Section 1
Boolean Functions
Find the values of these expressions.a) $1 \cdot \overline{0}$b) $1+\overline{1}$c) $\overline{0} \cdot 0$d) $\overline{(1+0)}$
Find the values, if any, of the Boolean variable $x$ that satisfy these equations.a) $x \cdot 1=0$b) $x+x=0$c) $x \cdot 1=x$d) $x \cdot \bar{x}=1$
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 to a $\mathbf{F}$, each 1 to a T, each Boolean sum into a disjunction, each Boolean product into a conjunction, each complementation into a negation, and the equals sign to a propositional equivalence sign.
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 to a $\mathbf{F}$, each 1 to a T, each Boolean sum into a disjunction, each Boolean product into a conjunction, each complementation into a negation, and the equals sign to a propositional equivalence sign.
Use a table to express the values of each of these Boolean functions.a) $F(x, y, z)=\bar{x} y$b) $F(x, y, z)=x+y z$c) $F(x, y, z)=x \bar{y}+\overline{(x y z)}$d) $F(x, y, z)=x(y z+\bar{y} \bar{z})$
Use a table to express the values of each of these Boolean functions.a) $F(x, y, z)=\bar{z}$b) $F(x, y, z)=\bar{x} y+\bar{y} z$c) $F(x, y, z)=x \bar{y} z+\overline{(x y z)}$d) $F(x, y, z)=\bar{y}(x z+\bar{x} \bar{z})$
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 .
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 .
What values of the Boolean variables $x$ and $y$ satisfy $x y=x+y$ ?
How many different Boolean functions are there of degree 7 ?
Prove the absorption law $x+x y=x$ using the other laws in Table 5 .
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 .
Show that $x \bar{y}+y \bar{z}+\bar{x} z=\bar{x} y+\bar{y} z+x \bar{z}$.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
The Boolean operator $\oplus$, called the XOR operator, is defined by $1 \oplus 1=0,1 \oplus 0=1,0 \oplus 1=1$, and $0 \oplus 0=0$.Simplify these expressions.a) $x \oplus 0$b) $x \oplus 1$c) $x \oplus x$d) $x \oplus \bar{x}$
Show that these identities hold.a) $x \oplus y=(x+y) \overline{(x y)}$b) $x \oplus y=(x \bar{y})+(\bar{x} y)$
Show that $x \oplus y=y \oplus x$.
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) $x \oplus(y+z)=(x \oplus y)+(x \oplus z)$
Find the duals of these Boolean expressions.a) $x+y$b) $\bar{x} \bar{y}$c) $x y z+\bar{x} \bar{y} \bar{z}$d) $x \bar{z}+x \cdot 0+\bar{x} \cdot 1$
Suppose that $F$ is a Boolean function represented by a Boolean expression in the variables $x_1, \ldots, x_n$. Show that $F^d\left(x_1, \ldots, x_n\right)=\overline{F\left(\bar{x}_1, \ldots, \bar{x}_n\right)}$.
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.]
How many different Boolean functions $F(x, y, z)$ are there such that $F(\bar{x}, \bar{y}, \bar{z})=F(x, y, z)$ for all values of the Boolean variables $x, y$, and $z$ ?
How many different Boolean functions $F(x, y, z)$ are there such that $F(\bar{x}, y, z)=F(x, \bar{y}, z)=F(x, y, \bar{z})$ for all values of the Boolean variables $x, y$, and $z$ ?
Show that you obtain De Morgan's laws for propositions (in Table 5 in Section 1.2) when you transform De Morgan'\$ laws for Boolean algebra in Table 6 into logical equivalences.
Show that you obtain the absorption laws for propositions (in Table 5 in Section 1.2) when you transform the absorption laws for Boolean algebra in Table 5 into logical equivalences.
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 $x$.
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 $\bar{x}$ such that $x \vee \bar{x}=1$ and $x \wedge \bar{x}=0$.
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.
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, $\overline{\bar{x}}=x$ for every element $x$.
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 $y, \overline{(x \vee y)}=\bar{x} \wedge \bar{y}$ and $\overline{(x \wedge y)}=\bar{x} \vee \bar{y}$.
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 $x \vee(y \wedge(x \vee z))=(x \vee y) \wedge(x \vee z)$.
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 $y=1$.
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 $\vee$ and $\wedge$ operators and interchanging the elements 0 and 1 , is also a valid identity.
Show that a complemented, distributive lattice is a Boolean algebra.