0195136047.pdf

6.1 DIGITAL BUILDING BLOCKS 285

Noting thatFhas an output of 0 that corresponds to maxtermsM 0 ,M 2 ,M 4 , andM 5 ,Fcan

therefore be expressed as

F(A,B,C)=M 0 ·M 2 ·M 4 ·M 5

=(A+B+C)·(A+B ̄+C)·(A ̄+B+C)·(A ̄+B+C) ̄

or in compact form as

F(A,B,C)=

∏

Mi( 0 , 2 , 4 , 5 )

where

∏

Mi( ) means the product of all the maxterms whose subscriptiis given inside the

parentheses.

Now coming back to the K map, a K map is a diagram made up of cells (squares), one for
each minterm of the function to be represented. Ann-variable K map, representing ann-variable
function, therefore has 2ncells. Figures 6.1.12, 6.1.13, and 6.1.14 show the two-variable, three-
variable, and four-variable K maps, respectively, in two different forms. Note particularly the
code used in listing the row and column headings when more than one variable is needed.
The K map provides an immediate view of the values of the function in graphical form.
Note that the arrangement of the cells in the K map is such that any two adjacent cells contain
minterms that vary only in one variable. 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 ann-variable
map, there will benminterms adjacent to any given minterm. For example, in Figure 6.1.14, the
cell corresponding tom 2 is adjacent to the cells corresponding tom 3 ,m 6 ,m 0 , andm 10. It can be
verified that any two adjacent cells contain minterms that vary only in one variable. Them 2 cell,

A ⋅ B

(a) (b)

AA

Binary

values

of A

Binary

values

of B

0

0

B

A

1

1

B

B A ⋅ B m 2

m 3

m 0

A ⋅ B A ⋅ B m 1

Figure 6.1.12Two-variable K maps.

A ⋅ B ⋅ C

(a)

AB

C

C

A ⋅ B ⋅ C

A ⋅ B ⋅ C

AB

A ⋅ B ⋅ C

A ⋅ B ⋅ C

AB

A ⋅ B ⋅ C

A ⋅ B ⋅ C

AB

A ⋅ B ⋅ C

(b)

1

0

00 01 11

AB

C 10

m 2

m 3

m 0

m 1

m 4

m 5

m 6

m 7

Figure 6.1.13Three-variable K maps.

A ⋅ B ⋅ C ⋅ D

(a)

AB
CD
CD
CD
CD

A ⋅ B ⋅ C ⋅ D

AB AB AB

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

A ⋅ B ⋅ C ⋅ D

(b)

01

00

10

11

00 01 11

AB

CD^10

m 4

m 5

m 0

m 1

m 8

m 9

m 12

m 13

m 7

m 6

m 3

m 2

m 11

m 10

m 15

m 14

Figure 6.1.14Four-variable K maps.