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Chapter 12: Q1E (page 841)

\(a)\) 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\(?\)

Short Answer

Expert verified

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

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

Step by step solution

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01

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 \(\bar x\).

02

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 \(\bar x\bar y\).

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Most popular questions from this chapter

In Exercises 1–5 find the output of the given circuit.

Find the sum-of-products expansions represented by each of these \(K{\bf{ - }}\)maps.

\(({\bf{a)}}\)

\({\bf{(b)}}\)

\({\bf{(c)}}\)

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

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

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\({\bf{?}}\)

use the laws in Definition \(1\) to show that the stated properties hold in every Boolean algebra.

Show that in a Boolean algebra, the complement of the element \(0\) is the element \(1\) and vice versa.
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