Deck 12: A: Boolean Algebra

Question 1

mark each statement TRUE or FALSE.
{+, ·} is a functionally complete set of operators.

Answer 1

False

Question 2

fill in the blanks.
When written as a product of maxterms (in the variables x and y), .

Answer 2

Question 3

fill in the blanks.
There are Boolean functions with 2 variables.

Answer 3

24

Question 4

mark each statement TRUE or FALSE.

Question 5

fill in the blanks.
The idempotent laws in a Boolean algebra state that and .

Question 6

mark each statement TRUE or FALSE.
Every Boolean function can be written using only the operators , +, and · .

Question 7

mark each statement TRUE or FALSE.

Question 8

mark each statement TRUE or FALSE.
The circuit diagrams for

Question 9

fill in the blanks.
Using can be written in terms of , +, and · as .

Question 10

mark each statement TRUE or FALSE.

Question 11

mark each statement TRUE or FALSE.

Question 12

fill in the blanks.
There are Boolean functions with 3 variables.

Question 13

mark each statement TRUE or FALSE.
When written as a sum of minterms in the variables x and

Question 14

mark each statement TRUE or FALSE.
When written as a sum of minterms in the variables x and

Question 15

mark each statement TRUE or FALSE.

Question 16

mark each statement TRUE or FALSE.

Question 17

fill in the blanks.
Using can be written in terms of , +, and · as .

Question 18

mark each statement TRUE or FALSE.

Question 19

fill in the blanks.
There are Boolean functions with 4 variables.

Question 20

mark each statement TRUE or FALSE.

Question 21

Write 1 as a sum-of-products in the variables x and y.

Question 22

Write as a sum-of-products in the variables x, y, and z.

Question 23

If = + , find (0,1,0,1) .

Question 24

Prove that F=G , where = and

Question 25

Write x + y as a sum-of-products in the variables x and y.

Question 26

If = + , find (1,0,1,1)

Question 27

If = + , find

Question 28

Find the sum-of-products expansion of the Boolean function f(x, y, z) that is 1 if and only if either x = z = 1 and y = 0, or x = 0 and y = z = 1.

Question 29

mark each statement TRUE or FALSE.
The circuit diagrams for

Question 30

Write x + z as a sum-of-products in the variables x, y, and z.

Question 31

Find a Boolean function

Question 32

Write as a sum-of-products in the variables x, y, and z.

Question 33

Find the sum-of-products expansion of the Boolean function f(x, y, z) that is 1 if and only if exactly two of the three variables have value 1.

Question 34

Show that the Boolean function only the definition of a Boolean algebra.

Question 35

Find the sum-of-products expansion of the Boolean function f(x, y) that is 1 if and only if either x = 0 and

Question 36

Write as a sum-of-products in the variables x and y.

Question 37

If = + , find

Question 38

If = + , find (1,1,1,1) .

Question 39

If = + , find (0,0,1,0).

Question 40

Write as a sum-of-products in the variables x, y, and z.

Question 41

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 42

Give a reason for each step in the proof that x + x = x is true in Boolean algebras. Your reasons should come from the following: associative laws for addition and multiplication, commutative laws for addition and multiplication, distributive law for multiplication over addition and distributive law for addition over multiplication, identity laws, unit property, and zero property.

Question 43

Prove that the set of real numbers, with addition and multiplication of real numbers as + and · , negation as complementation, and the real numbers 0 and 1 as the 0 and the 1 respectively, is not a Boolean algebra.

Question 44

Using only the five properties associative laws, commutative laws, distributive laws, identity laws, and com- plement laws, prove that is true in all Boolean algebras.

Question 45

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 46

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 47

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 48

Using only the five properties associative laws, commutative laws, distributive laws, identity laws, and com- plement laws, prove that x + (x y) = x is true in all Boolean algebras.

Question 49

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 50

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 51

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 52

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 53

Give a reason for each step in the proof that x + xy = x is true in Boolean algebras. Your reasons should come from the following: associative laws for addition and multiplication, commutative laws for addition and multiplication, distributive law for multiplication over addition and distributive law for addition over multiplication, identity laws, unit property, zero property, and idempotent laws.

Question 54

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 55

Give a reason for each step in the proof that x + 1 = x is true in Boolean algebras. Your reasons should come from the following: associative laws for addition and multiplication, commutative laws for addition and multiplication, distributive law for multiplication over addition and distributive law for addition over multiplication, identity laws, unit property, and zero property.

Question 56

Using only the five properties associative laws, commutative laws, distributive laws, identity laws, and com- plement laws, prove that x + 1 = 1 is true in all Boolean algebras.

Question 57

Using only the five properties associative laws, commutative laws, distributive laws, identity laws, and com- plement laws, prove that x + x = x is true in all Boolean algebras.

Question 58

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 59

determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.

Question 60

Show that the Boolean function only the definition of a Boolean algebra.

Question 61

Draw a logic gate diagram for F .

Question 62

Use the Quine-McCluskey method to simplify the Boolean expression

Question 63

Draw a logic gate diagram for F .

Question 64

Construct a circuit using inverters, OR gates, and AND gates that gives an output of 1 if and only if three people on a committee do not all vote the same.

Question 65

Draw a logic gate diagram for the Boolean function

Question 66

Use the Quine-McCluskey method to simplify the Boolean expression

Question 67

Show that F can be simplified to give