решение идз 3 из пособия вариант 7
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МИНИСТЕРСТВО НАУКИ И ВЫСШЕГО ОБРАЗОВАНИЯ РФ
ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ БЮДЖЕТНОЕ ОБРАЗОВАТЕЛЬНОЕ УЧРЕЖДЕНИЕ ВЫСШЕГО ОБРАЗОВАНИЯ «ЛИПЕЦКИЙ ГОСУДАРСТВЕННЫЙ ТЕХНИЧЕСКИЙ УНИВЕРСИТЕТ»
Институт (факультет)
Компьютерных наук
Кафедра
Прикладной математики
Индивидуальное домашнее заданий №3
По дисциплине: «Дискретная математика».
На тему: «Алгебра Жегалкина.»
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Студент
АС-23-2
Корниенко А.С.
группа
подпись, дата
фамилия, инициалы
Руководитель
Жихорева С.В.
ученая степень, ученое звание
подпись, дата
фамилия, инициалы
Липецк 2024
Вариант 7.
№1 Построить полиномы Жегалкина для булевых функций
а) f(x,y) = x ∧ y | x = ¬((x ∧ y) ∧ x)
f(0,0) = a00 = 1 ⇒ a00 = 1
f(0,1) = a00 ⊕ a01 = 1 ⊕ a01 = 1 ⇒ a01 = 0
f(1,0) = a00 ⊕ a10 = 1 ⊕ a10 = 1 ⇒ a10 = 0
f(1,1) = a00 ⊕ a01 ⊕ a10 ⊕ a11 = 1 ⊕ 0 ⊕ 0 ⊕ a11 = 0 ⇒ a11 = 1
f(x,y) = 1 ⊕ xy
б) f(x,y) = x ∨ y ∧ ¬x | y = ¬((x ∨ y ¬x) ∧ y)
f(0,0) = a00 = 1 ⇒ a00 = 1
f(0,1) = a00 ⊕ a01 = 1 ⊕ a01 = 0 ⇒ a01 = 1
f(1,0) = a00 ⊕ a10 = 1 ⊕ a10 = 1 ⇒ a10 = 0
f(1,1) = a00 ⊕ a01 ⊕ a10 ⊕ a11 = 1 ⊕ 1 ⊕ 0 ⊕ a11 = 0 ⇒ a11 = 0
f(x,y) = 1 ⊕ y
в) f(x,y) = ¬x ~ y ⊕ ¬y ↓ x ∧ y = ¬((¬x ~ y ⊕ ¬y) ∨ ( x ∧ y))
f(0,0) = a00 = 0 ⇒ a00 = 0
f(0,1) = a00 ⊕ a01 = 0 ⊕ a01 = 0 ⇒ a01 = 0
f(1,0) = a00 ⊕ a10 = 0 ⊕ a10 = 1 ⇒ a10 = 1
f(1,1) = a00 ⊕ a01 ⊕ a10 ⊕ a11 = 0 ⊕ 0 ⊕ 1 ⊕ a11 = 0 ⇒ a11 = 1
f(x,y) = x ⊕ xy
№2
а) f(x,y,z) = x∧z∨y
f(0,0,0) = a000 = 0 ⇒ a000 = 0
f(0,0,1) = a000 ⊕ a001 = 0 ⊕ a001 = 0 ⇒ a001 = 0
f(0,1,0) = a000 ⊕ a010 = 0 ⊕ a010 = 1 ⇒ a010 = 1
f(1,0,0) = a000 ⊕ a100 = 0 ⊕ a100 = 0 ⇒ a100 = 0
f(0,1,1) = a000 ⊕ a001 ⊕ a010 ⊕ a011 = 0 ⊕ 0 ⊕ 1 ⊕ a011 = 1 ⇒ a011 = 0
f(1,0,1) = a000 ⊕ a001 ⊕ a100 ⊕ a101 = 0 ⊕ 0 ⊕ 0 ⊕ a101 = 1 ⇒ a101 = 1
f(1,1,0) = a000 ⊕ a010 ⊕ a100 ⊕ a110 = 0 ⊕ 1 ⊕ 0 ⊕ a110 = 1 ⇒ a110 = 0
f(1,1,1) = a000 ⊕ a001 ⊕ a010 ⊕ a100 ⊕ a011 ⊕ a101 ⊕ a110 ⊕ a111 = 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ a111 = 1 ⇒ a111 = 1
f(x,y,z) = y ⊕ xz ⊕ xyz
б) f(x,y,z) = ¬x ↓ y ⊕ ¬z ↑ y = ¬(¬x ∨ (y ⊕ ¬z ↑ y))
f(0,0,0) = a000 = 0 ⇒ a000 = 0
f(0,0,1) = a000 ⊕ a001 = 0 ⊕ a001 = 0 ⇒ a001 = 0
f(0,1,0) = a000 ⊕ a010 = 0 ⊕ a010 = 0 ⇒ a010 = 0
f(1,0,0) = a000 ⊕ a100 = 0 ⊕ a100 = 0 ⇒ a100 = 0
f(0,1,1) = a000 ⊕ a001 ⊕ a010 ⊕ a011 = 0 ⊕ 0 ⊕ 0 ⊕ a011 = 0 ⇒ a011 = 0
f(1,0,1) = a000 ⊕ a001 ⊕ a100 ⊕ a101 = 0 ⊕ 0 ⊕ 0 ⊕ a101 = 0 ⇒ a101 = 0
f(1,1,0) = a000 ⊕ a010 ⊕ a100 ⊕ a110 = 0 ⊕ 0 ⊕ 0 ⊕ a110 = 0 ⇒ a110 = 0
f(1,1,1) = a000 ⊕ a001 ⊕ a010 ⊕ a100 ⊕ a011 ⊕ a101 ⊕ a110 ⊕ a111 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ a111 = 1 ⇒ a111 = 1
f(x,y,z) = xyz
в) f(x,y,z) = z⊕¬y∧x→x∨¬z
f(0,0,0) = a000 = 1 ⇒ a000 = 1
f(0,0,1) = a000 ⊕ a001 = 1 ⊕ a001 = 0 ⇒ a001 = 1
f(0,1,0) = a000 ⊕ a010 = 1 ⊕ a010 = 1 ⇒ a010 = 0
f(1,0,0) = a000 ⊕ a100 = 1 ⊕ a100 = 1 ⇒ a100 = 0
f(0,1,1) = a000 ⊕ a001 ⊕ a010 ⊕ a011 = 1 ⊕ 1 ⊕ 0 ⊕ a011 = 0 ⇒ a011 = 0
f(1,0,1) = a000 ⊕ a001 ⊕ a100 ⊕ a101 = 1 ⊕ 1 ⊕ 0 ⊕ a101 = 1 ⇒ a101 = 1
f(1,1,0) = a000 ⊕ a010 ⊕ a100 ⊕ a110 = 1 ⊕ 0 ⊕ 0 ⊕ a110 = 1 ⇒ a110 = 0
f(1,1,1) = a000 ⊕ a001 ⊕ a010 ⊕ a100 ⊕ a011 ⊕ a101 ⊕ a110 ⊕ a111 = 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ a111 = 1 ⇒ a111 = 0
f(x,y,z) = 1 ⊕ z ⊕ xz
№3
а) f(x,y,z,t) = y∨t≡x↓z
f(0,0,0,0) = a0000 = 0 ⇒ a0000 = 0
f(0,0,0,1) = a0000 ⊕ a0001 = 0 ⊕ a0001 = 1 ⇒ a0001 = 1
f(0,0,1,0) = a0000 ⊕ a0010 = 0 ⊕ a0010 = 1 ⇒ a0010 = 1
f(0,1,0,0) = a0000 ⊕ a0100 = 0 ⊕ a0100 = 1 ⇒ a0100 = 1
f(1,0,0,0) = a0000 ⊕ a1000 = 0 ⊕ a1000 = 1 ⇒ a1000 = 1
f(0,0,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0011 = 0 ⊕ 1 ⊕ 1 ⊕ a0011 = 0 ⇒ a0011 = 0
f(0,1,0,1) = a0000 ⊕ a0001 ⊕ a0100 ⊕ a0101 = 0 ⊕ 1 ⊕ 1 ⊕ a0101 = 1 ⇒ a0101 = 1
f(0,1,1,0) = a0000 ⊕ a0010 ⊕ a0100 ⊕ a0110 = 0 ⊕ 1 ⊕ 1 ⊕ a0110 = 0 ⇒ a0110 = 0
f(1,0,0,1) = a0000 ⊕ a0001 ⊕ a1000 ⊕ a1001 = 0 ⊕ 1 ⊕ 1 ⊕ a1001 = 0 ⇒ a1001 = 0
f(1,0,1,0) = a0000 ⊕ a0010 ⊕ a1000 ⊕ a1010 = 0 ⊕ 1 ⊕ 1 ⊕ a1010 = 1 ⇒ a1010 = 1
f(1,1,0,0) = a0000 ⊕ a0100 ⊕ a1000 ⊕ a1100 = 0 ⊕ 1 ⊕ 1 ⊕ a1100 = 0 ⇒ a1100 = 0
f(0,1,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0100 ⊕ a0011 ⊕ a0101 ⊕ a0110 ⊕ a0111 = 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ a0111 = 0 ⇒ a0111 = 0
f(1,0,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a1000 ⊕ a0011 ⊕ a1001 ⊕ a1010 ⊕ a1011 = 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ a1011 = 0 ⇒ a1011 = 0
f(1,1,0,1) = a0000 ⊕ a0001 ⊕ a0100 ⊕ a1000 ⊕ a0101 ⊕ a1001 ⊕ a1100 ⊕ a1101 = 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ a1101 = 0 ⇒ a1101 = 0
f(1,1,1,0) = a0000 ⊕ a0010 ⊕ a0100 ⊕ a1000 ⊕ a0110 ⊕ a1010 ⊕ a1100 ⊕ a1110 = 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ a1110 = 0 ⇒ a1110 = 0
f(1,1,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0100 ⊕ a1000 ⊕ a0011 ⊕ a0101 ⊕ a0110 ⊕ a1001 ⊕ a1010 ⊕ a1100 ⊕ a0111 ⊕ a1011 ⊕ a1101 ⊕ a1110 ⊕ a1111 = 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ a1111 = 0 ⇒ a1111 = 0
f(x,y,z,t) = z ⊕ y ⊕ x ⊕ t ⊕ xz ⊕ ty
б) f(x,y,z,t) = y∧x⊕z≡t→x
f(0,0,0,0) = a0000 = 0 ⇒ a0000 = 0
f(0,0,0,1) = a0000 ⊕ a0001 = 0 ⊕ a0001 = 1 ⇒ a0001 = 1
f(0,0,1,0) = a0000 ⊕ a0010 = 0 ⊕ a0010 = 0 ⇒ a0010 = 0
f(0,1,0,0) = a0000 ⊕ a0100 = 0 ⊕ a0100 = 0 ⇒ a0100 = 0
f(1,0,0,0) = a0000 ⊕ a1000 = 0 ⊕ a1000 = 1 ⇒ a1000 = 1
f(0,0,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0011 = 0 ⊕ 1 ⊕ 0 ⊕ a0011 = 1 ⇒ a0011 = 0
f(0,1,0,1) = a0000 ⊕ a0001 ⊕ a0100 ⊕ a0101 = 0 ⊕ 1 ⊕ 0 ⊕ a0101 = 1 ⇒ a0101 = 0
f(0,1,1,0) = a0000 ⊕ a0010 ⊕ a0100 ⊕ a0110 = 0 ⊕ 0 ⊕ 0 ⊕ a0110 = 1 ⇒ a0110 = 1
f(1,0,0,1) = a0000 ⊕ a0001 ⊕ a1000 ⊕ a1001 = 0 ⊕ 1 ⊕ 1 ⊕ a1001 = 0 ⇒ a1001 = 0
f(1,0,1,0) = a0000 ⊕ a0010 ⊕ a1000 ⊕ a1010 = 0 ⊕ 0 ⊕ 1 ⊕ a1010 = 1 ⇒ a1010 = 0
f(1,1,0,0) = a0000 ⊕ a0100 ⊕ a1000 ⊕ a1100 = 0 ⊕ 0 ⊕ 1 ⊕ a1100 = 0 ⇒ a1100 = 1
f(0,1,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0100 ⊕ a0011 ⊕ a0101 ⊕ a0110 ⊕ a0111 = 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ a0111 = 0 ⇒ a0111 = 0
f(1,0,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a1000 ⊕ a0011 ⊕ a1001 ⊕ a1010 ⊕ a1011 = 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ a1011 = 0 ⇒ a1011 = 0
f(1,1,0,1) = a0000 ⊕ a0001 ⊕ a0100 ⊕ a1000 ⊕ a0101 ⊕ a1001 ⊕ a1100 ⊕ a1101 = 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ a1101 = 1 ⇒ a1101 = 0
f(1,1,1,0) = a0000 ⊕ a0010 ⊕ a0100 ⊕ a1000 ⊕ a0110 ⊕ a1010 ⊕ a1100 ⊕ a1110 = 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ a1110 = 1 ⇒ a1110 = 0
f(1,1,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0100 ⊕ a1000 ⊕ a0011 ⊕ a0101 ⊕ a0110 ⊕ a1001 ⊕ a1010 ⊕ a1100 ⊕ a0111 ⊕ a1011 ⊕ a1101 ⊕ a1110 ⊕ a1111 = 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ a1111 = 0 ⇒ a1111 = 0
f(x,y,z,t) = z ⊕ t ⊕ xy ⊕ tx
в) f(x,y,z,t) = y→ x∨t⊕z∧y≡x
f(0,0,0,0) = a0000 = 0 ⇒ a0000 = 0
f(0,0,0,1) = a0000 ⊕ a0001 = 0 ⊕ a0001 = 0 ⇒ a0001 = 0
f(0,0,1,0) = a0000 ⊕ a0010 = 0 ⊕ a0010 = 1 ⇒ a0010 = 1
f(0,1,0,0) = a0000 ⊕ a0100 = 0 ⊕ a0100 = 1 ⇒ a0100 = 1
f(1,0,0,0) = a0000 ⊕ a1000 = 0 ⊕ a1000 = 0 ⇒ a1000 = 0
f(0,0,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0011 = 0 ⊕ 0 ⊕ 1 ⊕ a0011 = 0 ⇒ a0011 = 1
f(0,1,0,1) = a0000 ⊕ a0001 ⊕ a0100 ⊕ a0101 = 0 ⊕ 0 ⊕ 1 ⊕ a0101 = 1 ⇒ a0101 = 0
f(0,1,1,0) = a0000 ⊕ a0010 ⊕ a0100 ⊕ a0110 = 0 ⊕ 1 ⊕ 1 ⊕ a0110 = 1 ⇒ a0110 = 1
f(1,0,0,1) = a0000 ⊕ a0001 ⊕ a1000 ⊕ a1001 = 0 ⊕ 0 ⊕ 0 ⊕ a1001 = 0 ⇒ a1001 = 0
f(1,0,1,0) = a0000 ⊕ a0010 ⊕ a1000 ⊕ a1010 = 0 ⊕ 1 ⊕ 0 ⊕ a1010 = 0 ⇒ a1010 = 1
f(1,1,0,0) = a0000 ⊕ a0100 ⊕ a1000 ⊕ a1100 = 0 ⊕ 1 ⊕ 0 ⊕ a1100 = 1 ⇒ a1100 = 0
f(0,1,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0100 ⊕ a0011 ⊕ a0101 ⊕ a0110 ⊕ a0111 = 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ a0111 = 0 ⇒ a0111 = 0
f(1,0,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a1000 ⊕ a0011 ⊕ a1001 ⊕ a1010 ⊕ a1011 = 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ a1011 = 1 ⇒ a1011 = 0
f(1,1,0,1) = a0000 ⊕ a0001 ⊕ a0100 ⊕ a1000 ⊕ a0101 ⊕ a1001 ⊕ a1100 ⊕ a1101 = 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ a1101 = 1 ⇒ a1101 = 0
f(1,1,1,0) = a0000 ⊕ a0010 ⊕ a0100 ⊕ a1000 ⊕ a0110 ⊕ a1010 ⊕ a1100 ⊕ a1110 = 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ a1110 = 1 ⇒ a1110 = 1
f(1,1,1,1) = a0000 ⊕ a0001 ⊕ a0010 ⊕ a0100 ⊕ a1000 ⊕ a0011 ⊕ a0101 ⊕ a0110 ⊕ a1001 ⊕ a1010 ⊕ a1100 ⊕ a0111 ⊕ a1011 ⊕ a1101 ⊕ a1110 ⊕ a1111 = 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 1 ⊕ a1111 = 0 ⇒ a1111 = 0
f(x,y,z,t) = y ⊕ x ⊕ yz ⊕ xy ⊕ ty ⊕ txy