Chapter 12: Q9E (page 818)
What values of the Boolean variables \({\bf{x}}\) and \({\bf{y}}\) satisfy \({\bf{xy = x + y}}\)\(?\)
The value of x=0 and y=0 or x=1 and y=1 will solve the given equation \(xy = x + y\).
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Draw the \({\bf{K}}\)-maps of these sum-of-products expansions in three variables.
\(\begin{array}{l}{\bf{a) x\bar y\bar z}}\\{\bf{b) \bar xyz + \bar x\bar y\bar z}}\\{\bf{c) xyz + xy\bar z + \bar xy\bar z + \bar x\bar yz}}\end{array}\)
Show that if \({\bf{F, G}}\), and \({\bf{H}}\) are Boolean functions of degree \({\bf{n}}\), then \({\bf{F + G}} \le {\bf{H}}\) if and only if \({\bf{F}} \le {\bf{H}}\) and \({\bf{G}} \le {\bf{H}}\).
Prove or disprove these equalities.
\(\begin{array}{l}a)\;x \oplus (y \oplus z){\bf{ = }}(x \oplus y) \oplus z\\b)\;x{\bf{ + }}(y \oplus z){\bf{ = }}(x{\bf{ + }}y) \oplus (x{\bf{ + }}z)\\c)\;x \oplus (y{\bf{ + }}z){\bf{ = }}(x \oplus y){\bf{ + }}(x \oplus z)\end{array}\)
Find the sum-of-products expansions represented by each of these \(K{\bf{ - }}\)maps.
\(({\bf{a)}}\)
\({\bf{(b)}}\)
\({\bf{(c)}}\)
Find the depth of
a)The circuit constructed in Example 2 for majority voting among three people.
b)The circuit constructed in Example 3 for a light controlled by two switches.
c)The half adder shown in Figure 8.
d)The full adder shown in Figure 9.
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