Таблица истинности для функции ¬(X1∨X2)∨X3≡1∧¬(X2∨X3)∨X4≡1∧¬(X3∨X4)∨X5≡1:


Промежуточные таблицы истинности:
X1∨X2:
X1X2X1∨X2
000
011
101
111

X2∨X3:
X2X3X2∨X3
000
011
101
111

X3∨X4:
X3X4X3∨X4
000
011
101
111

¬(X1∨X2):
X1X2X1∨X2¬(X1∨X2)
0001
0110
1010
1110

¬(X2∨X3):
X2X3X2∨X3¬(X2∨X3)
0001
0110
1010
1110

¬(X3∨X4):
X3X4X3∨X4¬(X3∨X4)
0001
0110
1010
1110

1∧(¬(X2∨X3)):
X2X3X2∨X3¬(X2∨X3)1∧(¬(X2∨X3))
00011
01100
10100
11100

1∧(¬(X3∨X4)):
X3X4X3∨X4¬(X3∨X4)1∧(¬(X3∨X4))
00011
01100
10100
11100

(¬(X1∨X2))∨X3:
X1X2X3X1∨X2¬(X1∨X2)(¬(X1∨X2))∨X3
000011
001011
010100
011101
100100
101101
110100
111101

(1∧(¬(X2∨X3)))∨X4:
X2X3X4X2∨X3¬(X2∨X3)1∧(¬(X2∨X3))(1∧(¬(X2∨X3)))∨X4
0000111
0010111
0101000
0111001
1001000
1011001
1101000
1111001

(1∧(¬(X3∨X4)))∨X5:
X3X4X5X3∨X4¬(X3∨X4)1∧(¬(X3∨X4))(1∧(¬(X3∨X4)))∨X5
0000111
0010111
0101000
0111001
1001000
1011001
1101000
1111001

((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4):
X1X2X3X4X1∨X2¬(X1∨X2)(¬(X1∨X2))∨X3X2∨X3¬(X2∨X3)1∧(¬(X2∨X3))(1∧(¬(X2∨X3)))∨X4((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4)
000001101111
000101101111
001001110000
001101110011
010010010001
010110010010
011010110000
011110110011
100010001110
100110001110
101010110000
101110110011
110010010001
110110010010
111010110000
111110110011

(((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4))≡((1∧(¬(X3∨X4)))∨X5):
X1X2X3X4X5X1∨X2¬(X1∨X2)(¬(X1∨X2))∨X3X2∨X3¬(X2∨X3)1∧(¬(X2∨X3))(1∧(¬(X2∨X3)))∨X4((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4)X3∨X4¬(X3∨X4)1∧(¬(X3∨X4))(1∧(¬(X3∨X4)))∨X5(((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4))≡((1∧(¬(X3∨X4)))∨X5)
000000110111101111
000010110111101111
000100110111110000
000110110111110011
001000111000010001
001010111000010010
001100111001110000
001110111001110011
010001001000101111
010011001000101111
010101001001010001
010111001001010010
011001011000010001
011011011000010010
011101011001110000
011111011001110011
100001000111001110
100011000111001110
100101000111010001
100111000111010010
101001011000010001
101011011000010010
101101011001110000
101111011001110011
110001001000101111
110011001000101111
110101001001010001
110111001001010010
111001011000010001
111011011000010010
111101011001110000
111111011001110011

((((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4))≡((1∧(¬(X3∨X4)))∨X5))≡1:
X1X2X3X4X5X1∨X2¬(X1∨X2)(¬(X1∨X2))∨X3X2∨X3¬(X2∨X3)1∧(¬(X2∨X3))(1∧(¬(X2∨X3)))∨X4((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4)X3∨X4¬(X3∨X4)1∧(¬(X3∨X4))(1∧(¬(X3∨X4)))∨X5(((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4))≡((1∧(¬(X3∨X4)))∨X5)((((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4))≡((1∧(¬(X3∨X4)))∨X5))≡1
0000001101111011111
0000101101111011111
0001001101111100000
0001101101111100111
0010001110000100011
0010101110000100100
0011001110011100000
0011101110011100111
0100010010001011111
0100110010001011111
0101010010010100011
0101110010010100100
0110010110000100011
0110110110000100100
0111010110011100000
0111110110011100111
1000010001110011100
1000110001110011100
1001010001110100011
1001110001110100100
1010010110000100011
1010110110000100100
1011010110011100000
1011110110011100111
1100010010001011111
1100110010001011111
1101010010010100011
1101110010010100100
1110010110000100011
1110110110000100100
1111010110011100000
1111110110011100111

Общая таблица истинности:

X1X2X3X4X5X1∨X2X2∨X3X3∨X4¬(X1∨X2)¬(X2∨X3)¬(X3∨X4)1∧(¬(X2∨X3))1∧(¬(X3∨X4))(¬(X1∨X2))∨X3(1∧(¬(X2∨X3)))∨X4(1∧(¬(X3∨X4)))∨X5((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4)(((¬(X1∨X2))∨X3)≡((1∧(¬(X2∨X3)))∨X4))≡((1∧(¬(X3∨X4)))∨X5)¬(X1∨X2)∨X3≡1∧¬(X2∨X3)∨X4≡1∧¬(X3∨X4)∨X5≡1
0000000011111111111
0000100011111111111
0001000111010110100
0001100111010111111
0010001110000100011
0010101110000101000
0011001110000110100
0011101110000111111
0100011000101001111
0100111000101001111
0101011100000010011
0101111100000011000
0110011100000100011
0110111100000101000
0111011100000110100
0111111100000111111
1000010001111011000
1000110001111011000
1001010101010010011
1001110101010011000
1010011100000100011
1010111100000101000
1011011100000110100
1011111100000111111
1100011000101001111
1100111000101001111
1101011100000010011
1101111100000011000
1110011100000100011
1110111100000101000
1111011100000110100
1111111100000111111

Логическая схема:

Совершенная дизъюнктивная нормальная форма (СДНФ):

По таблице истинности:
X1X2X3X4X5F
000001
000011
000100
000111
001001
001010
001100
001111
010001
010011
010101
010110
011001
011010
011100
011111
100000
100010
100101
100110
101001
101010
101100
101111
110001
110011
110101
110110
111001
111010
111100
111111
Fсднф = ¬X1∧¬X2∧¬X3∧¬X4∧¬X5 ∨ ¬X1∧¬X2∧¬X3∧¬X4∧X5 ∨ ¬X1∧¬X2∧¬X3∧X4∧X5 ∨ ¬X1∧¬X2∧X3∧¬X4∧¬X5 ∨ ¬X1∧¬X2∧X3∧X4∧X5 ∨ ¬X1∧X2∧¬X3∧¬X4∧¬X5 ∨ ¬X1∧X2∧¬X3∧¬X4∧X5 ∨ ¬X1∧X2∧¬X3∧X4∧¬X5 ∨ ¬X1∧X2∧X3∧¬X4∧¬X5 ∨ ¬X1∧X2∧X3∧X4∧X5 ∨ X1∧¬X2∧¬X3∧X4∧¬X5 ∨ X1∧¬X2∧X3∧¬X4∧¬X5 ∨ X1∧¬X2∧X3∧X4∧X5 ∨ X1∧X2∧¬X3∧¬X4∧¬X5 ∨ X1∧X2∧¬X3∧¬X4∧X5 ∨ X1∧X2∧¬X3∧X4∧¬X5 ∨ X1∧X2∧X3∧¬X4∧¬X5 ∨ X1∧X2∧X3∧X4∧X5
Логическая cхема:

Совершенная конъюнктивная нормальная форма (СКНФ):

По таблице истинности:
X1X2X3X4X5F
000001
000011
000100
000111
001001
001010
001100
001111
010001
010011
010101
010110
011001
011010
011100
011111
100000
100010
100101
100110
101001
101010
101100
101111
110001
110011
110101
110110
111001
111010
111100
111111
Fскнф = (X1∨X2∨X3∨¬X4∨X5) ∧ (X1∨X2∨¬X3∨X4∨¬X5) ∧ (X1∨X2∨¬X3∨¬X4∨X5) ∧ (X1∨¬X2∨X3∨¬X4∨¬X5) ∧ (X1∨¬X2∨¬X3∨X4∨¬X5) ∧ (X1∨¬X2∨¬X3∨¬X4∨X5) ∧ (¬X1∨X2∨X3∨X4∨X5) ∧ (¬X1∨X2∨X3∨X4∨¬X5) ∧ (¬X1∨X2∨X3∨¬X4∨¬X5) ∧ (¬X1∨X2∨¬X3∨X4∨¬X5) ∧ (¬X1∨X2∨¬X3∨¬X4∨X5) ∧ (¬X1∨¬X2∨X3∨¬X4∨¬X5) ∧ (¬X1∨¬X2∨¬X3∨X4∨¬X5) ∧ (¬X1∨¬X2∨¬X3∨¬X4∨X5)
Логическая cхема:

Построение полинома Жегалкина:

По таблице истинности функции
X1X2X3X4X5Fж
000001
000011
000100
000111
001001
001010
001100
001111
010001
010011
010101
010110
011001
011010
011100
011111
100000
100010
100101
100110
101001
101010
101100
101111
110001
110011
110101
110110
111001
111010
111100
111111

Построим полином Жегалкина:
Fж = C00000 ⊕ C10000∧X1 ⊕ C01000∧X2 ⊕ C00100∧X3 ⊕ C00010∧X4 ⊕ C00001∧X5 ⊕ C11000∧X1∧X2 ⊕ C10100∧X1∧X3 ⊕ C10010∧X1∧X4 ⊕ C10001∧X1∧X5 ⊕ C01100∧X2∧X3 ⊕ C01010∧X2∧X4 ⊕ C01001∧X2∧X5 ⊕ C00110∧X3∧X4 ⊕ C00101∧X3∧X5 ⊕ C00011∧X4∧X5 ⊕ C11100∧X1∧X2∧X3 ⊕ C11010∧X1∧X2∧X4 ⊕ C11001∧X1∧X2∧X5 ⊕ C10110∧X1∧X3∧X4 ⊕ C10101∧X1∧X3∧X5 ⊕ C10011∧X1∧X4∧X5 ⊕ C01110∧X2∧X3∧X4 ⊕ C01101∧X2∧X3∧X5 ⊕ C01011∧X2∧X4∧X5 ⊕ C00111∧X3∧X4∧X5 ⊕ C11110∧X1∧X2∧X3∧X4 ⊕ C11101∧X1∧X2∧X3∧X5 ⊕ C11011∧X1∧X2∧X4∧X5 ⊕ C10111∧X1∧X3∧X4∧X5 ⊕ C01111∧X2∧X3∧X4∧X5 ⊕ C11111∧X1∧X2∧X3∧X4∧X5

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

Таким образом, полином Жегалкина будет равен:
Fж = 1 ⊕ X1 ⊕ X4 ⊕ X1∧X2 ⊕ X1∧X3 ⊕ X2∧X4 ⊕ X3∧X5 ⊕ X4∧X5 ⊕ X1∧X2∧X3 ⊕ X2∧X3∧X4 ⊕ X3∧X4∧X5
Логическая схема, соответствующая полиному Жегалкина:

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