Chapter 12

Evaluate the boolean expression \(x\left(y z^{\prime}+y^{\prime} z\right)\) at the ordered triplets (1,0,1) and (1,1,1)

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

Using Example \(12.2,\) evaluate each. $$6+10$$

When will the combinatorial circuit for each boolean expression produce 1 as the output? $$x^{\prime}$$

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

When will the combinatorial circuit for each boolean expression produce 1 as the output? $$x+y$$

Simplify the boolean expression represented by each Karnaugh map. $$\begin{aligned}&\begin{array}{lllll}\quad yz \quad y z^{\prime} \quad y^{\prime} z^{\prime}\quad y^{\prime}z \end{array} \\ &\begin{array}{lllll}x \\\ x^{\prime}\end{array} \quad\begin{array}{|c|c|c|c|c|}\hline d & & 1 & d\\\\\hline & d & d & 1\\\ \hline\end{array}\end{aligned}$$

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

How many constant boolean functions can be defined from \(B^{n}\) to \(B\) with \(B\) a two-element boolean algebra?

Compute the NAND gate output from inputing each pair of bits. $$0,0$$

When will the combinatorial circuit for each boolean expression produce 1 as the output? $$x y$$

Simplify the boolean expression represented by each Karnaugh map. $$\begin{aligned}&\begin{array}{lllll}\qquad yz \quad y z^{\prime} \quad y^{\prime} z^{\prime}\quad y^{\prime}z \end{array} \\ &\begin{array}{lllll}wx \\\ wx^{\prime} \\ w^{\prime} x^{\prime}\\\ w^{\prime} x\end{array} \quad\begin{array}{|c|c|c|c|}\hline & & 1 & 1 \\\\\hline & & d & d \\\\\hline d & d & d & d \\\\\hline 1 & 1 & d & d \\\\\hline\end{array}\end{aligned}$$

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

Find the number of boolean functions that can be defined from \(B^{n}\) to \(B,\) where \(B\) is a two-element boolean algebra.

Using Example \(12.2,\) evaluate each. $$(2+3) + 5$$

Simplify the boolean expression represented by each Karnaugh map. $$\begin{aligned}&\begin{array}{lllll}\qquad yz \quad y z^{\prime} \quad y^{\prime} z^{\prime}\quad y^{\prime}z \end{array} \\ &\begin{array}{lllll}wx \\\ wx^{\prime} \\ w^{\prime} x^{\prime}\\\ w^{\prime} x\end{array} \quad\begin{array}{|l|l|l|l|} \hline 1 & d & & \\ \hline d & 1 & & \\ \hline & & d & d \\ \hline & & 1 & 1 \\ \hline \end{array}\end{aligned}$$

Compute the NAND gate output from inputing each pair of bits. $$0,1$$

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

Determine if each is a boolean expression, where each variable is boolean. $$\left(\left(x y^{\prime}\right)^{\prime}\right)^{\prime}$$

Compute the NAND gate output from inputing each pair of bits. $$1,0$$

Simplify the boolean expression represented by each Karnaugh map. $$\begin{aligned}&\begin{array}{lllll}\qquad yz \quad y z^{\prime} \quad y^{\prime} z^{\prime}\quad y^{\prime}z \end{array} \\ &\begin{array}{lllll}wx \\\ wx^{\prime} \\ w^{\prime} x^{\prime}\\\ w^{\prime} x\end{array} \quad\begin{array}{|c|c|c|c|} \hline d & 1 & 1 & 1 \\ \hline & & & \\ \hline & & & d \\ \hline 1 & & 1 & d \\ \hline \end{array}\end{aligned}$$

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

Determine if each is a boolean expression, where each variable is boolean. $$ x^{\prime}+y z $$

Simplify the boolean expression represented by each Karnaugh map. $$\begin{aligned}&\begin{array}{lllll}\qquad yz \quad y z^{\prime} \quad y^{\prime} z^{\prime}\quad y^{\prime}z \end{array} \\ &\begin{array}{lllll}wx \\\ wx^{\prime} \\ w^{\prime} x^{\prime}\\\ w^{\prime} x\end{array} \quad\begin{array}{|c|c|c|c|}\hline & d & & \\\\\hline 1 & d & d & 1 \\\\\hline & & 1 & d \\ \hline d & 1 & 1 & d \\\\\hline\end{array}\end{aligned}$$

Compute the NAND gate output from inputing each pair of bits. $$1,1$$

Find the minimum number of edges that must be removed from each complete graph, so the resulting graph is planar. $$x^{\prime}+y z$$

Determine if each is a boolean expression, where each variable is boolean. $$ \left(x y+y^{\prime} z^{\prime}\right)^{\prime} $$

Simplify the boolean expression defined by each table. $$\begin{array}{|lll||l|} \hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{z} & \boldsymbol{f} \\ \hline 0 & 0 & 0 & 1 \\\0 & 0 & 1 & 1 \\\0 & 1 & 0 & 0 \\\0 & 1 & 1 & 0 \\\1 & 0 & 0 & d \\\1 & 0 & 1 & d \\\1 & 1 & 0 & 1 \\\1 & 1 & 1 & 1 \\\ \hline\end{array}$$

Simplify the boolean expression defined by each table. $$\begin{array}{|lll||l|} \hline x & y & z & f \\ \hline 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & d \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & d \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ \hline\end{array}$$

Find the minimum number of edges that must be removed from each complete graph, so the resulting graph is planar. $$x\left(y z^{\prime}\right)^{\prime}$$

Simplify each boolean expression using the laws of boolean algebra. $$(x+y)(y+z)(z+x)$$

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

Using the BCD expansions of the decimal numbers 0 through 9, develop a boolean function \(f(w, x, y, z)\) such that the decimal value of \((w x y z)_{\text {two }}\) is divisible by: Two

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

Using the BCD expansions of the decimal numbers 0 through 9, develop a boolean function \(f(w, x, y, z)\) such that the decimal value of \((w x y z)_{\text {two }}\) is divisible by: Three

U11. 5 13\. 35 15\. 2 \begin{tabular}{l} 17\. 70 \\ \hline \end{tabular} 19\. 10 21\. \(5^{\prime}=70 / 5=14\) 23\. \(5 \odot(5 \oplus 7)=5 \odot 35=5\) \(\therefore\left(5^{\prime}\right)^{\prime}=14^{\prime}=70 / 14=5\) 25\. \(2^{n}\) 27\. yes 29\. no 31\. yes 33\. yes 35 . \begin{tabular}{|c|cccc|} \hline\(+\) & 0 & 1 & \(a\) & \(b\) \\ \hline 0 & 0 & 1 & \(a\) & \(b\) \\ 1 & 1 & 1 & 1 & 1 \\ \(a\) & \(a\) & 1 & \(a\) & 1 \\ \(b\) & \(b\) & 1 & 1 & \(b\) \\ \hline \end{tabular} \begin{tabular}{|c|cccc|} \hline & \(\mathbf{0}\) & \(\mathbf{1}\) & \(\boldsymbol{a}\) & \(\boldsymbol{b}\) \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & \(a\) & \(b\) \\ \(a\) & 0 & \(a\) & \(a\) & 0 \\ \(b\) & 0 & \(b\) & 0 & \(b\) \\ \hline \end{tabular} \(0^{\prime}=1,1^{\prime}=0, a^{\prime}=b, b^{\prime}=a\) 37\. \(x+x y=x\) 39\. \((x+y)^{\prime}=x^{\prime} y^{\prime}\)sing Example \(12.2,\) evaluate each. $$5^{\prime} \oplus 10^{\prime}$$

Simplify each boolean expression using the laws of boolean algebra. $$(x+y)\left(x^{\prime}+y\right)\left(x+y^{\prime}\right)$$

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

Using the BCD expansions of the decimal numbers 0 through 9, develop a boolean function \(f(w, x, y, z)\) such that the decimal value of \((w x y z)_{\text {two }}\) is divisible by: Two and three

Using Example \(12.2,\) evaluate each. $$2+3.5$$

Construct a logic table for each gate. \(x \downarrow y\)

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

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

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

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. $$7 \oplus 5$$

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. $$2 \odot 7$$

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

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

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