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Q1E

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Discrete Mathematics and its Applications
Found in: Page 841
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

\(a)\) Draw a \(K{\bf{ - }}\)map for a function in two variables and put a \(1\) in the cell representing \(\bar xy\).

\(b)\) What are the minterms represented by cells adjacent to this cell\(?\)

\((a)\) The K-map for a function in two variables

\((b)\)The adjacent cells have minterms \(xy\) and \(\bar x\bar y\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1:Definition

(a)

To reduce the number of terms in a Boolean expression representing a circuit, it is necessary to find terms to combine. There is a graphical method, called a Karnaugh map or K-map, for finding terms to combine for Boolean functions involving a relatively small number of variables. It will first illustrate how K-maps are used to simplify expansions of Boolean functions in two variables. It will continue by showing how K-maps can be used to minimize Boolean functions in three variables and then in four variables. Then it will describe the concepts that can be used to extend K-maps to minimize Boolean functions in more than four variables.

A \(K{\bf{ - }}\)map for a function in two variables is basically a table with two columns \(y\) and \(\bar y\) and two rows \(x\) and \(\bar x\).

Step 2: Adjacent cells

(b)

Itplaces a \(1\) in the cell corresponding to \(\bar xy\) (which is the in the cell in the row \(\bar x\) and in the column \(y\)).

The minterms represented by the adjacent cells (to the cell with a \(1\)) are then the Boolean product of the row title and the column title of the cell.

It then notes that the adjacent cells have minterms \(xy\) and \(\bar x\bar y\).

Most popular questions for Math Textbooks

Show that these identities hold.

\(\begin{array}{c}a)\;x \oplus y{\bf{ = (x + }}y)\overline {(xy)} \\b)\;x \oplus y{\bf{ = (x\bar y) + }}(\bar xy)\end{array}\)

Construct a circuit for a full subtractor using AND gates, OR gates, and inverters. A full subtractor has two bits and a borrow as input, and produces as output a difference bit and a borrow.

Use a table to express the values of each of these Boolean functions.

\(\begin{array}{l}(a)F(x,y,z) = \bar xy\\(b)F(x,y,z) = x + yz\\(c)F(x,y,z) = x\bar y + \overline {(xyz)} \\(d)F(x,y,z) = x(yz + \bar y\bar z)\end{array}\)

Find the duals of these Boolean expressions.

\(\begin{array}{l}{\bf{a) x + y}}\\{\bf{b) \bar x\bar y}}\\{\bf{c) xyz + \bar x\bar y\bar z}}\\{\bf{d) x\bar z + x \times 0 + \bar x \times 1}}\end{array}\)

Show that \(x \odot y{\bf{ = }}\overline {(x \oplus y)} \).
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