12.1 Boolean Functions

Question 1

Boolean Complement

Answer 1

-

x ≡ ¬x

Question 2

x + y=

Answer 2

x ∨ y

Question 3

0 + 0 =

Answer 3

0

Question 4

0 + 1 =

Answer 4

1

Question 5

Boolean Sum

Answer 5

x ∨ y, only 0 if 0+0

Question 6

Boolean Product

Answer 6

x ∧ y, only 1 if 1*1

Question 7

Boolean Product

Answer 7

x ∧ y, only 1 if 1*1

Question 8

Boolean Functions

Answer 8

is the set of all possible
n-tuples of 0s and 1s. The variable x is called a Boolean variable if it assumes values only from
B, that is, if its only possible values are 0 and 1. A function from Bn
to B is called a Boolean
function of degree n

Question 9

Boolean Expressions

Answer 9

The Boolean expressionsin the variables, x1, x2, . . . , xn are defined recursively as 0, 1, x1, x2, . . . , xn. . Each Boolean expression represents a Boolean function.

Question 10

Duality

Answer 10

The dual of a Boolean expression is obtained by interchanging Boolean sums and Boolean products
and interchanging 0s and 1s.

Question 11

• Identity laws:

Answer 11

x ∨ 0 = x

x ∧ 1 = x

Question 12

• Complement laws:

Answer 12

x ∨ ¬x = 1

x ∧ ¬x = 0

Question 13

• Complement laws:

Answer 13

x ∨ ¬x = 1

x ∧ ¬x = 0

Question 14

Associative laws:

Answer 14

(x ∨ y) ∨ z = x ∨ (y ∨ z)

x ∧ y) ∧ z = x ∧ (y ∧ z

Question 15

Commutative laws:

Answer 15

x ∨ y = y ∨ x

x ∧ y = y ∧ x

Question 16

Commutative laws:

Answer 16

x ∨ y = y ∨ x

x ∧ y = y ∧ x

Question 17

Distributive laws:

Answer 17

x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)