Page 77 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes

Use the K-map to simplify the expression y = ABC + BC + AB.
Solution:
In this problem we are not given the truth table from which to fill in the K-map instead, we
must fill in the K-map by taking each of the product terms in the expression and placing
1s in the corresponding squares.

The first term, ABC tells us to enter a 1 in the ABC square of the map (see 4.23). The
second term, BC, tells us to enter a 1 in each square that contains a BC, in its label. In
Figure 4.23 this would be the ABC and ABC squares. Likewise, the AB term tells us to
place a 1 in the ABC and ABC squares. All other squares will be filled with 0s.

Now the K-map can be looped for simplification. The result is y = (A + BC ,) as shown in
the Figure 4.23.
Figure 4.22: Example K-map

“Don’t Care” Conditions

Some logic circuits can be designed so that there are certain input conditions for which
there are no specified output levels, usually because these input conditions will never
occur. In other words, there will be certain combinations of input levels where we “don’t
care” whether the output is HIGH or LOW. This is illustrated in the truth table of Figure
4.24 (a).
Figure 4.23: “Don’t Care” Conditions should be Changed to 0 or 1 to Produce
K-map Looping that Yields the Simplest Expression

C C C C
AB C z
0 0 0 0 AB 0 0 AB 0 0
0 0 1 0
0 1 0 0 AB 0 x AB 0 0
0 1 1 x "don't
1 0 0 x care" AB 1 1 AB 1 1 z= A
1 0 1 1
1 1 0 1 AB x 1 AB 1 1
1 1 1 1
(a) (b) (c)
Here the output z is not specified as either 0 or 1 for the conditions A,B,C = 1, 0, 0 and
A, B, C = 0, 1, 1. Instead, an x is shown for these conditions. T represents the “don’t care”
condition. A “don’t care” condition can come about several reasons, the most common
being that in some situations certain input combinations can never occur, and so there is
no specified output for these conditions.

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