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6.1 DIGITAL BUILDING BLOCKS 289

as adon’t-care outputbecause of a physical constraint. As another example, Figure 6.1.16 shows
the logic circuitry in a black box with four input lines representing BCD data. For such a circuit,
six of the input combinations (1010, 1011, 1100, 1101, 1110, and1111) are notvalid BCDs, and
therefore the output of this circuitry cannot be specified because of a logic constraint. Such outputs
are hence labeled as don’t-care outputs. A don’t-care output or condition is generally represented
by the letterdin the truth table and the K map.
Theds can be treated either as 0s or 1s when the subcubes are formed in the K map, depending
on which results in a greater simplification, thereby helping in the formation of the smallest number
of maximal subcubes. Note that some or all of theds can be treated as 0s or 1s. However, one
should not form a subcube that contains onlyds.

A Black box

CB (logic circuitry) F(A, B, C)

Figure 6.1.15Physical constraint for a don’t-care condi-

tion.

A

Black box

(logic circuitry)

BCD data B F(A, B, C, D)

C

D

Figure 6.1.16Logic constraint for a don’t-

care condition.

EXAMPLE 6.1.12

A Boolean functionF(A, B, C, D) is specified by the truth table of Figure E6.1.12 (a). Obtain:
(a) a minimum SOP expression, and (b) a minimum POS expression.

Solution

The four-variable K map ofFis shown in Figure E6.1.12(b).

(a) The prime implicants (with 1 cells) forFare marked in Figure E6.1.12(c). The minimum

SOP expression is then given by

F=C ̄·D ̄+B·C ̄

AB

00

00

00

00

F

1

0

0

0

01

01

01

0

D 0 1 0 1 0 1 0

1 1

C 0 0 1 1 0 0 1 1

1

1

0

0

10

10

10

10

1

0

d

d

11

11

11

1

0 1 0 1 0 1 0

1 1

0 0 1 1 0 0 1 1

d

d

d

d

(a) (b)

01

11

10

00

00 01 11

AB

CD^10

1 11 d

0 10 d

d

d

d

d

0

0

0

0

Figure E6.1.12