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Chapter 12: Q12RE (page 844)

\(a)\)What is a don't care condition\(?\)

\(b)\)Explain how don't care conditions can be used to build a circuit using \(OR\) gates, \(AND\) gates, and inverters that produces an output of \({\bf{1}}\) if a decimal digit is \(6\) or greater, and an output of \({\bf{0}}\) if this digit is less than \(6\).

Short Answer

Expert verified

\((a)\)A don't care condition is a combination of input values for which it doesn't care about the output, because they are not possible or never occur.

\((b)\) The sum-of-products expansion is \(w{\bf{ + xy}}\).

Step by step solution

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01

Don’t care condition

(a)

A don't care condition is a combination of input values for which you don't care about the output, because they are not possible or never occur.

02

Sum-of-product expansion

(b)

A \(K{\bf{ - }}\)map for a function in four variables has 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 \({\bf{z}}\) and four \(wx,w\bar x,\bar w\bar x\) and \(\bar wx\); which contains all possible combinations of \({\bf{y}}\) and \({\bf{z}}\).

You place a \(d\) in all cells in the first row and in the first two cells in the second row.

You place a \({\bf{1}}\) in the cell corresponding to a digit that is larger than \(6\) (The numbers \(0\) to \(6\) are represented by

\(0000/\bar w\bar x\bar y\bar z,0001/\bar w\bar x\bar yz,0010/\bar w\bar xy\bar z,0011/\bar w\bar xyz,0100/\bar wx\bar y\bar z,0101/\bar wx\bar yz,0110/\bar wxy\bar z\)

The largest block containing \(w\bar x\bar yz\) is \(w\), because all values in the rows with \(wx\) and \(w\bar x\) are \(d\) or \({\bf{1}}\).

The largest block containing \(\bar wxyz\) is \({\bf{xy}}\), because all cells corresponding to both \({\bf{y}}\) and \(x\) have a value of \({\bf{1}}\) or \(d\).

The corresponding sum-of-products expansion is then the sum of the two blocks:

Therefore, the sum-of-products expansion \({\bf{ = }}w{\bf{ + xy}}\).

03

Circuit

A Boolean sum is represented by an \(OR\) gate and a Boolean product is represented by an \(AND\) gate.

Note: No inverters are needed.

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Most popular questions from this chapter

Find the sum-of-products expansion of the Boolean function F (\({{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{,}}{{\bf{x}}_{\bf{3}}}{\bf{,}}{{\bf{x}}_{\bf{4}}}{\bf{,}}{{\bf{x}}_{\bf{5}}}\)) that has the value 1 if and only if three or more of the variables \({{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{,}}{{\bf{x}}_{\bf{3}}}{\bf{,}}{{\bf{x}}_{\bf{4}}}{\bf{,}}{{\bf{x}}_{\bf{5}}}\) have the value 1.

Which of these functions are self-dual?

\(\begin{array}{l}\left. {\bf{a}} \right)\;{\bf{F}}\left( {{\bf{x,y}}} \right) = x\\\left. {\bf{b}} \right)\;{\bf{F}}\left( {{\bf{x,y}}} \right) = {\bf{xy + \bar x\bar y}}\\\left. {\bf{c}} \right)\;{\bf{F}}\left( {{\bf{x,y}}} \right) = {\bf{x + y}}\\\left. {\bf{d}} \right)\;{\bf{F}}\left( {{\bf{x,y}}} \right) = {\bf{xy + \bar xy}}\end{array}\)

Find the sum-of-products expansion of the Boolean function\({\bf{F}}\left( {{\bf{w, x, y, z}}} \right)\) that has the value 1 if and only if an odd number of w, x, y, and z have the value 1.

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

Use a \(3\)- cube \({{\bf{Q}}_{\bf{3}}}\) to represent each of the Boolean functions in Exercise \(6\) by displaying a black circle at each vertex that corresponds to a \(3\)-tuple where this function has the value \(1\) .
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