Chapter 12

Construct a logic table for each boolean expression. $$(x \uparrow x) \downarrow(y \uparrow y)$$

Construct a logic table for each boolean expression. $$(x \downarrow x) \downarrow(y \downarrow y)$$

Using the laws of boolean algebra, find the DNF of each boolean function. $$f(x, y)=x \uparrow y$$

Using the laws of boolean algebra, find the DNF of each boolean function. $$f(x, y)=x \downarrow y$$

Verify each. $$x+y=(x \uparrow x) \uparrow(y \uparrow y)$$

Verify each. $$x y=(x \downarrow x) \downarrow(y \downarrow y)$$

Give a counterexample to disprove each statement. $$x \downarrow(y \downarrow z)=(x \downarrow y) \downarrow z$$

Verify that each set of boolean operators is functionally complete. $$\\{+, '\\}$$

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Evaluate each. $$ 1 \oplus(1 \oplus 1) $$

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Evaluate each. $$ 1 \oplus(0 \oplus 1) $$

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Evaluate each. $$ (1 \oplus 0) \oplus 1 $$

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Evaluate each. $$ 1 \uparrow(0 \oplus 1) $$

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Evaluate each. $$ 1 \downarrow(0 \oplus 1) $$

Use the following definition of the binary operator XOR, denoted by \(\oplus,\) for Exercises \(69-81\) $$x \oplus y=\left\\{\begin{array}{ll} 1 & \text { if exactly one of the bits } x \text { and } y \text { is } 1 \\ 0 & \text { otherwise } \end{array}\right.$$ Evaluate each. $$(1 \uparrow 0) \oplus(1 \downarrow 1)$$

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Is \(\\{\oplus\\}\) functionally complete?

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Find the DNF of the boolean function \(f(x, y)=x \oplus y\)

Find the DNF of the boolean function \(f(x, y)=x \oplus y\)

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Prove each. $$ x \oplus x=0 $$

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Prove each. $$ x \oplus y=y \oplus x $$

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Prove each. $$ x \oplus y=(x+y)(x y)^{\prime} $$

Use the following definition of the binary operator XOR, denoted by \(\oplus,\) for Exercises \(69-81\) $$x \oplus y=\left\\{\begin{array}{ll} 1 & \text { if exactly one of the bits } x \text { and } y \text { is } 1 \\ 0 & \text { otherwise } \end{array}\right.$$ Prove each. $$x \oplus(y \oplus z)=(x \oplus y) \oplus z$$

Write an algorithm to find the CNF of a boolean function \(f\)