 => С00101 = 1 ⊕ 1 ⊕ 0 ⊕ 0 = 0
Fж(00011) = С00000 ⊕ С00010 ⊕ С00001 ⊕ С00011 = 0 => С00011 = 1 ⊕ 1 ⊕ 0 ⊕ 0 = 0
Fж(11100) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00100 ⊕ С11000 ⊕ С10100 ⊕ С01100 ⊕ С11100 = 0 => С11100 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 = 0
Fж(11010) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00010 ⊕ С11000 ⊕ С10010 ⊕ С01010 ⊕ С11010 = 0 => С11010 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 = 0
Fж(11001) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00001 ⊕ С11000 ⊕ С10001 ⊕ С01001 ⊕ С11001 = 1 => С11001 = 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 = 0
Fж(10110) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С00010 ⊕ С10100 ⊕ С10010 ⊕ С00110 ⊕ С10110 = 1 => С10110 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 = 0
Fж(10101) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С00001 ⊕ С10100 ⊕ С10001 ⊕ С00101 ⊕ С10101 = 1 => С10101 = 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(10011) = С00000 ⊕ С10000 ⊕ С00010 ⊕ С00001 ⊕ С10010 ⊕ С10001 ⊕ С00011 ⊕ С10011 = 1 => С10011 = 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(01110) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С01100 ⊕ С01010 ⊕ С00110 ⊕ С01110 = 0 => С01110 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 = 1
Fж(01101) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С00001 ⊕ С01100 ⊕ С01001 ⊕ С00101 ⊕ С01101 = 1 => С01101 = 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 = 0
Fж(01011) = С00000 ⊕ С01000 ⊕ С00010 ⊕ С00001 ⊕ С01010 ⊕ С01001 ⊕ С00011 ⊕ С01011 = 1 => С01011 = 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 = 0
Fж(00111) = С00000 ⊕ С00100 ⊕ С00010 ⊕ С00001 ⊕ С00110 ⊕ С00101 ⊕ С00011 ⊕ С00111 = 0 => С00111 = 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 = 0
Fж(11110) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С11000 ⊕ С10100 ⊕ С10010 ⊕ С01100 ⊕ С01010 ⊕ С00110 ⊕ С11100 ⊕ С11010 ⊕ С10110 ⊕ С01110 ⊕ С11110 = 0 => С11110 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 = 0
Fж(11101) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00100 ⊕ С00001 ⊕ С11000 ⊕ С10100 ⊕ С10001 ⊕ С01100 ⊕ С01001 ⊕ С00101 ⊕ С11100 ⊕ С11001 ⊕ С10101 ⊕ С01101 ⊕ С11101 = 1 => С11101 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(11011) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00010 ⊕ С00001 ⊕ С11000 ⊕ С10010 ⊕ С10001 ⊕ С01010 ⊕ С01001 ⊕ С00011 ⊕ С11010 ⊕ С11001 ⊕ С10011 ⊕ С01011 ⊕ С11011 = 1 => С11011 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(10111) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С00010 ⊕ С00001 ⊕ С10100 ⊕ С10010 ⊕ С10001 ⊕ С00110 ⊕ С00101 ⊕ С00011 ⊕ С10110 ⊕ С10101 ⊕ С10011 ⊕ С00111 ⊕ С10111 = 1 => С10111 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(01111) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С00001 ⊕ С01100 ⊕ С01010 ⊕ С01001 ⊕ С00110 ⊕ С00101 ⊕ С00011 ⊕ С01110 ⊕ С01101 ⊕ С01011 ⊕ С00111 ⊕ С01111 = 1 => С01111 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(11111) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С00001 ⊕ С11000 ⊕ С10100 ⊕ С10010 ⊕ С10001 ⊕ С01100 ⊕ С01010 ⊕ С01001 ⊕ С00110 ⊕ С00101 ⊕ С00011 ⊕ С11100 ⊕ С11010 ⊕ С11001 ⊕ С10110 ⊕ С10101 ⊕ С10011 ⊕ С01110 ⊕ С01101 ⊕ С01011 ⊕ С00111 ⊕ С11110 ⊕ С11101 ⊕ С11011 ⊕ С10111 ⊕ С01111 ⊕ С11111 = 1 => С11111 = 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0

Таким образом, полином Жегалкина будет равен:
Fж = 1 ⊕ XYP ⊕ Z ⊕ P ⊕ Y ⊕ XYP∧Z ⊕ Z∧P ⊕ Z∧Y ⊕ Z∧X ⊕ P∧Y ⊕ Z∧P∧Y
Логическая схема, соответствующая полиному Жегалкина:

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