Boolean Algebra and Simplification Techniques
215
The product-of-sums expression that tells about the combination of prime implicants required to
account for all the terms is given by the expression
L + SSSSM + SSSSN + SSSSL + PPPPT TTTP + T TTTQ + T T
(6.38)
After obvious simplification, this reduces to the expression
TTTL+SSSSM + SSSSN + SSSSL + PP
= TTTLM + LS + MS + SSSSLN + PN + LS + PSS
= TTTLM + SSSSLN + PN + LS + PSS
= TTTLMN + LMPN + LMS + LMPS + LNS + PNS + LS + PSS
= TTTLMN + LMPN + LS + PSS
= TLMN + TLMPN + TLS + TPS
(6.39)
0001
0010
0011
1001
1100
1101
1110
Prime implicants
−001
L
−010
M
0−11
N
1−01
P
1−10
Q
−111
R
00−−
S
11−−
T
The sum-of-products Boolean expression (6.39) states that all the input combinations can be accounted
for by the prime implicants (T , L, M, N N or (T , L, M, PP N N or (T , L, SS or (T , P, SS. The most
optimum expression would result from either TLS or TPS. Therefore, the minimized Boolean function
is given by
ffAA BB CC DD = AAB + BBCCD + AAB
(6.40)
or by
ffAA BB CC DD = AAB + AAB + AACCD
(6.41)
Example 6.10
A logic system has three inputs A, B and C and two outputs Y 1 and Y 2 . The output functions Y 1 and Y 2
are expressed by Y 1 = AABBC + BBC + AAC + AABBC + AABBC and Y 2 = AAB + AAC + AABBC. Determine
the minimized output logic functions using the Quine–McCluskey tabular method.
Solution
The expanded forms of Y 1 and Y 2 are written as follows:
Y 1 = AABBC + AABBC + AABBC + AABBC + AABBC + AABBC + AABBC
= AABBC + AABBC + AABBC + AABBC + AABBC + AABBC
Y 2 = AABBC + AABBC + AABBC + AABBC + AABBC