Chapter 12

Boolean Algebra

Educators

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Problem 1

Find the values of these expressions.
$$\begin{array}{llll}{\text { a) } 1 \cdot \overline{0}} & {\text { b) } 1+\overline{1}} & {\text { c) } \overline{0} \cdot 0} & {\text { d) }(1+0)}\end{array}$$

Adithya R.
Numerade Educator

Problem 2

Find the values, if any, of the Boolean variable $x$ that satisfy these equations.
$$\begin{array}{ll}{\text { a) } x \cdot 1=0} & {\text { b) } x+x=0} \\ {\text { c) } x \cdot 1=x} & {\text { d) } x \cdot \overline{x}=1}\end{array}$$

WZ
Wen Z.
Numerade Educator

Problem 3

a) Show that $(1 \cdot 1)+(\overline{0 \cdot 1}+0)=1$
b) Translate the equation in part (a) into a propositional equivalence by changing each 0 into an $\mathbf{F}$ , each 1 into a $\mathbf{T}$ , each Boolean sum into a disjunction, each Boolean product into a conjunction, each complementation into a negation, and the equals sign into a propositional equivalence sign.

Adithya R.
Numerade Educator

Problem 4

a) Show that $(\overline{1} \cdot \overline{0})+(1 \cdot \overline{0})=1$
b) Translate the equation in part (a) into a propositional equivalence by changing each 0 into an $\mathbf{F}$ , each 1 into a $\mathbf{T}$ , each Boolean sum into a disjunction, each Boolean product into a conjunction, each complementation into a negation, and the equals sign into a propositional equivalence sign.

WZ
Wen Z.
Numerade Educator

Problem 5

Use a table to express the values of each of these Boolean functions.
a) $F(x, y, z)=\overline{x} y$
b) $F(x, y, z)=x+y z$
c) $F(x, y, z)=x \overline{y}+\overline{(x y z)}$
d) $F(x, y, z)=x(y z+\overline{y} \overline{z})$

Adithya R.
Numerade Educator

Problem 6

Use a table to express the values of each of these Boolean functions.
a) $F(x, y, z)=\overline{z}$
b) $F(x, y, z)=\overline{x} y+\overline{y} z$
c) $F(x, y, z)=x \overline{y} z+\overline{(x y z)}$
d) $F(x, y, z)=\overline{y}(x z+\overline{x} \overline{z})$

WZ
Wen Z.
Numerade Educator

Problem 7

Use a 3 -cube $Q_{3}$ to represent each of the Boolean functions in Exercise 5 by displaying a black circle at each vertex that corresponds to a 3 -tuple where this function has the value $1 .$

Adithya R.
Numerade Educator

Problem 8

Use a 3 -cube $Q_{3}$ to represent each of the Boolean functions in Exercise 6 by displaying a black circle at each vertex that corresponds to a 3 -tuple where this function has the value $1 .$

WZ
Wen Z.
Numerade Educator

Problem 9

What values of the Boolean variables $x$ and $y$ satisfy $x y=x+y ?$

Adithya R.
Numerade Educator

Problem 10

How many different Boolean functions are there of degree 7$?$

WZ
Wen Z.
Numerade Educator

Problem 11

Prove the absorption law $x+x y=x$ using the other laws in Table $5 .$

Adithya R.
Numerade Educator

Problem 12

Show that $F(x, y, z)=x y+x z+y z$ has the value 1 if and only if at least two of the variables $x, y,$ and $z$ have the value $1 .$

WZ
Wen Z.
Numerade Educator

Problem 13

Show that $x \overline{y}+y \overline{z}+\overline{x} z=\overline{x} y+\overline{y} z+x \overline{z}$

Adithya R.
Numerade Educator

Problem 14

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the law of the double complement.

WZ
Wen Z.
Numerade Educator

Problem 15

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the idempotent laws.

Sam S.
Numerade Educator

Problem 16

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the identity laws.

WZ
Wen Z.
Numerade Educator

Problem 17

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the domination laws.

Sam S.
Numerade Educator

Problem 18

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the commutative laws.

WZ
Wen Z.
Numerade Educator

Problem 19

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the associative laws.

Sam S.
Numerade Educator

Problem 20

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the first distributive law in Table $5 .$

WZ
Wen Z.
Numerade Educator

Problem 21

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify De Morgan's laws.

Sam S.
Numerade Educator

Problem 22

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the unit property.

WZ
Wen Z.
Numerade Educator

Problem 23

Exercises $14-23$ deal with the Boolean algebra $\{0,1\}$ with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8 .
Verify the zero property.

Sam S.
Numerade Educator

Problem 24

The Boolean operator $\oplus,$ called the $X O R$ operator, is defined by $1 \oplus 1=0,1 \oplus 0=1,0 \oplus 1=1,$ and $0 \oplus 0=0$.
Simplify these expressions.
$$\begin{array}{ll}{\text { a) } x \oplus 0} & {\text { b) } x \oplus 1} \\ {\text { c) } x \oplus x} & {\text { d) } x \oplus \overline{x}}\end{array}$$

WZ
Wen Z.
Numerade Educator

Problem 25

The Boolean operator $\oplus,$ called the $X O R$ operator, is defined by $1 \oplus 1=0,1 \oplus 0=1,0 \oplus 1=1,$ and $0 \oplus 0=0$.
Show that these identities hold.
a) $x \oplus y=(x+y)(x y)$
b) $x \oplus y=(x \overline{y})+(\overline{x} y)$

Adithya R.
Numerade Educator

Problem 26

The Boolean operator $\oplus,$ called the $X O R$ operator, is defined by $1 \oplus 1=0,1 \oplus 0=1,0 \oplus 1=1,$ and $0 \oplus 0=0$.
Show that $x \oplus y=y \oplus x$

WZ
Wen Z.
Numerade Educator

Problem 27

The Boolean operator $\oplus,$ called the $X O R$ operator, is defined by $1 \oplus 1=0,1 \oplus 0=1,0 \oplus 1=1,$ and $0 \oplus 0=0$.
Prove or disprove these equalities.
a) $x \oplus(y \oplus z)=(x \oplus y) \oplus z$
b) $x+(y \oplus z)=(x+y) \oplus(x+z)$
c) $x \oplus(y+z)=(x \oplus y)+(x \oplus z)$

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Problem 28

Find the duals of these Boolean expressions.
$$\begin{array}{ll}{\text { a) } x+y} & {\text { b) } \overline{x} \overline{y}} \\ {\text { c) } x y z+\overline{x} \overline{y} \overline{z}} & {\text { d) } x \overline{z}+x \cdot 0+\overline{x} \cdot 1}\end{array}$$

WZ
Wen Z.
Numerade Educator

Problem 29

Suppose that $F$ is a Boolean function represented by a Boolean expression in the variables $x_{1}, \ldots, x_{n} .$ Show that $F^{d}\left(x_{1}, \ldots, x_{n}\right)=\overline{F\left(\overline{x}_{1}, \ldots, \overline{x}_{n}\right)}$

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Problem 30

Show that if $F$ and $G$ are Boolean functions represented by Boolean expressions in $n$ variables and $F=G$ , then $F^{d}=G^{d}$ , where $F^{d}$ and $G^{d}$ are the Boolean functions represented by the duals of the Boolean expressions representing $F$ and $G,$ respectively. [Hint: Use the result of
Exercise $29 . ]$

WZ
Wen Z.
Numerade Educator

Problem 31

How many different Boolean functions $F(x, y, z)$ are there such that $F(\overline{x}, \overline{y}, \overline{z})=F(x, y, z)$ for all values of the Boolean variables $x, y,$ and $z ?$

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Problem 32

How many different Boolean functions $F(x, y, z)$ are there such that $F(\overline{x}, y, z)=F(x, \overline{y}, z)=F(x, y, \overline{z})$ for all values of the Boolean variables $x, y,$ and $z ?$

WZ
Wen Z.
Numerade Educator

Problem 33

Show that you obtain De Morgan's laws for propositions (in Table 6 in Section 1.3 ) when you transform De
Morgan's laws for Boolean algebra in Table 6 into logical equivalences.

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Problem 34

Show that you obtain the absorption laws for propositions (in Table 6 in Section 1.3 ) when you transform the absorption laws for Boolean algebra in Table 6 into logical equivalences.

WZ
Wen Z.
Numerade Educator

Problem 35

In Exercises $35-42,$ use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
Show that in a Boolean algebra, the idempotent laws $x \vee x=x$ and $x \wedge x=x$ hold for every element $x .$

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Problem 36

In Exercises $35-42,$ use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
Show that in a Boolean algebra, every element $x$ has a unique complement $\overline{x}$ such that $x \vee \overline{x}=1$ and $x \wedge \overline{x}=0$ .

WZ
Wen Z.
Numerade Educator

Problem 37

In Exercises $35-42,$ use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
Show that in a Boolean algebra, the complement of the element 0 is the element 1 and vice versa.

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Problem 38

In Exercises $35-42,$ use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
Prove that in a Boolean algebra, the law of the double complement holds; that is, $\overline{\overline{x}}=x$ for every element $x .$

WZ
Wen Z.
Numerade Educator

Problem 39

In Exercises $35-42,$ use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
Show that De Morgan's laws hold in a Boolean algebra. That is, show that for all $x$ and $y, \overline{(x \vee y)}=\overline{x} \wedge \overline{y}$ and $\frac{1}{(x \wedge y)}=\overline{x} \vee \overline{y}$

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Problem 40

In Exercises $35-42,$ use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
Show that in a Boolean algebra, the modular properties hold. That is, show that $x \wedge(y \vee(x \wedge z))=(x \wedge y) \vee(x \wedge$ $z )$ and $x \vee(y \wedge(x \vee z))=(x \vee y) \wedge(x \vee z)$

WZ
Wen Z.
Numerade Educator

Problem 41

In Exercises $35-42,$ use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
Show that in a Boolean algebra, if $x \vee y=0,$ then $x=0$ and $y=0,$ and that if $x \wedge y=1,$ then $x=1$ and $y=1$

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Problem 42

In Exercises $35-42,$ use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
Show that in a Boolean algebra, the dual of an identity, obtained by interchanging the $\mathrm{V}$ and $\wedge$ operators and interchanging the elements 0 and $1,$ is also a valid identity.

WZ
Wen Z.
Numerade Educator

Problem 43

Show that a complemented, distributive lattice is a Boolean algebra.

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