Question

\({\bf{a)}}\)Draw a \(K{\bf{ - }}\)map for a function in three variables. Put a \(1\) in the cell that represents \(\bar xy\bar z\).

\({\bf{b)}}\)Which minterms are represented by cells adjacent to this cell\(?\)

Short answer

\(({\bf{a)}}\) K-map for a function in three variables,

\(({\bf{b)}}\) The adjacent cells have minterms\(xy\bar z,\bar xyz\) and \(\bar x\bar y\bar z\).

Step-by-step solution

Step 1:Definition

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. You will continue by showing how K-maps can be used to minimize Boolean functions in three variables and then in four variables. Then you 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 three variables is a table with four columns \(yz,y\bar z,\bar y\bar z\) and \(\bar yz\); which contains all possible combinations of \({\bf{y}}\) and \(z\)and two rows \({\bf{x}}\) and \({\bf{\bar x}}\).

Placing the values

(a)

It places a \(1\) in the cell corresponding to \(\bar xy\bar z\) (which is the in the cell in the row \({\bf{\bar x}}\) and in the column \(y\bar z\).)

Step 3:Using the Boolean product

(b)

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.

Then note that the adjacent cells have minterms\(xy\bar z,\bar xyz\) and \(\bar x\bar y\bar z\).