Chapter 12: Q26E (page 818)
Show that \({\bf{x}} \oplus {\bf{y = y}} \oplus {\bf{x}}\).
The given \(x \oplus y = y \oplus x\)is proved.
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Use a \({\bf{K}}\)-map to find a minimal expansion as a Boolean sum of Boolean products of each of these functions in the variables \({\bf{w, x, y}}\) and \({\bf{z}}\).
\(\begin{array}{l}{\bf{a) wxyz + wx\bar yz + wx\bar y\bar z + w\bar xy\bar z + w\bar x\bar yz}}\\{\bf{b) wxy\bar z + wx\bar yz + w\bar xyz + \bar wx\bar yz + \bar w\bar xy\bar z + \bar w\bar x\bar yz}}\\{\bf{c) wxyz + wxy\bar z + wx\bar yz + w\bar x\bar yz + w\bar x\bar y\bar z + \bar wx\bar yz + \bar w\bar xy\bar z + \bar w\bar x\bar yz}}\\{\bf{d) wxyz + wxy\bar z + wx\bar yz + w\bar xyz + w\bar xy\bar z + \bar wxyz + \bar w\bar xyz + \bar w\bar xy\bar z + \bar w\bar x\bar yz}}\end{array}\)
Show that \({\bf{x}} \odot {\bf{y = xy + \bar x\bar y}}\).
Exercises 14-23 deal with the Boolean algebra \(\left\{ {{\bf{0,1}}} \right\}\) with addition,multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example \(8\).
22. Verify the unit property.
Construct a multiplexer using AND gates, OR gates, andinverters that has as input the four bits\({{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{,}}{{\bf{x}}_{\bf{3}}}\)and the two control bits\({{\bf{c}}_{\bf{o}}}\)and\({{\bf{c}}_{\bf{1}}}\). Set up the circuit so that\({{\bf{x}}_{\bf{i}}}\)is the output, where iis the value of the two-bit integer\({{\bf{(}}{{\bf{c}}_{\bf{1}}}{{\bf{c}}_{\bf{o}}}{\bf{)}}_{\bf{2}}}\).The depthof a combinatorial circuit can be defined by specifyingthat the depth of the initial input is 0 and if a gate has ndifferent inputs at depths\({{\bf{d}}_{\bf{1}}}{\bf{,}}{{\bf{d}}_{\bf{2}}}{\bf{,}}.....{\bf{,}}{{\bf{d}}_{\bf{n}}}\),respectively, then its outputs have depth equal to max\({\bf{(}}{{\bf{d}}_{\bf{1}}}{\bf{,}}{{\bf{d}}_{\bf{2}}}{\bf{,}}.....{\bf{,}}{{\bf{d}}_{\bf{n}}}{\bf{) + 1}}\); this value is also defined to be the depth of the gate. The depth of a combinatorial circuit is the maximum depth of the gates in the circuit.
Prove or disprove these equalities.
\(\begin{array}{l}a)\;x \oplus (y \oplus z){\bf{ = }}(x \oplus y) \oplus z\\b)\;x{\bf{ + }}(y \oplus z){\bf{ = }}(x{\bf{ + }}y) \oplus (x{\bf{ + }}z)\\c)\;x \oplus (y{\bf{ + }}z){\bf{ = }}(x \oplus y){\bf{ + }}(x \oplus z)\end{array}\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
