Для функции (A∨B)∧¬(A∧¬B)∧X∧O∧R∧B:


Промежуточные таблицы истинности:
A∨B:
ABA∨B
000
011
101
111

¬B:
B¬B
01
10

A∧(¬B):
AB¬BA∧(¬B)
0010
0100
1011
1100

¬(A∧(¬B)):
AB¬BA∧(¬B)¬(A∧(¬B))
00101
01001
10110
11001

(A∨B)∧(¬(A∧(¬B))):
ABA∨B¬BA∧(¬B)¬(A∧(¬B))(A∨B)∧(¬(A∧(¬B)))
0001010
0110011
1011100
1110011

((A∨B)∧(¬(A∧(¬B))))∧X:
ABXA∨B¬BA∧(¬B)¬(A∧(¬B))(A∨B)∧(¬(A∧(¬B)))((A∨B)∧(¬(A∧(¬B))))∧X
000010100
001010100
010100110
011100111
100111000
101111000
110100110
111100111

(((A∨B)∧(¬(A∧(¬B))))∧X)∧O:
ABXOA∨B¬BA∧(¬B)¬(A∧(¬B))(A∨B)∧(¬(A∧(¬B)))((A∨B)∧(¬(A∧(¬B))))∧X(((A∨B)∧(¬(A∧(¬B))))∧X)∧O
00000101000
00010101000
00100101000
00110101000
01001001100
01011001100
01101001110
01111001111
10001110000
10011110000
10101110000
10111110000
11001001100
11011001100
11101001110
11111001111

((((A∨B)∧(¬(A∧(¬B))))∧X)∧O)∧R:
ABXORA∨B¬BA∧(¬B)¬(A∧(¬B))(A∨B)∧(¬(A∧(¬B)))((A∨B)∧(¬(A∧(¬B))))∧X(((A∨B)∧(¬(A∧(¬B))))∧X)∧O((((A∨B)∧(¬(A∧(¬B))))∧X)∧O)∧R
0000001010000
0000101010000
0001001010000
0001101010000
0010001010000
0010101010000
0011001010000
0011101010000
0100010011000
0100110011000
0101010011000
0101110011000
0110010011100
0110110011100
0111010011110
0111110011111
1000011100000
1000111100000
1001011100000
1001111100000
1010011100000
1010111100000
1011011100000
1011111100000
1100010011000
1100110011000
1101010011000
1101110011000
1110010011100
1110110011100
1111010011110
1111110011111

(((((A∨B)∧(¬(A∧(¬B))))∧X)∧O)∧R)∧B:
ABXORA∨B¬BA∧(¬B)¬(A∧(¬B))(A∨B)∧(¬(A∧(¬B)))((A∨B)∧(¬(A∧(¬B))))∧X(((A∨B)∧(¬(A∧(¬B))))∧X)∧O((((A∨B)∧(¬(A∧(¬B))))∧X)∧O)∧R(((((A∨B)∧(¬(A∧(¬B))))∧X)∧O)∧R)∧B
00000010100000
00001010100000
00010010100000
00011010100000
00100010100000
00101010100000
00110010100000
00111010100000
01000100110000
01001100110000
01010100110000
01011100110000
01100100111000
01101100111000
01110100111100
01111100111111
10000111000000
10001111000000
10010111000000
10011111000000
10100111000000
10101111000000
10110111000000
10111111000000
11000100110000
11001100110000
11010100110000
11011100110000
11100100111000
11101100111000
11110100111100
11111100111111

Общая таблица истинности:

ABXORA∨B¬BA∧(¬B)¬(A∧(¬B))(A∨B)∧(¬(A∧(¬B)))((A∨B)∧(¬(A∧(¬B))))∧X(((A∨B)∧(¬(A∧(¬B))))∧X)∧O((((A∨B)∧(¬(A∧(¬B))))∧X)∧O)∧R(A∨B)∧¬(A∧¬B)∧X∧O∧R∧B
00000010100000
00001010100000
00010010100000
00011010100000
00100010100000
00101010100000
00110010100000
00111010100000
01000100110000
01001100110000
01010100110000
01011100110000
01100100111000
01101100111000
01110100111100
01111100111111
10000111000000
10001111000000
10010111000000
10011111000000
10100111000000
10101111000000
10110111000000
10111111000000
11000100110000
11001100110000
11010100110000
11011100110000
11100100111000
11101100111000
11110100111100
11111100111111

Логическая схема:

Совершенная дизъюнктивная нормальная форма (СДНФ):

По таблице истинности:
ABXORF
000000
000010
000100
000110
001000
001010
001100
001110
010000
010010
010100
010110
011000
011010
011100
011111
100000
100010
100100
100110
101000
101010
101100
101110
110000
110010
110100
110110
111000
111010
111100
111111
Fсднф = ¬A∧B∧X∧O∧R ∨ A∧B∧X∧O∧R
Логическая cхема:

Совершенная конъюнктивная нормальная форма (СКНФ):

По таблице истинности:
ABXORF
000000
000010
000100
000110
001000
001010
001100
001110
010000
010010
010100
010110
011000
011010
011100
011111
100000
100010
100100
100110
101000
101010
101100
101110
110000
110010
110100
110110
111000
111010
111100
111111
Fскнф = (A∨B∨X∨O∨R) ∧ (A∨B∨X∨O∨¬R) ∧ (A∨B∨X∨¬O∨R) ∧ (A∨B∨X∨¬O∨¬R) ∧ (A∨B∨¬X∨O∨R) ∧ (A∨B∨¬X∨O∨¬R) ∧ (A∨B∨¬X∨¬O∨R) ∧ (A∨B∨¬X∨¬O∨¬R) ∧ (A∨¬B∨X∨O∨R) ∧ (A∨¬B∨X∨O∨¬R) ∧ (A∨¬B∨X∨¬O∨R) ∧ (A∨¬B∨X∨¬O∨¬R) ∧ (A∨¬B∨¬X∨O∨R) ∧ (A∨¬B∨¬X∨O∨¬R) ∧ (A∨¬B∨¬X∨¬O∨R) ∧ (¬A∨B∨X∨O∨R) ∧ (¬A∨B∨X∨O∨¬R) ∧ (¬A∨B∨X∨¬O∨R) ∧ (¬A∨B∨X∨¬O∨¬R) ∧ (¬A∨B∨¬X∨O∨R) ∧ (¬A∨B∨¬X∨O∨¬R) ∧ (¬A∨B∨¬X∨¬O∨R) ∧ (¬A∨B∨¬X∨¬O∨¬R) ∧ (¬A∨¬B∨X∨O∨R) ∧ (¬A∨¬B∨X∨O∨¬R) ∧ (¬A∨¬B∨X∨¬O∨R) ∧ (¬A∨¬B∨X∨¬O∨¬R) ∧ (¬A∨¬B∨¬X∨O∨R) ∧ (¬A∨¬B∨¬X∨O∨¬R) ∧ (¬A∨¬B∨¬X∨¬O∨R)
Логическая cхема:

Построение полинома Жегалкина:

По таблице истинности функции
ABXORFж
000000
000010
000100
000110
001000
001010
001100
001110
010000
010010
010100
010110
011000
011010
011100
011111
100000
100010
100100
100110
101000
101010
101100
101110
110000
110010
110100
110110
111000
111010
111100
111111

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

Так как Fж(00000) = 0, то С00000 = 0.

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

Таким образом, полином Жегалкина будет равен:
Fж = B∧X∧O∧R
Логическая схема, соответствующая полиному Жегалкина:

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