Deck 12: Boolean Algebra

Question 1

The idempotent laws in a Boolean algebra state that and .

Answer 1

x + x = x, x · x = x.

Question 2

In questions mark each statement TRUE or FALSE.

Answer 2

True

Question 3

In questions mark each statement TRUE or FALSE.

Answer 3

True

Question 4

In questions mark each statement TRUE or FALSE.

Question 5

There are Boolean functions with 2 variables.

Question 6

There are Boolean functions with 3 variables.

Question 7

In questions mark each statement TRUE or FALSE.

Question 8

Using can be written in terms of

Question 9

There are Boolean functions with 4 variables.

Question 10

In questions mark each statement TRUE or FALSE.

Question 11

In questions mark each statement TRUE or FALSE.

Question 12

In questions mark each statement TRUE or FALSE.

Question 13

In questions mark each statement TRUE or FALSE.

Question 14

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

Question 15

Using PPP can be written in terms of PPP

Question 16

In questions mark each statement TRUE or FALSE.

Question 17

In questions mark each statement TRUE or FALSE.

Question 18

In questions mark each statement TRUE or FALSE.

Question 19

When written as a sum of minterms (in the variables x and y ),

Question 20

In questions mark each statement TRUE or FALSE.

Question 21

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 22

Find the sum-of-products expansion of the Boolean function that is 1 if and only if either and

Question 23

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

Question 24

If , find

Question 25

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 26

(a) Find a Boolean function such that and
otherwise.
(b) Write using only . and

Question 27

Write as a sum-of-products in the variables

Question 28

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

Question 29

In questions mark each statement TRUE or FALSE.

Question 30

If , find

Question 31

Write as a sum-of-products in the variables

Question 32

Find a Boolean function such that and

Question 33

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

Question 34

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

Question 35

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

Question 36

In questions mark each statement TRUE or FALSE.

Question 37

If , find

Question 38

Write as a sum-of-products in the variables

Question 39

Write as a sum-of-products in the variables

Question 40

If , find

Question 41

In questions determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.
x(x + y) = x + y x.

Question 42

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

Question 43

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

Question 44

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

Question 45

If find

Question 46

If find

Question 47

Show that the Boolean function F given by simplifies to by using only the definition of a Boolean algebra.

Question 48

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

Question 49

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

Question 50

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 51

Show that the Boolean function F given by simplifies to by using only the difinition of a booleab algebra .

Question 52

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

Question 53

Prove that F=G , where and

Question 54

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

Question 55

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

Question 56

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

Question 57

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

Question 58

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

Question 59

In questions determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.
(x x + 1) = (x + 1)(x + 1).

Question 60

Using only the five properties associative laws, commutative laws, distributive laws, identity laws, and complement laws, prove that x x = x is true in all boolean algebras .

Question 61

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 62

Let . Draw a logic gate diagram for F .

Question 63

Use a Karnaugh map to minimize the sum-of-products expression

Question 64

Use the Quine-McCluskey method to simplify the Boolean expression

Question 65

A circuit is to be built that takes the numbers 0 through 9 as inputs (1 = 0001, 2 = 0010, . . . , 9 = 1001). Let be the Boolean function that produces an output of 1 if and only if the input is an odd number.
Find a Karnaugh map for G and use the map and don't care conditions to find a simple expression for
G.

Question 66

Procomplemenve that thetation,setandof realthenurealmbners,umwithbers 0additionand 1 aands them0ultiplicationand the 1 respof realectivnelyum, bisersnotasa+Boandolean· , negationalgebra. as

Question 67

Draw a logic gate diagram for the Boolean function

Question 68

Let https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/. Show that F can be simplified to give

Question 69

Give a reason for each step in the proof that 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 70

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 71

Use the Quine-McCluskey method to simplify the Boolean expression x y z + x y z + x y z + x y z + x y z

Question 72

A circuit is to be built that takes the numbers 0 through 9 as inputs (1 = 0001, 2 = 0010, . . . , 9 = 1001). Let be the Boolean function that produces an output of 1 if and only if the input is an even number.
Find a Karnaugh map for F and use the map and don't care conditions to find a simple expression for
F.

Question 73

Let F(x, y, z) = y (x z) + y x + y z. Use a Karnaugh map to simplify the function F .

Question 74

Use a Karnaugh map to minimize the sum-of-products expression

Question 75

Let Draw a logic gate diagram for F .

Question 76

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 77

In questions determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent
Boolean variables.
(0 + x)(1 + x) = x x.