
4.9 FACTORIZATION, RESUBSTITUTION, AND DECOMPOSITION METHODS 179
As a practical example of the application of Shannon’s expansion theorem, consider the
function
F(A, B, C, D) =
m(1, 3, 4, 5, 9, 10, 13, 14, 15), (4.71)
which is represented in the truth table of Fig. 4.48a and in the K-map of Fig. 4.48b. Applying
Eq. (4.69) for decomposition with respect to variables C and D gives the cofactors
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
F
0
=(A, B, 0, 0) =
¯
AB
F
1
=(A, B, 0, 1) = 1
F
2
=(A, B, 1, 0) = A
F
3
=(A, B, 1, 1) = A B
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
,
from which the function F can be written as
F
CD
(A, B, C, D) =
¯
C
¯
D(
¯
AB) +
¯
CD(1) +C
¯
D(A) +CD(A B)
=
¯
AB
¯
C
¯
D +
¯
CD + AC
¯
D + (A B)CD,
which could have been deduced ...