Chapter 12

Construct a logic table for each boolean function defined by each boolean expression. $$x y z+x(y z)^{\prime}$$

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$(5 \oplus 7)^{\prime}$$

Find the DNF of each boolean function. $$f(x, y, z)=x \uparrow(y \uparrow z)$$

Find the DNF of each boolean function. $$f(x, y, z)=x \uparrow(y \uparrow z)$$

Simplify each boolean expression using the laws of boolean algebra. $$w x^{\prime} y z+w x^{\prime} y z^{\prime}+w^{\prime} x^{\prime} y z^{\prime}+w^{\prime} x y z^{\prime}$$

Construct a logic table for each boolean function defined by each boolean expression. $$x^{\prime} y z^{\prime}+x^{\prime}(y z)^{\prime}$$

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$5^{\prime} \odot 7^{\prime}$$

Find the DNF of each boolean function. $$f(x, y, z)=(x \uparrow y) \uparrow z$$

Find the DNF of each boolean function. $$f(x, y, z)=(x \uparrow y) \uparrow z$$

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$\left(7 \odot 2^{\gamma}\right.$$

Suppose the bits \(x, y,\) and \(z\) are input into a NAND gate. List the possible output in a logic table.

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$7^{\prime} \oplus 2^{\prime}$$

Using a logic table, verify each. $$(x+y)^{\prime}=x^{\prime} y^{\prime}$$

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$10 \oplus 10$$

Mark each statement as true or false. $$x \uparrow y=y \uparrow x$$

Find the boolean expression represented by each Karnaugh map.

Using a logic table, verify each. $$(x y)^{\prime}=x^{\prime}+y^{\prime}$$

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$7 \odot 7$$

Mark each statement as true or false. $$x\downarrow y=y \downarrow x$$

Mark each statement as true or false. $$x \downarrow y=y \downarrow x$$

Using a logic table, verify each. $$(x+y)^{\prime} \neq x^{\prime}+y^{\prime}$$

Mark each statement as true or false. $$x \uparrow(y \uparrow z)=(x \uparrow y) \uparrow z$$

Using the boolean algebra \(D_{70},\) verify each. $$\left(5^{\prime}\right)^{\prime}=5$$

Using a logic table, verify each. $$(x y)^{\prime} \neq x^{\prime} y^{\prime}$$

Using the boolean algebra \(D_{70},\) verify each. $$7+(7 .5)=7$$

Using only NAND gates, design a combinatorial circuit that receives \(x\) and \(y\) as input signals and outputs: $$x^{\prime}$$

Mark each statement as true or false. $$x \downarrow(y \downarrow z)=(x \downarrow y) \downarrow z$$

Display each sum of minterms in a Karnaugh map. $$x^{\prime} y+x y^{\prime}$$

Mark each statement as true or false. $$x \downarrow(y \downarrow z)=(x \downarrow y) \downarrow z$$

Using the boolean algebra \(D_{70},\) verify each. $$7 \oplus(7 \odot 5)=7$$

Using a Karnaugh map, simplify each sum of minterms. $$x y+x y^{\prime}$$

Is the equality relation on the set of boolean expressions in \(n\) variables an equivalence relation?

Using a Karnaugh map, simplify each sum of minterms. $$x y+x y^{\prime}+x^{\prime} y^{\prime}$$

List all minterms in two boolean variables \(x\) and \(y\)

Using only NAND gates, design a combinatorial circuit that receives \(x\) and \(y\) as input signals and outputs: $$x y$$

Give all minterms three boolean variables \(x, y,\) and \(z\) can generate.

How many minterms can \(n\) boolean variables produce?

Define the operations \(+, \cdot,\) and \(^{\prime}\) on \(B=\\{0,1\\}\) as follows: \(x+y=\) \(\max \\{x, y\\}, x \cdot y=\min \\{x, y\\}, 0^{\prime}=1,\) and \(1^{\prime}=0 .\) Is \(\left\langle B,+, \cdot,,^{\prime}, 0,1\right\rangle \mathrm{a}\) boolean algebra?

Find the DNFs of the boolean functions in Exercises \(27-34\) $$ \begin{array}{|c|c|c|}\hline x & {y} & {f(x, y)} \\ \hline 0 & {0} & {0} \\\ {0} & {1} & {0} \\ {1} & {0} & {0} \\ {1} & {1} & {1} \\ \hline\end{array} $$

Find the DNFs of the boolean functions $$\begin{array}{|ll||c|} \hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y}) \\ \hline 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ \hline \end{array}$$

Find the DNFs of the boolean functions in Exercises \(27-34\) $$ \begin{array}{|c|c|c|}\hline x & {y} & {f(x, y)} \\ \hline 0 & {0} & {0} \\\ {0} & {1} & {1} \\ {1} & {0} & {1} \\ {1} & {1} & {1} \\ \hline\end{array} $$

Determine if \(\left\langle S,+,^{\prime}, \quad, 0,1\right\rangle\) is a boolean algebra for each subset \(S\) of the boolean algebra \(D_{30}\) . $$ \\{1,6,10,30\\} $$

Make a combinatorial circuit for a hallway light fixture controlled by two switches \(x\) and \(y .\) Assume the light is off when both switches are.

Find the DNFs of the boolean functions $$\begin{array}{|ll||c|} \hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y}) \\ \hline 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \hline \end{array}$$

Using a Karnaugh map, simplify each boolean expression. $$x y^{\prime} z+x y^{\prime} z^{\prime}$$

Find the DNFs of the boolean functions in Exercises \(27-34\) $$ \begin{array}{|c|c|c|}\hline x & {y} & {f(x, y)} \\ \hline 0 & {0} & {1} \\\ {0} & {1} & {0} \\ {1} & {0} & {0} \\ {1} & {1} & {1} \\ \hline\end{array} $$

Find the DNFs of the boolean functions $$\begin{array}{|ll||c|} \hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y}) \\ \hline 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ \hline \end{array}$$

Using a Karnaugh map, simplify each boolean expression. $$x y z+x y^{\prime} z+x^{\prime} y z+x^{\prime} y^{\prime} z$$

Find the DNFs of the boolean functions in Exercises \(27-34\) $$ \begin{array}{|c|c|c|}\hline x & {y} & {f(x, y)} \\ \hline 0 & {0} & {1} \\\ {0} & {1} & {1} \\ {1} & {0} & {1} \\ {1} & {1} & {0} \\ \hline\end{array} $$

Find the DNFs of the boolean functions $$\begin{array}{|ll||c|} \hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y}) \\ \hline 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \hline \end{array}$$