0195136047.pdf

284 DIGITAL BUILDING BLOCKS AND COMPUTER SYSTEMS

function, its SOP form can be obtained by taking the sum of the minterms that correspond to a 1

in the output column of the table.

Any Boolean function can also be expressed, algebraically, as a product (AND) of maxterms.

An expression of this form is known as acanonical product of sums. Given a truth table for a logic

function, its POS form can be obtained by taking the product of the maxterms that correspond to

a 0 (zero) in the output column of the table. Let us now consider an example to illustrate the use

of minterms and maxterms.

TABLE 6.1.4Minterms and Maxterms for Three Variables

A B C i Minterm mi Maxterm Mi

0000 A ̄·B ̄·CA ̄ +B+C

0011 A ̄·B ̄·CA+B+C ̄

0102 A ̄·B·CA ̄ +B ̄+C

0113 A ̄·B·CA+B ̄+C ̄

1004 A·B ̄·C ̄ A ̄+B+C

1015 A·B ̄·C A ̄+B+C ̄

1106 A·B·C ̄ A ̄+B ̄+C

1117 A·B·C A ̄+B ̄+C ̄

EXAMPLE 6.1.8

Given the truth table in Table E6.1.8 for an arbitrary Boolean functionF, expressFas a sum of

minterms and a product of maxterms.

TABLE E6.1.8

mi ABCF

m 0 0000

m 1 0011

m 2 0100

m 3 0111

m 4 1000

m 5 1010

m 6 1101

m 7 1111

Solution

Noting thatFhas an output of 1 that corresponds to mintermsm 1 ,m 3 ,m 6 , andm 7 ,Fcan be

expressed as

F(A,B,C)=m 1 +m 3 +m 6 +m 7

=A ̄·B ̄·C+A ̄·B·C+A·B·C ̄+A·B·C

or in a compact form as

F(A,B,C)=

∑

mi( 1 , 3 , 6 , 7 )

where

∑

mi( ) means the sum of all the minterms whose subscriptiis given inside the parentheses.