Для функции A∨¬B∧¬C∨¬D∧¬E:


Промежуточные таблицы истинности:
¬B:
B¬B
01
10

¬C:
C¬C
01
10

¬D:
D¬D
01
10

¬E:
E¬E
01
10

(¬B)∧(¬C):
BC¬B¬C(¬B)∧(¬C)
00111
01100
10010
11000

(¬D)∧(¬E):
DE¬D¬E(¬D)∧(¬E)
00111
01100
10010
11000

A∨((¬B)∧(¬C)):
ABC¬B¬C(¬B)∧(¬C)A∨((¬B)∧(¬C))
0001111
0011000
0100100
0110000
1001111
1011001
1100101
1110001

(A∨((¬B)∧(¬C)))∨((¬D)∧(¬E)):
ABCDE¬B¬C(¬B)∧(¬C)A∨((¬B)∧(¬C))¬D¬E(¬D)∧(¬E)(A∨((¬B)∧(¬C)))∨((¬D)∧(¬E))
0000011111111
0000111111001
0001011110101
0001111110001
0010010001111
0010110001000
0011010000100
0011110000000
0100001001111
0100101001000
0101001000100
0101101000000
0110000001111
0110100001000
0111000000100
0111100000000
1000011111111
1000111111001
1001011110101
1001111110001
1010010011111
1010110011001
1011010010101
1011110010001
1100001011111
1100101011001
1101001010101
1101101010001
1110000011111
1110100011001
1111000010101
1111100010001

Общая таблица истинности:

ABCDE¬B¬C¬D¬E(¬B)∧(¬C)(¬D)∧(¬E)A∨((¬B)∧(¬C))A∨¬B∧¬C∨¬D∧¬E
0000011111111
0000111101011
0001011011011
0001111001011
0010010110101
0010110100000
0011010010000
0011110000000
0100001110101
0100101100000
0101001010000
0101101000000
0110000110101
0110100100000
0111000010000
0111100000000
1000011111111
1000111101011
1001011011011
1001111001011
1010010110111
1010110100011
1011010010011
1011110000011
1100001110111
1100101100011
1101001010011
1101101000011
1110000110111
1110100100011
1111000010011
1111100000011

Логическая схема:

Совершенная дизъюнктивная нормальная форма (СДНФ):

По таблице истинности:
ABCDEF
000001
000011
000101
000111
001001
001010
001100
001110
010001
010010
010100
010110
011001
011010
011100
011110
100001
100011
100101
100111
101001
101011
101101
101111
110001
110011
110101
110111
111001
111011
111101
111111
Fсднф = ¬A∧¬B∧¬C∧¬D∧¬E ∨ ¬A∧¬B∧¬C∧¬D∧E ∨ ¬A∧¬B∧¬C∧D∧¬E ∨ ¬A∧¬B∧¬C∧D∧E ∨ ¬A∧¬B∧C∧¬D∧¬E ∨ ¬A∧B∧¬C∧¬D∧¬E ∨ ¬A∧B∧C∧¬D∧¬E ∨ A∧¬B∧¬C∧¬D∧¬E ∨ A∧¬B∧¬C∧¬D∧E ∨ A∧¬B∧¬C∧D∧¬E ∨ A∧¬B∧¬C∧D∧E ∨ A∧¬B∧C∧¬D∧¬E ∨ A∧¬B∧C∧¬D∧E ∨ A∧¬B∧C∧D∧¬E ∨ A∧¬B∧C∧D∧E ∨ A∧B∧¬C∧¬D∧¬E ∨ A∧B∧¬C∧¬D∧E ∨ A∧B∧¬C∧D∧¬E ∨ A∧B∧¬C∧D∧E ∨ A∧B∧C∧¬D∧¬E ∨ A∧B∧C∧¬D∧E ∨ A∧B∧C∧D∧¬E ∨ A∧B∧C∧D∧E
Логическая cхема:

Совершенная конъюнктивная нормальная форма (СКНФ):

По таблице истинности:
ABCDEF
000001
000011
000101
000111
001001
001010
001100
001110
010001
010010
010100
010110
011001
011010
011100
011110
100001
100011
100101
100111
101001
101011
101101
101111
110001
110011
110101
110111
111001
111011
111101
111111
Fскнф = (A∨B∨¬C∨D∨¬E) ∧ (A∨B∨¬C∨¬D∨E) ∧ (A∨B∨¬C∨¬D∨¬E) ∧ (A∨¬B∨C∨D∨¬E) ∧ (A∨¬B∨C∨¬D∨E) ∧ (A∨¬B∨C∨¬D∨¬E) ∧ (A∨¬B∨¬C∨D∨¬E) ∧ (A∨¬B∨¬C∨¬D∨E) ∧ (A∨¬B∨¬C∨¬D∨¬E)
Логическая cхема:

Построение полинома Жегалкина:

По таблице истинности функции
ABCDEFж
000001
000011
000101
000111
001001
001010
001100
001110
010001
010010
010100
010110
011001
011010
011100
011110
100001
100011
100101
100111
101001
101011
101101
101111
110001
110011
110101
110111
111001
111011
111101
111111

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

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 = 1 => С10000 = 1 ⊕ 1 = 0
Fж(01000) = С00000 ⊕ С01000 = 1 => С01000 = 1 ⊕ 1 = 0
Fж(00100) = С00000 ⊕ С00100 = 1 => С00100 = 1 ⊕ 1 = 0
Fж(00010) = С00000 ⊕ С00010 = 1 => С00010 = 1 ⊕ 1 = 0
Fж(00001) = С00000 ⊕ С00001 = 1 => С00001 = 1 ⊕ 1 = 0
Fж(11000) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С11000 = 1 => С11000 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(10100) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С10100 = 1 => С10100 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(10010) = С00000 ⊕ С10000 ⊕ С00010 ⊕ С10010 = 1 => С10010 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(10001) = С00000 ⊕ С10000 ⊕ С00001 ⊕ С10001 = 1 => С10001 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(01100) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С01100 = 1 => С01100 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(01010) = С00000 ⊕ С01000 ⊕ С00010 ⊕ С01010 = 0 => С01010 = 1 ⊕ 0 ⊕ 0 ⊕ 0 = 1
Fж(01001) = С00000 ⊕ С01000 ⊕ С00001 ⊕ С01001 = 0 => С01001 = 1 ⊕ 0 ⊕ 0 ⊕ 0 = 1
Fж(00110) = С00000 ⊕ С00100 ⊕ С00010 ⊕ С00110 = 0 => С00110 = 1 ⊕ 0 ⊕ 0 ⊕ 0 = 1
Fж(00101) = С00000 ⊕ С00100 ⊕ С00001 ⊕ С00101 = 0 => С00101 = 1 ⊕ 0 ⊕ 0 ⊕ 0 = 1
Fж(00011) = С00000 ⊕ С00010 ⊕ С00001 ⊕ С00011 = 1 => С00011 = 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(11100) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00100 ⊕ С11000 ⊕ С10100 ⊕ С01100 ⊕ С11100 = 1 => С11100 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(11010) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00010 ⊕ С11000 ⊕ С10010 ⊕ С01010 ⊕ С11010 = 1 => С11010 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 = 1
Fж(11001) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00001 ⊕ С11000 ⊕ С10001 ⊕ С01001 ⊕ С11001 = 1 => С11001 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 = 1
Fж(10110) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С00010 ⊕ С10100 ⊕ С10010 ⊕ С00110 ⊕ С10110 = 1 => С10110 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 = 1
Fж(10101) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С00001 ⊕ С10100 ⊕ С10001 ⊕ С00101 ⊕ С10101 = 1 => С10101 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 = 1
Fж(10011) = С00000 ⊕ С10000 ⊕ С00010 ⊕ С00001 ⊕ С10010 ⊕ С10001 ⊕ С00011 ⊕ С10011 = 1 => С10011 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(01110) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С01100 ⊕ С01010 ⊕ С00110 ⊕ С01110 = 0 => С01110 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 = 1
Fж(01101) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С00001 ⊕ С01100 ⊕ С01001 ⊕ С00101 ⊕ С01101 = 0 => С01101 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 = 1
Fж(01011) = С00000 ⊕ С01000 ⊕ С00010 ⊕ С00001 ⊕ С01010 ⊕ С01001 ⊕ С00011 ⊕ С01011 = 0 => С01011 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 = 1
Fж(00111) = С00000 ⊕ С00100 ⊕ С00010 ⊕ С00001 ⊕ С00110 ⊕ С00101 ⊕ С00011 ⊕ С00111 = 0 => С00111 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 = 1
Fж(11110) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С11000 ⊕ С10100 ⊕ С10010 ⊕ С01100 ⊕ С01010 ⊕ С00110 ⊕ С11100 ⊕ С11010 ⊕ С10110 ⊕ С01110 ⊕ С11110 = 1 => С11110 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 = 1
Fж(11101) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00100 ⊕ С00001 ⊕ С11000 ⊕ С10100 ⊕ С10001 ⊕ С01100 ⊕ С01001 ⊕ С00101 ⊕ С11100 ⊕ С11001 ⊕ С10101 ⊕ С01101 ⊕ С11101 = 1 => С11101 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 = 1
Fж(11011) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00010 ⊕ С00001 ⊕ С11000 ⊕ С10010 ⊕ С10001 ⊕ С01010 ⊕ С01001 ⊕ С00011 ⊕ С11010 ⊕ С11001 ⊕ С10011 ⊕ С01011 ⊕ С11011 = 1 => С11011 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 = 1
Fж(10111) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С00010 ⊕ С00001 ⊕ С10100 ⊕ С10010 ⊕ С10001 ⊕ С00110 ⊕ С00101 ⊕ С00011 ⊕ С10110 ⊕ С10101 ⊕ С10011 ⊕ С00111 ⊕ С10111 = 1 => С10111 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 = 1
Fж(01111) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С00001 ⊕ С01100 ⊕ С01010 ⊕ С01001 ⊕ С00110 ⊕ С00101 ⊕ С00011 ⊕ С01110 ⊕ С01101 ⊕ С01011 ⊕ С00111 ⊕ С01111 = 0 => С01111 = 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 = 1
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 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 = 1

Таким образом, полином Жегалкина будет равен:
Fж = 1 ⊕ B∧D ⊕ B∧E ⊕ C∧D ⊕ C∧E ⊕ A∧B∧D ⊕ A∧B∧E ⊕ A∧C∧D ⊕ A∧C∧E ⊕ B∧C∧D ⊕ B∧C∧E ⊕ B∧D∧E ⊕ C∧D∧E ⊕ A∧B∧C∧D ⊕ A∧B∧C∧E ⊕ A∧B∧D∧E ⊕ A∧C∧D∧E ⊕ B∧C∧D∧E ⊕ A∧B∧C∧D∧E
Логическая схема, соответствующая полиному Жегалкина:

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