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Q30E

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Discrete Mathematics and its Applications
Found in: Page 843
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

Find a minimal sum-of-products expansion, given the \({\bf{K}}\)-map shown with don't care conditions indicated with \({\bf{d}}'{\bf{s}}\).

Minimal sum-of-products expansion \({\bf{\bar z + wx}}\)

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1:Definition

In some circuits we care only about the output for some combinations of input values, because other combinations of input values are not possible or never occur. This gives us freedom in producing a simple circuit with the desired output because the output values for all those combinations that never occur can be arbitrarily chosen. The values of the function for these combinations are called don’t care conditions.

Step 2: Largest block

Given:

The largest block in the graph consists of two columns \({\bf{\bar yz}}\) and \({\bf{\bar y\bar z}}\) (as these two columns contain only \(1{\bf{'s}}\)and \({\bf{d's}}\)), while the two columns can be notated as \({\bf{\bar z}}\) (as the two columns contain the only possible outcomes with \({\bf{\bar z}}\) ).

Only the element \({\bf{wx \bar yz}}\) was not included in the previous block, we still need to include it in a block. The largest block that contains \({\bf{wx\bar yz}}\) is the row \({\bf{wx}}\) (as the entire row contains \({\bf{d's}}\)and \(1{\bf{'s}}\)).

Step 3: Minimal sum-of-products

The minimal sum-of-products expansion is then the sum of these blocks.

Hence, the Minimal sum-of-products expansion \({\bf{\bar z + wx}}\).

Most popular questions for Math Textbooks

Show that if \({\bf{F}}\) and \({\bf{G}}\) are Boolean functions represented by Boolean expressions in \({\bf{n}}\) variables and \({\bf{F = G}}\), then \({{\bf{F}}^{\bf{d}}}{\bf{ = }}{{\bf{G}}^{\bf{d}}}\), where \({{\bf{F}}^{\bf{d}}}\) and \({{\bf{G}}^{\bf{d}}}\) are the Boolean functions represented by the duals of the Boolean expressions representing \({\bf{F}}\) and \({\bf{G}}\), respectively. (Hint: Use the result of Exercise \(29\).)

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 i is the value of the two-bit integer\({{\bf{(}}{{\bf{c}}_{\bf{1}}}{{\bf{c}}_{\bf{o}}}{\bf{)}}_{\bf{2}}}\).The depth of a combinatorial circuit can be defined by specifyingthat the depth of the initial input is 0 and if a gate has n different 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.

Is it always true that \((x \odot y) \odot z{\bf{ = }}x \odot (y \odot z)\)\(?\)

Draw the \(K{\bf{ - }}\)maps of these sum-of-products expansions in two variables.

\(\begin{array}{l}{\bf{a)x\bar y}}\\{\bf{b)xy + \bar x\bar y}}\\{\bf{c)xy + x\bar y + \bar xy + \bar x\bar y}}\end{array}\)

Show that cells in a \({\bf{K}}\)-map for Boolean functions in five variables represent minterms that differ in exactly one literal if and only if they are adjacent or are in cells that become adjacent when the top and bottom rows and cells in the first and eighth columns, the first and fourth columns, the second and seventh columns, the third and sixth columns, and the fifth and eighth columns are considered adjacent.
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