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Q19E

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Found in: Page 842

### Discrete Mathematics and its Applications

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

# Which rows and which columns of a $$4{\bf{ \ast }}16$$ map for Boolean functions in six variables using the Gray codes $${\bf{1111}},{\bf{1110}},{\bf{1010}},{\bf{1011}},{\bf{1001}},{\bf{1000}},{\bf{0000}},{\bf{0001}},{\bf{0011}},{\bf{0010}},{\bf{0110}},{\bf{0111}},{\bf{0101}},{\bf{0100}},{\bf{1100}},{\bf{1101}}$$ to label the columns and $${\bf{11}},{\bf{10}},{\bf{00}},{\bf{01}}$$ to label the rows need to be considered adjacent so that cells that represent min-terms that differ in exactly one literal are considered adjacent$$?$$

Minterms that differ in exactly one literal are adjacent in a $$K{\bf{ - }}$$map or becomes adjacent when:

Top and bottom row are considered adjacent

First and Twelfth column are considered adjacent

Second and Eleventh column are considered adjacent

Third and Tenth column are considered adjacent

Fourth and Nineth column are considered adjacent

Fifth and Eighth column are considered adjacent

Thirteenth and Sixteenth column are considered adjacent

First and Fourth column are considered adjacent

Fifth and Sixteenth column are considered adjacent

Sixth and Fifteenth column are considered adjacent

Seventh and Fourteenth column are considered adjacent

Eighth and Thirteenth column are considered adjacent

Nineth and Twelfth column are considered adjacent

First and Sixteenth column are considered adjacent

Second and Fifteenth column are considered adjacent

Third and Sixth column are considered adjacent

Seventh and Tenth column are considered adjacent

Eleventh and Fourteenth column are considered adjacent

See the 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. You 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.

## Step 2: Mapping

The described $$4{\bf{ \times }}16$$ map in the exercise prompt is given below (the second table should be to the right and adjacent of the first table):

## Step 3:Minterms

If the minterms differ at the first digit in the rows (thus $${\bf{11}}$$ is replaced by $${\bf{01}}$$ or $${\bf{10}}$$ is replaced by $${\bf{01}}$$ to obtain other minterm), then we note that the cells are adjacent in the second and third row of the table or that the cells become adjacent when the top and bottom row are considered adjacent. If the minterms differ at the second digit in the rows, then we note that the cells are adjacent in the first and second row of the table or are adjacent in the third and fourth row of the table.

## Step 7:Minterms differ at the fourth digit

If the minterms differ at the fourth digit in the columns, then it is note that the cells are adjacent in the first and second column or the cells are adjacent in the third and fourth column or the cells are adjacent in the fifth and sixth column or the cells are adjacent in the seventh and eighth column or the cells are adjacent in the ninth and tenth column or the cells are adjacent in the eleventh and twelfth column or the cells are adjacent in the thirteenth and fourteenth column or the cells are adjacent in the fifteenth and sixteenth column.

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