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N=3 Z XYZ XYZ XYZ XYZ 1

WX YZ

YZ 00

WX 00

WXYZ WXYZ WXYZ WXYZ

YZ WXYZ WXYZ WXYZ WXYZ 01 N=4 YZ WXYZ WXYZ WXYZ WXYZ 11

XY 1 1

XY 1 1

XY 0 0

Karnaugh map

YZ WXYZ WXYZ WXYZ WXYZ 10

Y 0 0 1 1 0 0 1 1

Z 0 1 0 1 0 1 0 1

Desired Function 0 0 1 1 0 0 1 1

Figure 1329 Two-, three-, and four-variable Karnaugh maps

0 0 0 0 1 1 1 1

X X X X

=0 =0 =1 =1

Y Y Y Y

=1 =1 =1 =1

Z Z Z Z

=0 =1 =0 =1

The same truth table is shown in Figure 1330 together with the corresponding Karnaugh map The Karnaugh map provides an immediate view of the values of the function in graphical form Further, the arrangement of the cells in the Karnaugh map is such that any two adjacent cells contain minterms that vary in only one variable This property, as will be veri ed shortly, is quite useful in the design of logic functions by means of logic gates, especially if we consider the map to be continuously wrapping around itself, as if the top and bottom, and right and left, edges were touching each other For the three-variable map given in Figure 1329, for example, the cell X Y Z is adjacent to X Y Z if we roll the map so that the right edge touches the left Note that these two cells differ only in the variable X, a property we earlier claimed adjacent cells have1

Truth table

Figure 1330 Truth table and Karnaugh map representations of a logic function

useful rule to remember is that in a two-variable map there are two minterms adjacent to any given minterm; in a three-variable map, three minterms are adjacent to any given minterm; in a four-variable map, the number is four, and so on

13

Digital Logic Circuits

WX YZ YZ YZ YZ 1 1 0 0

WX 0 1 0 1

WX 0 0 1 0

WX 0 1 0 1

1 cell subcubes WX YZ YZ YZ YZ 1 1 0 0 WX 0 1 0 1 WX 0 0 1 0 WX 0 1 0 1

Shown in Figure 1331 is a more complex, four-variable logic function, which will serve as an example in explaining how Karnaugh maps can be used directly to implement a logic function First, we de ne a subcube as a set of 2m adjacent cells with logical value 1, for m = 1, 2, 3, , N Thus, a subcube can consist of 1, 2, 4, 8, 16, 32, cells All possible subcubes for the four-variable map of Figure 1331 are shown in Figure 1332 Note that there are no four-cell subcubes in this particular case Note also that there is some overlap between subcubes Examples of four-cell and eight-cell subcubes are shown in Figure 1333 for an arbitrary expression

X 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

Y 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

Y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

Z 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Desired Function 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 1

WX YZ YZ YZ YZ 1 1 0 0

WX 0 1 0 1

WX 0 0 1 0

WX 0 1 0 1

2 cell subcubes

Figure 1332 One- and two-cell subcubes for the Karnaugh map of Figure 1331

Truth table for four-variable expression WX YZ YZ YZ YZ 1 1 1 1 WX 0 0 0 0 (a) WX YZ YZ YZ YZ 1 0 0 1 WX 1 0 0 1 (b) WX 1 0 0 1 WX 1 0 0 1 WX 0 1 1 0 WX 0 1 1 0

Figure 1331 Karnaugh map for a four-variable expression

In general, one tries to nd the largest possible subcubes to cover all of the 1 entries in the map How do maps and subcubes help in the realization of logic functions, then The use of maps and subcubes in minimizing logic expressions is best explained by considering the following rule of Boolean algebra: Y X+Y X =Y where the variable Y could represent a product of logic variables (for example, we could similarly write (Z W ) X + (Z W ) X = Z W with Y = Z W ) This rule is easily proven by factoring Y : Y (X + X) and observing that X + X = 1, always Then it should be clear that the variable X need not appear in the expression at all Let us apply this rule to a more complex logic expression, to verify that it can also apply to this case Consider the logic expression W X Y Z+W X Y Z+W X Y Z+W X Y Z and factor it as follows: W Z Y (X + X) + W Y Z (X + X) = W Z Y + W Y Z = Y Z (W + W ) = Y Z