Distributive law: Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz). The representation of distributive law is as follows: x + (y ⋅ z) = (x + y) (x + z) x (y + z) = x ⋅ y + x ⋅ z Inverse law: The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0. The representation of inverse law is as follows: x + x' = 1 x ⋅ x' = 0 Identity law: The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”. The representation of identity law is as follows: 0 + x = x 1 ⋅ x = x

Question

Want to see more full solutions like this?

Answer 1

Question

Chapter 3, Problem 18E

a.

Program Plan Intro

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

a.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = y⋅(x⋅z' + x'⋅z) + y'⋅(x⋅z' + x'⋅z)= y⋅x⋅z' + y⋅x'⋅z + y'⋅(x⋅z' + x'⋅z) (by Distributive law)= y⋅x⋅z' + y⋅x'⋅z + y'⋅x⋅z' + y'⋅x'⋅z (by Distributive law)= x⋅z'(y + y') + x'⋅z(y + y') (by Distributive law)

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

Therefore, the simplified expression is “x⋅z' + x'⋅z”.

b.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

b.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = x⋅(y'⋅z + y) + x'⋅(y + z')'= x⋅(y'⋅z + y) + x'⋅(y'⋅z) (by DeMorgan's law)= x⋅(y'⋅z + y) + x'⋅y'⋅z= x⋅y'⋅z + x⋅y + x'⋅y'⋅z (by Destributive law)

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

Therefore, the simplified expression is “y'⋅z + x⋅y”.

c.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Idempotent law:

The sum of the variable (x) and the same variable (x) is “x” and the product of the variable (x) and the same variable (x) is “x”.

The representation of idempotent law is as follows:

x + x = xx ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Null law:

The sum of the variable (x) and the value 1 is “1” and the product of the variable (x) and the value 0 is “0”.

The representation of null law is as follows:

1 + x = 10 ⋅ x = 0

c.

Expert Solution

Explanation of Solution

Simplification:

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

=x⋅y'⋅z + (x'⋅y + z)= x⋅y'⋅z⋅x'⋅y + x⋅y'⋅z⋅z (by Distributive law)= x⋅x'⋅y'⋅y⋅z + x⋅y'⋅z⋅z= (0)⋅(0)⋅z + x⋅y'⋅z⋅z (by Inverse law)

=x⋅y'⋅z⋅z (by Null law)=x⋅y'⋅z (by Idempotent law)

Therefore, the simplified expression is “x⋅y'⋅z”.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Students have asked these similar questions

I need help writing pseudocode for calculating class score average by putting in 5 test scores, and showing the average from all 5 inputs and the GPA score.Starting with 3 functions outside of a main module. The functions are getScore(), getGPAPoint(Integer score), and getAverage(). Function getscore is an input for a grade as a class percentage. Function getGPAPoint will calculate the score into a GPA point and return as a float (values of 90-100 as 4.0, 80-89 as 3.0, 70-79 as 2.0, 60-69 as 1.0, and anything below 60 as 0.) Function averageGPA will finally make a call to both previous functions when the user inputs numbers 5 times that then calculates the average (add up all the scores, divide by 5) and the average grade alongside displaying the average GPA. End result is a main module that makes a proper call to the averageGPA function and display its results. Need help with this!

Please original work Why is integration between data collection and business analysis so important to success in an organization that uses business analytics? How can a company that is just starting to use business analytics set up its program for success right from the beginning? Please cite in text references and add weblinks

How to make a 1 bit adder with CLA?

Answer 2

Question

Chapter 3, Problem 18E

a.

Program Plan Intro

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

a.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = y⋅(x⋅z' + x'⋅z) + y'⋅(x⋅z' + x'⋅z)= y⋅x⋅z' + y⋅x'⋅z + y'⋅(x⋅z' + x'⋅z) (by Distributive law)= y⋅x⋅z' + y⋅x'⋅z + y'⋅x⋅z' + y'⋅x'⋅z (by Distributive law)= x⋅z'(y + y') + x'⋅z(y + y') (by Distributive law)

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

Therefore, the simplified expression is “x⋅z' + x'⋅z”.

b.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

b.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = x⋅(y'⋅z + y) + x'⋅(y + z')'= x⋅(y'⋅z + y) + x'⋅(y'⋅z) (by DeMorgan's law)= x⋅(y'⋅z + y) + x'⋅y'⋅z= x⋅y'⋅z + x⋅y + x'⋅y'⋅z (by Destributive law)

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

Therefore, the simplified expression is “y'⋅z + x⋅y”.

c.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Idempotent law:

The sum of the variable (x) and the same variable (x) is “x” and the product of the variable (x) and the same variable (x) is “x”.

The representation of idempotent law is as follows:

x + x = xx ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Null law:

The sum of the variable (x) and the value 1 is “1” and the product of the variable (x) and the value 0 is “0”.

The representation of null law is as follows:

1 + x = 10 ⋅ x = 0

c.

Expert Solution

Explanation of Solution

Simplification:

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

=x⋅y'⋅z + (x'⋅y + z)= x⋅y'⋅z⋅x'⋅y + x⋅y'⋅z⋅z (by Distributive law)= x⋅x'⋅y'⋅y⋅z + x⋅y'⋅z⋅z= (0)⋅(0)⋅z + x⋅y'⋅z⋅z (by Inverse law)

=x⋅y'⋅z⋅z (by Null law)=x⋅y'⋅z (by Idempotent law)

Therefore, the simplified expression is “x⋅y'⋅z”.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Question

Chapter 3, Problem 18E

a.

Program Plan Intro

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

a.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = y⋅(x⋅z' + x'⋅z) + y'⋅(x⋅z' + x'⋅z)= y⋅x⋅z' + y⋅x'⋅z + y'⋅(x⋅z' + x'⋅z) (by Distributive law)= y⋅x⋅z' + y⋅x'⋅z + y'⋅x⋅z' + y'⋅x'⋅z (by Distributive law)= x⋅z'(y + y') + x'⋅z(y + y') (by Distributive law)

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

Therefore, the simplified expression is “x⋅z' + x'⋅z”.

b.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

b.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = x⋅(y'⋅z + y) + x'⋅(y + z')'= x⋅(y'⋅z + y) + x'⋅(y'⋅z) (by DeMorgan's law)= x⋅(y'⋅z + y) + x'⋅y'⋅z= x⋅y'⋅z + x⋅y + x'⋅y'⋅z (by Destributive law)

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

Therefore, the simplified expression is “y'⋅z + x⋅y”.

c.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Idempotent law:

The sum of the variable (x) and the same variable (x) is “x” and the product of the variable (x) and the same variable (x) is “x”.

The representation of idempotent law is as follows:

x + x = xx ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Null law:

The sum of the variable (x) and the value 1 is “1” and the product of the variable (x) and the value 0 is “0”.

The representation of null law is as follows:

1 + x = 10 ⋅ x = 0

c.

Expert Solution

Explanation of Solution

Simplification:

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

=x⋅y'⋅z + (x'⋅y + z)= x⋅y'⋅z⋅x'⋅y + x⋅y'⋅z⋅z (by Distributive law)= x⋅x'⋅y'⋅y⋅z + x⋅y'⋅z⋅z= (0)⋅(0)⋅z + x⋅y'⋅z⋅z (by Inverse law)

=x⋅y'⋅z⋅z (by Null law)=x⋅y'⋅z (by Idempotent law)

Therefore, the simplified expression is “x⋅y'⋅z”.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Question

Chapter 3, Problem 18E

a.

Program Plan Intro

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

a.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = y⋅(x⋅z' + x'⋅z) + y'⋅(x⋅z' + x'⋅z)= y⋅x⋅z' + y⋅x'⋅z + y'⋅(x⋅z' + x'⋅z) (by Distributive law)= y⋅x⋅z' + y⋅x'⋅z + y'⋅x⋅z' + y'⋅x'⋅z (by Distributive law)= x⋅z'(y + y') + x'⋅z(y + y') (by Distributive law)

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

Therefore, the simplified expression is “x⋅z' + x'⋅z”.

b.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

b.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = x⋅(y'⋅z + y) + x'⋅(y + z')'= x⋅(y'⋅z + y) + x'⋅(y'⋅z) (by DeMorgan's law)= x⋅(y'⋅z + y) + x'⋅y'⋅z= x⋅y'⋅z + x⋅y + x'⋅y'⋅z (by Destributive law)

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

Therefore, the simplified expression is “y'⋅z + x⋅y”.

c.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Idempotent law:

The sum of the variable (x) and the same variable (x) is “x” and the product of the variable (x) and the same variable (x) is “x”.

The representation of idempotent law is as follows:

x + x = xx ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Null law:

The sum of the variable (x) and the value 1 is “1” and the product of the variable (x) and the value 0 is “0”.

The representation of null law is as follows:

1 + x = 10 ⋅ x = 0

c.

Expert Solution

Explanation of Solution

Simplification:

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

=x⋅y'⋅z + (x'⋅y + z)= x⋅y'⋅z⋅x'⋅y + x⋅y'⋅z⋅z (by Distributive law)= x⋅x'⋅y'⋅y⋅z + x⋅y'⋅z⋅z= (0)⋅(0)⋅z + x⋅y'⋅z⋅z (by Inverse law)

=x⋅y'⋅z⋅z (by Null law)=x⋅y'⋅z (by Idempotent law)

Therefore, the simplified expression is “x⋅y'⋅z”.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 3, Problem 18E

a.

Program Plan Intro

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

a.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = y⋅(x⋅z' + x'⋅z) + y'⋅(x⋅z' + x'⋅z)= y⋅x⋅z' + y⋅x'⋅z + y'⋅(x⋅z' + x'⋅z) (by Distributive law)= y⋅x⋅z' + y⋅x'⋅z + y'⋅x⋅z' + y'⋅x'⋅z (by Distributive law)= x⋅z'(y + y') + x'⋅z(y + y') (by Distributive law)

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

Therefore, the simplified expression is “x⋅z' + x'⋅z”.

b.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

b.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = x⋅(y'⋅z + y) + x'⋅(y + z')'= x⋅(y'⋅z + y) + x'⋅(y'⋅z) (by DeMorgan's law)= x⋅(y'⋅z + y) + x'⋅y'⋅z= x⋅y'⋅z + x⋅y + x'⋅y'⋅z (by Destributive law)

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

Therefore, the simplified expression is “y'⋅z + x⋅y”.

c.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Idempotent law:

The sum of the variable (x) and the same variable (x) is “x” and the product of the variable (x) and the same variable (x) is “x”.

The representation of idempotent law is as follows:

x + x = xx ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Null law:

The sum of the variable (x) and the value 1 is “1” and the product of the variable (x) and the value 0 is “0”.

The representation of null law is as follows:

1 + x = 10 ⋅ x = 0

c.

Expert Solution

Explanation of Solution

Simplification:

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

=x⋅y'⋅z + (x'⋅y + z)= x⋅y'⋅z⋅x'⋅y + x⋅y'⋅z⋅z (by Distributive law)= x⋅x'⋅y'⋅y⋅z + x⋅y'⋅z⋅z= (0)⋅(0)⋅z + x⋅y'⋅z⋅z (by Inverse law)

=x⋅y'⋅z⋅z (by Null law)=x⋅y'⋅z (by Idempotent law)

Therefore, the simplified expression is “x⋅y'⋅z”.

Answer 6

Chapter 3, Problem 18E

a.

Program Plan Intro

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

a.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = y⋅(x⋅z' + x'⋅z) + y'⋅(x⋅z' + x'⋅z)= y⋅x⋅z' + y⋅x'⋅z + y'⋅(x⋅z' + x'⋅z) (by Distributive law)= y⋅x⋅z' + y⋅x'⋅z + y'⋅x⋅z' + y'⋅x'⋅z (by Distributive law)= x⋅z'(y + y') + x'⋅z(y + y') (by Distributive law)

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

Therefore, the simplified expression is “x⋅z' + x'⋅z”.

Answer 7

Chapter 3, Problem 18E

a.

Program Plan Intro

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

Answer 8

a.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = y⋅(x⋅z' + x'⋅z) + y'⋅(x⋅z' + x'⋅z)= y⋅x⋅z' + y⋅x'⋅z + y'⋅(x⋅z' + x'⋅z) (by Distributive law)= y⋅x⋅z' + y⋅x'⋅z + y'⋅x⋅z' + y'⋅x'⋅z (by Distributive law)= x⋅z'(y + y') + x'⋅z(y + y') (by Distributive law)

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

Therefore, the simplified expression is “x⋅z' + x'⋅z”.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

b.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

b.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = x⋅(y'⋅z + y) + x'⋅(y + z')'= x⋅(y'⋅z + y) + x'⋅(y'⋅z) (by DeMorgan's law)= x⋅(y'⋅z + y) + x'⋅y'⋅z= x⋅y'⋅z + x⋅y + x'⋅y'⋅z (by Destributive law)

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

Therefore, the simplified expression is “y'⋅z + x⋅y”.

Answer 13

b.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Identity law:

The sum of the variable (x) and the value 0 is “x” and the product of the variable (x) and the value 1 is “x”.

The representation of identity law is as follows:

0 + x = x1 ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Answer 14

b.

Expert Solution

Explanation of Solution

Simplification:

F(x, y, z) = x⋅(y'⋅z + y) + x'⋅(y + z')'= x⋅(y'⋅z + y) + x'⋅(y'⋅z) (by DeMorgan's law)= x⋅(y'⋅z + y) + x'⋅y'⋅z= x⋅y'⋅z + x⋅y + x'⋅y'⋅z (by Destributive law)

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

Therefore, the simplified expression is “y'⋅z + x⋅y”.

Answer 15

b.

Expert Solution

Answer 16

b.

Expert Solution

Answer 17

Expert Solution

Answer 18

c.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Idempotent law:

The sum of the variable (x) and the same variable (x) is “x” and the product of the variable (x) and the same variable (x) is “x”.

The representation of idempotent law is as follows:

x + x = xx ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Null law:

The sum of the variable (x) and the value 1 is “1” and the product of the variable (x) and the value 0 is “0”.

The representation of null law is as follows:

1 + x = 10 ⋅ x = 0

c.

Expert Solution

Explanation of Solution

Simplification:

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

=x⋅y'⋅z + (x'⋅y + z)= x⋅y'⋅z⋅x'⋅y + x⋅y'⋅z⋅z (by Distributive law)= x⋅x'⋅y'⋅y⋅z + x⋅y'⋅z⋅z= (0)⋅(0)⋅z + x⋅y'⋅z⋅z (by Inverse law)

=x⋅y'⋅z⋅z (by Null law)=x⋅y'⋅z (by Idempotent law)

Therefore, the simplified expression is “x⋅y'⋅z”.

Answer 19

c.

Program Plan Intro

De Morgan’s Law:

The complement of the intersection of two sets is the union of their complements.

The complement of the union of two sets is the intersection of their complements.

The representation of the De Morgan’s law as follows:

(x⋅y)' = x' + y'(x + y)' = x' ⋅ y'

Distributive law:

Consider three variables x, y, and z. The multiplication of variable (x) with the sum of two variables (y and z) is same as the sum of the products (xy and xz).

The representation of distributive law is as follows:

x + (y⋅z) = (x + y) (x + z)x (y + z) = x⋅y + x⋅z

Idempotent law:

The sum of the variable (x) and the same variable (x) is “x” and the product of the variable (x) and the same variable (x) is “x”.

The representation of idempotent law is as follows:

x + x = xx ⋅ x = x

Inverse law:

The sum of the variable (x) and the complement of the variable (x’) is 1 and the product of the variable (x) and the complement of the variable (x’) is 0.

The representation of inverse law is as follows:

x + x' = 1x ⋅ x' = 0

Null law:

The sum of the variable (x) and the value 1 is “1” and the product of the variable (x) and the value 0 is “0”.

The representation of null law is as follows:

1 + x = 10 ⋅ x = 0

Answer 20

c.

Expert Solution

Explanation of Solution

Simplification:

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