Chapter 12: Q1RE (page 844)
Define a Boolean function of degree \({\bf{n}}\).
A function \({\bf{f}}\) in \({\bf{n}}\) variables taking two values \(0\) and \(1\) is called a Boolean function.
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Use a table to express the values of each of these Boolean functions.
\(\begin{array}{l}{\bf{a) F(x,y,z) = \bar z}}\\{\bf{b) F(x,y,z) = \bar xy + \bar yz}}\\{\bf{c) F(x,y,z) = x\bar yz + }}\overline {{\bf{(xyz)}}} \\{\bf{d) F(x,y,z) = \bar y(xz + \bar x\bar z)}}\end{array}\)
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 \({\bf{x}} \wedge {\bf{(y}} \vee {\bf{(x}} \wedge {\bf{z)) = (x}} \wedge {\bf{y)}} \vee {\bf{(x}} \wedge {\bf{z)}}\) and \({\bf{x}} \vee {\bf{(y}} \wedge {\bf{(x}} \vee {\bf{z)) = (x}} \vee {\bf{y)}} \wedge {\bf{(x}} \vee {\bf{z)}}\).
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 \(x \vee y{\bf{ = }}0\), then \(x{\bf{ = }}0\) and \(y{\bf{ = }}0\), and that if \(x \wedge y{\bf{ = }}1\), then \(x{\bf{ = }}1\) and \(y{\bf{ = }}1\).
In Exercises 1–5 find the output of the given circuit.
How many cells in a \({\bf{K}}\)-map for Boolean functions with six variables are needed to represent \({{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{\bar x}}_{\bf{1}}}{{\bf{x}}_{\bf{6}}}{\bf{, }}{{\bf{\bar x}}_{\bf{1}}}{{\bf{x}}_{\bf{2}}}{{\bf{\bar x}}_{\bf{6}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{{\bf{x}}_{\bf{3}}}{{\bf{x}}_{\bf{4}}}{{\bf{x}}_{\bf{5}}}\), and \({{\bf{x}}_{\bf{1}}}{{\bf{\bar x}}_{\bf{2}}}{{\bf{x}}_{\bf{4}}}{{\bf{\bar x}}_{\bf{5}}}\), respectively\({\bf{?}}\)
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