Для функции D∧C∧¬B∧¬A∨D∧¬B∧A∨E∧¬C∧B∧A∨C∧B∧A:


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

¬A:
A¬A
01
10

¬C:
C¬C
01
10

D∧C:
DCD∧C
000
010
100
111

(D∧C)∧(¬B):
DCBD∧C¬B(D∧C)∧(¬B)
000010
001000
010010
011000
100010
101000
110111
111100

((D∧C)∧(¬B))∧(¬A):
DCBAD∧C¬B(D∧C)∧(¬B)¬A((D∧C)∧(¬B))∧(¬A)
000001010
000101000
001000010
001100000
010001010
010101000
011000010
011100000
100001010
100101000
101000010
101100000
110011111
110111100
111010010
111110000

D∧(¬B):
DB¬BD∧(¬B)
0010
0100
1011
1100

(D∧(¬B))∧A:
DBA¬BD∧(¬B)(D∧(¬B))∧A
000100
001100
010000
011000
100110
101111
110000
111000

E∧(¬C):
EC¬CE∧(¬C)
0010
0100
1011
1100

(E∧(¬C))∧B:
ECB¬CE∧(¬C)(E∧(¬C))∧B
000100
001100
010000
011000
100110
101111
110000
111000

((E∧(¬C))∧B)∧A:
ECBA¬CE∧(¬C)(E∧(¬C))∧B((E∧(¬C))∧B)∧A
00001000
00011000
00101000
00111000
01000000
01010000
01100000
01110000
10001100
10011100
10101110
10111111
11000000
11010000
11100000
11110000

C∧B:
CBC∧B
000
010
100
111

(C∧B)∧A:
CBAC∧B(C∧B)∧A
00000
00100
01000
01100
10000
10100
11010
11111

(((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A):
DCBAD∧C¬B(D∧C)∧(¬B)¬A((D∧C)∧(¬B))∧(¬A)¬BD∧(¬B)(D∧(¬B))∧A(((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A)
0000010101000
0001010001000
0010000100000
0011000000000
0100010101000
0101010001000
0110000100000
0111000000000
1000010101100
1001010001111
1010000100000
1011000000000
1100111111101
1101111001111
1110100100000
1111100000000

((((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A))∨(((E∧(¬C))∧B)∧A):
DCBAED∧C¬B(D∧C)∧(¬B)¬A((D∧C)∧(¬B))∧(¬A)¬BD∧(¬B)(D∧(¬B))∧A(((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A)¬CE∧(¬C)(E∧(¬C))∧B((E∧(¬C))∧B)∧A((((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A))∨(((E∧(¬C))∧B)∧A)
0000001010100010000
0000101010100011000
0001001000100010000
0001101000100011000
0010000010000010000
0010100010000011100
0011000000000010000
0011100000000011111
0100001010100000000
0100101010100000000
0101001000100000000
0101101000100000000
0110000010000000000
0110100010000000000
0111000000000000000
0111100000000000000
1000001010110010000
1000101010110011000
1001001000111110001
1001101000111111001
1010000010000010000
1010100010000011100
1011000000000010000
1011100000000011111
1100011111110100001
1100111111110100001
1101011100111100001
1101111100111100001
1110010010000000000
1110110010000000000
1111010000000000000
1111110000000000000

(((((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A))∨(((E∧(¬C))∧B)∧A))∨((C∧B)∧A):
DCBAED∧C¬B(D∧C)∧(¬B)¬A((D∧C)∧(¬B))∧(¬A)¬BD∧(¬B)(D∧(¬B))∧A(((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A)¬CE∧(¬C)(E∧(¬C))∧B((E∧(¬C))∧B)∧A((((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A))∨(((E∧(¬C))∧B)∧A)C∧B(C∧B)∧A(((((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A))∨(((E∧(¬C))∧B)∧A))∨((C∧B)∧A)
0000001010100010000000
0000101010100011000000
0001001000100010000000
0001101000100011000000
0010000010000010000000
0010100010000011100000
0011000000000010000000
0011100000000011111001
0100001010100000000000
0100101010100000000000
0101001000100000000000
0101101000100000000000
0110000010000000000100
0110100010000000000100
0111000000000000000111
0111100000000000000111
1000001010110010000000
1000101010110011000000
1001001000111110001001
1001101000111111001001
1010000010000010000000
1010100010000011100000
1011000000000010000000
1011100000000011111001
1100011111110100001001
1100111111110100001001
1101011100111100001001
1101111100111100001001
1110010010000000000100
1110110010000000000100
1111010000000000000111
1111110000000000000111

Общая таблица истинности:

DCBAE¬B¬A¬CD∧C(D∧C)∧(¬B)((D∧C)∧(¬B))∧(¬A)D∧(¬B)(D∧(¬B))∧AE∧(¬C)(E∧(¬C))∧B((E∧(¬C))∧B)∧AC∧B(C∧B)∧A(((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A)((((D∧C)∧(¬B))∧(¬A))∨((D∧(¬B))∧A))∨(((E∧(¬C))∧B)∧A)D∧C∧¬B∧¬A∨D∧¬B∧A∨E∧¬C∧B∧A∨C∧B∧A
000001110000000000000
000011110000010000000
000101010000000000000
000111010000010000000
001000110000000000000
001010110000011000000
001100010000000000000
001110010000011100011
010001100000000000000
010011100000000000000
010101000000000000000
010111000000000000000
011000100000000010000
011010100000000010000
011100000000000011001
011110000000000011001
100001110001000000000
100011110001010000000
100101010001100000111
100111010001110000111
101000110000000000000
101010110000011000000
101100010000000000000
101110010000011100011
110001101111000000111
110011101111000000111
110101001101100000111
110111001101100000111
111000101000000010000
111010101000000010000
111100001000000011001
111110001000000011001

Логическая схема:

Совершенная дизъюнктивная нормальная форма (СДНФ):

По таблице истинности:
DCBAEF
000000
000010
000100
000110
001000
001010
001100
001111
010000
010010
010100
010110
011000
011010
011101
011111
100000
100010
100101
100111
101000
101010
101100
101111
110001
110011
110101
110111
111000
111010
111101
111111
Fсднф = ¬D∧¬C∧B∧A∧E ∨ ¬D∧C∧B∧A∧¬E ∨ ¬D∧C∧B∧A∧E ∨ D∧¬C∧¬B∧A∧¬E ∨ D∧¬C∧¬B∧A∧E ∨ D∧¬C∧B∧A∧E ∨ D∧C∧¬B∧¬A∧¬E ∨ D∧C∧¬B∧¬A∧E ∨ D∧C∧¬B∧A∧¬E ∨ D∧C∧¬B∧A∧E ∨ D∧C∧B∧A∧¬E ∨ D∧C∧B∧A∧E
Логическая cхема:

Совершенная конъюнктивная нормальная форма (СКНФ):

По таблице истинности:
DCBAEF
000000
000010
000100
000110
001000
001010
001100
001111
010000
010010
010100
010110
011000
011010
011101
011111
100000
100010
100101
100111
101000
101010
101100
101111
110001
110011
110101
110111
111000
111010
111101
111111
Fскнф = (D∨C∨B∨A∨E) ∧ (D∨C∨B∨A∨¬E) ∧ (D∨C∨B∨¬A∨E) ∧ (D∨C∨B∨¬A∨¬E) ∧ (D∨C∨¬B∨A∨E) ∧ (D∨C∨¬B∨A∨¬E) ∧ (D∨C∨¬B∨¬A∨E) ∧ (D∨¬C∨B∨A∨E) ∧ (D∨¬C∨B∨A∨¬E) ∧ (D∨¬C∨B∨¬A∨E) ∧ (D∨¬C∨B∨¬A∨¬E) ∧ (D∨¬C∨¬B∨A∨E) ∧ (D∨¬C∨¬B∨A∨¬E) ∧ (¬D∨C∨B∨A∨E) ∧ (¬D∨C∨B∨A∨¬E) ∧ (¬D∨C∨¬B∨A∨E) ∧ (¬D∨C∨¬B∨A∨¬E) ∧ (¬D∨C∨¬B∨¬A∨E) ∧ (¬D∨¬C∨¬B∨A∨E) ∧ (¬D∨¬C∨¬B∨A∨¬E)
Логическая cхема:

Построение полинома Жегалкина:

По таблице истинности функции
DCBAEFж
000000
000010
000100
000110
001000
001010
001100
001111
010000
010010
010100
010110
011000
011010
011101
011111
100000
100010
100101
100111
101000
101010
101100
101111
110001
110011
110101
110111
111000
111010
111101
111111

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

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 = 1 => С11000 = 0 ⊕ 0 ⊕ 0 ⊕ 1 = 1
Fж(10100) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С10100 = 0 => С10100 = 0 ⊕ 0 ⊕ 0 ⊕ 0 = 0
Fж(10010) = С00000 ⊕ С10000 ⊕ С00010 ⊕ С10010 = 1 => С10010 = 0 ⊕ 0 ⊕ 0 ⊕ 1 = 1
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 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 = 1
Fж(11010) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00010 ⊕ С11000 ⊕ С10010 ⊕ С01010 ⊕ С11010 = 1 => С11010 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 = 1
Fж(11001) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00001 ⊕ С11000 ⊕ С10001 ⊕ С01001 ⊕ С11001 = 1 => С11001 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(10110) = С00000 ⊕ С10000 ⊕ С00100 ⊕ С00010 ⊕ С10100 ⊕ С10010 ⊕ С00110 ⊕ С10110 = 0 => С10110 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 = 1
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 = 1 => С10011 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
Fж(01110) = С00000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С01100 ⊕ С01010 ⊕ С00110 ⊕ С01110 = 1 => С01110 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 1
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 = 1 => С00111 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 = 1
Fж(11110) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00100 ⊕ С00010 ⊕ С11000 ⊕ С10100 ⊕ С10010 ⊕ С01100 ⊕ С01010 ⊕ С00110 ⊕ С11100 ⊕ С11010 ⊕ С10110 ⊕ С01110 ⊕ С11110 = 1 => С11110 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 = 1
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 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 = 0
Fж(11011) = С00000 ⊕ С10000 ⊕ С01000 ⊕ С00010 ⊕ С00001 ⊕ С11000 ⊕ С10010 ⊕ С10001 ⊕ С01010 ⊕ С01001 ⊕ С00011 ⊕ С11010 ⊕ С11001 ⊕ С10011 ⊕ С01011 ⊕ С11011 = 1 => С11011 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 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 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 = 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 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 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 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 = 0

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

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