a. Sum-of-product: Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1. If the value is 1 write the input variables; otherwise write its complement. The OR and AND terms are the final combination of the expression.

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Answer 1

Question

Chapter 3, Problem 42E

Program Plan Intro

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

Expert Solution

Explanation of Solution

Truth table:

The given truth table is as follows:

x	y	z	F1	F2
0	0	0	1	0
0	0	1	1	0
0	1	0	1	1
0	1	1	0	1
1	0	0	0	0
1	0	1	0	0
1	1	0	0	1
1	1	1	0	1

Sum-of-product form of F1:

The sum-of-product form of F1 for the above table is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z

Therefore, the sum-of-product form of F1 and F2 are expressed.

Explanation of Solution

b.

Simplification of F1:

The simplifying the sum-of-product form of F1 is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'= x'⋅y'⋅(z' + z) + x'⋅y⋅z'                        (by Distributive law)= x'⋅y'⋅(1) + x'⋅y⋅z'                              (by Inverse law)= x'⋅y' + x'⋅y⋅z'                                     (by Identity law)

                   = x'⋅(y' + y⋅z')                                      (by Distributive law)= x'⋅(y' + y' + z)                                    (by DeMorgan's law)= x'⋅(y' + z)                                          (by Idempotent law)= x'⋅y' + x'⋅z                                         (by Distributive law)

Therefore, the simplified Boolean expression F1 is x'⋅y' + x'⋅z

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z=x'⋅y⋅(z' + z) + x⋅y⋅(z' + z)                              (by Distributive law)=x'⋅y⋅(1) + x⋅y⋅(1)                                            (by Inverse law)=x'⋅y + x⋅y                                                         (by Identity law)

                   =(x' + x)⋅y                                                          (by Distributive law)=(1)⋅y                                                                 (by Inverse law)=y                                                                       (by Identity law)

Therefore, the simplified Boolean expression F2 is y

Explanation of Solution

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows:

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Answer 2

Question

Chapter 3, Problem 42E

Program Plan Intro

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

Expert Solution

Explanation of Solution

Truth table:

The given truth table is as follows:

x	y	z	F1	F2
0	0	0	1	0
0	0	1	1	0
0	1	0	1	1
0	1	1	0	1
1	0	0	0	0
1	0	1	0	0
1	1	0	0	1
1	1	1	0	1

Sum-of-product form of F1:

The sum-of-product form of F1 for the above table is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z

Therefore, the sum-of-product form of F1 and F2 are expressed.

Explanation of Solution

b.

Simplification of F1:

The simplifying the sum-of-product form of F1 is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'= x'⋅y'⋅(z' + z) + x'⋅y⋅z'                        (by Distributive law)= x'⋅y'⋅(1) + x'⋅y⋅z'                              (by Inverse law)= x'⋅y' + x'⋅y⋅z'                                     (by Identity law)

                   = x'⋅(y' + y⋅z')                                      (by Distributive law)= x'⋅(y' + y' + z)                                    (by DeMorgan's law)= x'⋅(y' + z)                                          (by Idempotent law)= x'⋅y' + x'⋅z                                         (by Distributive law)

Therefore, the simplified Boolean expression F1 is x'⋅y' + x'⋅z

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z=x'⋅y⋅(z' + z) + x⋅y⋅(z' + z)                              (by Distributive law)=x'⋅y⋅(1) + x⋅y⋅(1)                                            (by Inverse law)=x'⋅y + x⋅y                                                         (by Identity law)

                   =(x' + x)⋅y                                                          (by Distributive law)=(1)⋅y                                                                 (by Inverse law)=y                                                                       (by Identity law)

Therefore, the simplified Boolean expression F2 is y

Explanation of Solution

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows:

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Answer 3

Question

Chapter 3, Problem 42E

Program Plan Intro

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

Expert Solution

Explanation of Solution

Truth table:

The given truth table is as follows:

x	y	z	F1	F2
0	0	0	1	0
0	0	1	1	0
0	1	0	1	1
0	1	1	0	1
1	0	0	0	0
1	0	1	0	0
1	1	0	0	1
1	1	1	0	1

Sum-of-product form of F1:

The sum-of-product form of F1 for the above table is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z

Therefore, the sum-of-product form of F1 and F2 are expressed.

Explanation of Solution

b.

Simplification of F1:

The simplifying the sum-of-product form of F1 is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'= x'⋅y'⋅(z' + z) + x'⋅y⋅z'                        (by Distributive law)= x'⋅y'⋅(1) + x'⋅y⋅z'                              (by Inverse law)= x'⋅y' + x'⋅y⋅z'                                     (by Identity law)

                   = x'⋅(y' + y⋅z')                                      (by Distributive law)= x'⋅(y' + y' + z)                                    (by DeMorgan's law)= x'⋅(y' + z)                                          (by Idempotent law)= x'⋅y' + x'⋅z                                         (by Distributive law)

Therefore, the simplified Boolean expression F1 is x'⋅y' + x'⋅z

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z=x'⋅y⋅(z' + z) + x⋅y⋅(z' + z)                              (by Distributive law)=x'⋅y⋅(1) + x⋅y⋅(1)                                            (by Inverse law)=x'⋅y + x⋅y                                                         (by Identity law)

                   =(x' + x)⋅y                                                          (by Distributive law)=(1)⋅y                                                                 (by Inverse law)=y                                                                       (by Identity law)

Therefore, the simplified Boolean expression F2 is y

Explanation of Solution

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows:

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Answer 4

Question

Chapter 3, Problem 42E

Program Plan Intro

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

Expert Solution

Explanation of Solution

Truth table:

The given truth table is as follows:

x	y	z	F1	F2
0	0	0	1	0
0	0	1	1	0
0	1	0	1	1
0	1	1	0	1
1	0	0	0	0
1	0	1	0	0
1	1	0	0	1
1	1	1	0	1

Sum-of-product form of F1:

The sum-of-product form of F1 for the above table is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z

Therefore, the sum-of-product form of F1 and F2 are expressed.

Explanation of Solution

b.

Simplification of F1:

The simplifying the sum-of-product form of F1 is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'= x'⋅y'⋅(z' + z) + x'⋅y⋅z'                        (by Distributive law)= x'⋅y'⋅(1) + x'⋅y⋅z'                              (by Inverse law)= x'⋅y' + x'⋅y⋅z'                                     (by Identity law)

                   = x'⋅(y' + y⋅z')                                      (by Distributive law)= x'⋅(y' + y' + z)                                    (by DeMorgan's law)= x'⋅(y' + z)                                          (by Idempotent law)= x'⋅y' + x'⋅z                                         (by Distributive law)

Therefore, the simplified Boolean expression F1 is x'⋅y' + x'⋅z

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z=x'⋅y⋅(z' + z) + x⋅y⋅(z' + z)                              (by Distributive law)=x'⋅y⋅(1) + x⋅y⋅(1)                                            (by Inverse law)=x'⋅y + x⋅y                                                         (by Identity law)

                   =(x' + x)⋅y                                                          (by Distributive law)=(1)⋅y                                                                 (by Inverse law)=y                                                                       (by Identity law)

Therefore, the simplified Boolean expression F2 is y

Explanation of Solution

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows:

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Answer 5

Chapter 3, Problem 42E

Program Plan Intro

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

Expert Solution

Explanation of Solution

Truth table:

The given truth table is as follows:

x	y	z	F1	F2
0	0	0	1	0
0	0	1	1	0
0	1	0	1	1
0	1	1	0	1
1	0	0	0	0
1	0	1	0	0
1	1	0	0	1
1	1	1	0	1

Sum-of-product form of F1:

The sum-of-product form of F1 for the above table is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z

Therefore, the sum-of-product form of F1 and F2 are expressed.

Explanation of Solution

b.

Simplification of F1:

The simplifying the sum-of-product form of F1 is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'= x'⋅y'⋅(z' + z) + x'⋅y⋅z'                        (by Distributive law)= x'⋅y'⋅(1) + x'⋅y⋅z'                              (by Inverse law)= x'⋅y' + x'⋅y⋅z'                                     (by Identity law)

                   = x'⋅(y' + y⋅z')                                      (by Distributive law)= x'⋅(y' + y' + z)                                    (by DeMorgan's law)= x'⋅(y' + z)                                          (by Idempotent law)= x'⋅y' + x'⋅z                                         (by Distributive law)

Therefore, the simplified Boolean expression F1 is x'⋅y' + x'⋅z

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z=x'⋅y⋅(z' + z) + x⋅y⋅(z' + z)                              (by Distributive law)=x'⋅y⋅(1) + x⋅y⋅(1)                                            (by Inverse law)=x'⋅y + x⋅y                                                         (by Identity law)

                   =(x' + x)⋅y                                                          (by Distributive law)=(1)⋅y                                                                 (by Inverse law)=y                                                                       (by Identity law)

Therefore, the simplified Boolean expression F2 is y

Explanation of Solution

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows:

Answer 6

Chapter 3, Problem 42E

Program Plan Intro

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

Expert Solution

Explanation of Solution

Truth table:

The given truth table is as follows:

x	y	z	F1	F2
0	0	0	1	0
0	0	1	1	0
0	1	0	1	1
0	1	1	0	1
1	0	0	0	0
1	0	1	0	0
1	1	0	0	1
1	1	1	0	1

Sum-of-product form of F1:

The sum-of-product form of F1 for the above table is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z

Therefore, the sum-of-product form of F1 and F2 are expressed.

Answer 7

Chapter 3, Problem 42E

Program Plan Intro

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

Answer 8

Expert Solution

Explanation of Solution

Truth table:

The given truth table is as follows:

x	y	z	F1	F2
0	0	0	1	0
0	0	1	1	0
0	1	0	1	1
0	1	1	0	1
1	0	0	0	0
1	0	1	0	0
1	1	0	0	1
1	1	1	0	1

Sum-of-product form of F1:

The sum-of-product form of F1 for the above table is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z

Therefore, the sum-of-product form of F1 and F2 are expressed.

Answer 9

Expert Solution

Answer 10

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

b.

Simplification of F1:

The simplifying the sum-of-product form of F1 is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'= x'⋅y'⋅(z' + z) + x'⋅y⋅z'                        (by Distributive law)= x'⋅y'⋅(1) + x'⋅y⋅z'                              (by Inverse law)= x'⋅y' + x'⋅y⋅z'                                     (by Identity law)

                   = x'⋅(y' + y⋅z')                                      (by Distributive law)= x'⋅(y' + y' + z)                                    (by DeMorgan's law)= x'⋅(y' + z)                                          (by Idempotent law)= x'⋅y' + x'⋅z                                         (by Distributive law)

Therefore, the simplified Boolean expression F1 is x'⋅y' + x'⋅z

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z=x'⋅y⋅(z' + z) + x⋅y⋅(z' + z)                              (by Distributive law)=x'⋅y⋅(1) + x⋅y⋅(1)                                            (by Inverse law)=x'⋅y + x⋅y                                                         (by Identity law)

                   =(x' + x)⋅y                                                          (by Distributive law)=(1)⋅y                                                                 (by Inverse law)=y                                                                       (by Identity law)

Therefore, the simplified Boolean expression F2 is y

Answer 13

Explanation of Solution

b.

Simplification of F1:

The simplifying the sum-of-product form of F1 is as follows:

F1(x, y, z) = x'⋅y'⋅z' + x'⋅y'⋅z + x'⋅y⋅z'= x'⋅y'⋅(z' + z) + x'⋅y⋅z'                        (by Distributive law)= x'⋅y'⋅(1) + x'⋅y⋅z'                              (by Inverse law)= x'⋅y' + x'⋅y⋅z'                                     (by Identity law)

                   = x'⋅(y' + y⋅z')                                      (by Distributive law)= x'⋅(y' + y' + z)                                    (by DeMorgan's law)= x'⋅(y' + z)                                          (by Idempotent law)= x'⋅y' + x'⋅z                                         (by Distributive law)

Therefore, the simplified Boolean expression F1 is x'⋅y' + x'⋅z

Sum-of-product form of F2:

The sum-of-product form of F2 for the above table is as follows:

F1(x, y, z) = x'⋅y⋅z' + x'⋅y⋅z + x⋅y⋅z' + x⋅y⋅z=x'⋅y⋅(z' + z) + x⋅y⋅(z' + z)                              (by Distributive law)=x'⋅y⋅(1) + x⋅y⋅(1)                                            (by Inverse law)=x'⋅y + x⋅y                                                         (by Identity law)

                   =(x' + x)⋅y                                                          (by Distributive law)=(1)⋅y                                                                 (by Inverse law)=y                                                                       (by Identity law)

Therefore, the simplified Boolean expression F2 is y

Answer 14

Explanation of Solution

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows:

Answer 15

Explanation of Solution

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows:

Answer 16

Ch. 3.3A - Prob. 1E Ch. 3.3A - Prob. 2E Ch. 3.3A - Prob. 3E Ch. 3.3A - Prob. 4E Ch. 3.3A - Prob. 5E Ch. 3.3A - Prob. 6E Ch. 3.3A - Prob. 7E Ch. 3.3A - Prob. 8E Ch. 3.3A - Prob. 9E Ch. 3.3A - Prob. 10E

a. Sum-of-product: Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1. If the value is 1 write the input variables; otherwise write its complement. The OR and AND terms are the final combination of the expression.

Concept explainers

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

Explanation of Solution

Explanation of Solution

Explanation of Solution

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows:

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Chapter 3 Solutions

a. Sum-of-product: Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1. If the value is 1 write the input variables; otherwise write its complement. The OR and AND terms are the final combination of the expression.

Concept explainers

a. Sum-of-product: Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1. If the value is 1 write the input variables; otherwise write its complement. The OR and AND terms are the final combination of the expression.

Explanation of Solution

Explanation of Solution

Explanation of Solution

c. Circuit diagram: The circuit diagram for the F1 and F2 is as follows:

Want to see more full solutions like this?

Chapter 3 Solutions

a.

Sum-of-product:

Sum-of-product is an equation written in the form of AND term for each input combination that produces output as logical 1.

If the value is 1 write the input variables; otherwise write its complement.

The OR and AND terms are the final combination of the expression.

c.

Circuit diagram:

The circuit diagram for the F1 and F2 is as follows: