Discrete Mathematics

Richard Johnsonbaugh

Chapter 11

Boolean Algebras and Combinatorial Circuits - all with Video Answers

Educators

Section 1

Combinatorial Circuits

Problem 1

Write the Boolean expression that represents the combinatorial circuit, write the logic table, and write the output of each gate symbolically as in Figure $11.1 .8 .$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 2

Write the Boolean expression that represents the combinatorial circuit, write the logic table, and write the output of each gate symbolically as in Figure $11.1 .8 .$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 3

Write the Boolean expression that represents the combinatorial circuit, write the logic table, and write the output of each gate symbolically as in Figure $11.1 .8 .$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 4

Write the Boolean expression that represents the combinatorial circuit, write the logic table, and write the output of each gate symbolically as in Figure $11.1 .8 .$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 5

Write the Boolean expression that represents the combinatorial circuit, write the logic table, and write the output of each gate symbolically as in Figure $11.1 .8 .$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 6

Write the Boolean expression that represents the combinatorial circuit, write the logic table, and write the output of each gate symbolically as in Figure $11.1 .8 .$
The circuit at the bottom of Figure $11.1 .7 .$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 7

Refer to the circuit.
Show that this circuit is not a combinatorial circuit.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 8

Refer to the circuit.
Show that if $x=0$, the output $y$ is uniquely determined.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 9

Refer to the circuit.
Show that if $x=1$, the output $y$ is undetermined.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 10

Find the value of the Boolean expressions for $$x_{1}=1, \quad x_{2}=1, \quad x_{3}=0, \quad x_{4}=1$$.
$\overline{x_{1} \wedge x_{2}}$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 11

Find the value of the Boolean expressions for $$x_{1}=1, \quad x_{2}=1, \quad x_{3}=0, \quad x_{4}=1$$.
$$
\left(x_{1} \wedge \bar{x}_{2}\right) \vee\left(x_{1} \vee \bar{x}_{3}\right)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 12

Find the value of the Boolean expressions for $$x_{1}=1, \quad x_{2}=1, \quad x_{3}=0, \quad x_{4}=1$$.
$$
x_{1} \vee\left(\bar{x}_{2} \wedge x_{3}\right)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 13

Find the value of the Boolean expressions for $$x_{1}=1, \quad x_{2}=1, \quad x_{3}=0, \quad x_{4}=1$$.
$$
\left(x_{1} \wedge\left(x_{2} \vee\left(x_{1} \wedge \bar{x}_{2}\right)\right)\right) \vee\left(\left(x_{1} \wedge \bar{x}_{2}\right) \vee\left(\overline{x_{1} \wedge \bar{x}_{3}}\right)\right)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 14

Find the value of the Boolean expressions for $$x_{1}=1, \quad x_{2}=1, \quad x_{3}=0, \quad x_{4}=1$$.
$$
\left(\left(\left(x_{1} \wedge x_{2}\right) \vee\left(x_{3} \wedge \bar{x}_{4}\right)\right) \vee\left(\overline{\left.\left(x_{1} \vee x_{3}\right) \wedge\left(\bar{x}_{2} \vee x_{3}\right)\right)}\right) \vee\left(x_{1} \wedge \bar{x}_{3}\right)\right.
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 15

Using Definition 11.1 .9 , show that each expression in Exercises $10-14$ is a Boolean expression.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 16

Tell whether the given expression is a Boolean expression. If it is a Boolean expression, use Definition 11.1 .9 to show that it is.
$$
x_{1} \wedge\left(x_{2} \vee x_{3}\right)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 17

Tell whether the given expression is a Boolean expression. If it is a Boolean expression, use Definition 11.1 .9 to show that it is.
$$
\left(x_{1}\right)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 18

Tell whether the given expression is a Boolean expression. If it is a Boolean expression, use Definition 11.1 .9 to show that it is.
$$
x_{1} \wedge \bar{x}_{2} \vee x_{3}
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 19

Tell whether the given expression is a Boolean expression. If it is a Boolean expression, use Definition 11.1 .9 to show that it is.
$$
\left(\left(x_{1} \wedge x_{2}\right) \vee \bar{x}_{3}\right.
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 20

Tell whether the given expression is a Boolean expression. If it is a Boolean expression, use Definition 11.1 .9 to show that it is.
$$
\left(\left(x_{1}\right)\right)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 21

Find the combinatorial circuit corresponding to each Boolean expression in Exercises $10-14$ and write the logic table.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 22

Draw a circuit with two switches $A$ and $B$ having the property that the circuit output is 1 precisely when both $A$ and $B$ are closed. This configuration is labeled $A \wedge B$ and is called a series circuit.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 23

Draw a circuit with two switches $A$ and $B$ having the property that the circuit output is 1 precisely when either $A$ or $B$ is closed. This configuration is labeled $A \vee B$ and is called a parallel circuit.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 24

Show that the circuit of Figure 11.1 .12 can be represented symbolically as $$(A \wedge B) \vee \bar{A} \vee(B \wedge C)$$.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 25

Represent each circuit in Exercises symbolically and give its switching table.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 26

Represent each circuit in Exercises symbolically and give its switching table.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 27

Represent each circuit in Exercises symbolically and give its switching table.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 28

Represent each circuit in Exercises symbolically and give its switching table.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 29

Represent the expressions in Exercises as switching circuits and write the switching tables.
$$
(A \vee \bar{B}) \wedge A
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 30

Represent the expressions in Exercises as switching circuits and write the switching tables.
$$
A \vee(\bar{B} \wedge C)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 31

Represent the expressions in Exercises as switching circuits and write the switching tables.
$$
(\bar{A} \wedge B) \vee(C \wedge A)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 32

Represent the expressions in Exercises as switching circuits and write the switching tables.
$$
(A \wedge((B \wedge \bar{C}) \vee(\bar{B} \wedge C))) \vee(\bar{A} \wedge B \wedge C)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 33

Represent the expressions in Exercises as switching circuits and write the switching tables.
$$
A \wedge((B \wedge C \wedge \bar{D}) \vee((\bar{B} \wedge C) \vee D) \vee(\bar{B} \wedge \bar{C} \wedge D)) \wedge(B \vee \bar{D})
$$

Adriano Chikande

Adriano Chikande

Numerade Educator