6.1 DIGITAL BUILDING BLOCKS 283AD CBA
B
(a)A
BADCBA
B
(b)
Figure E6.1.7The basic logic gates described earlier are used to implement more advanced functions and
are often combined to form logic blocks (or modules) which are available in compact IC packages.
Although SSI packages were common basic units at one time, the trend now is to integration on an
even larger scale. Entire digital systems, such as those to be discussed later, are now available in IC
form, and those ICs in turn become blocks for building even larger systems. The ICs are mounted
in packages known as DIPs (dual in-line packages), each with 8, 14, or more wires (pins) meant
to be plugged into corresponding pin sockets. The actual gates, with the same logic diagrams,
can be made with several different kinds of internal construction, giving rise to differentlogic
families.In general blocks of one family are compatible with other blocks of the same family, but
not with those of other families.
Karnaugh Maps and Logic Design
Even though a truth table uniquely represents a logic function, it is clear that the same function may
appear in different algebraic forms. While the Boolean identities can be used for the simplification
of a given algebraic form, it is desirable to have a systematic process that guarantees the minimum
form with a minimum number of components. The map method, known as theKarnaugh mapor,
simply, theK map, which is a modified form of the truth table, provides a convenient procedure for
obtaining a minimum SOP or POS of a Boolean expression. The K maps are usually restricted to
up to five variables, since they become too cumbersome to manipulate for a larger number of vari-
ables. Before we get into the details of K maps, it is necessary to introducemintermandmaxterm.
When a product term contains each of thenvariables of a function, it is called aminterm.
Fornvariables, there are 2npossible minterms. Theith minterm ofnvariables is denoted bymi,
where the subscripti,0≤i≤ 2 n−1, represents the decimal equivalent of the binary number
obtained when a variable in minterm is replaced by 1 and its complement is replaced by 0.
When a sum term contains each of thenvariables of a function, it is called amaxterm. The
ith maxterm ofnvariables, denoted byMi, is the complement of theith mintermmiof the same
nvariables; that is to say,Mi=mi. Table 6.1.4 lists the eight possible maxterms and minterms
of the three variablesA, B,andC.
Any Boolean function can be expressed, algebraically, as a sum (OR) of minterms. An
expression of this form is known as acanonical sum of products. Given a truth table for a logic