196
Digital Electronics
Y
Z
X
X
Y
Z
X.Y+X.Z
(a)
Y
Z
X
X+Y.Z
X
Y
Z
(X+Y).(X+Z)
(b)
X.(Y+Z)
Figure 6.2 Distributive laws.
Theorem 7(b) can be used to simplify A + BBBBA + BBBBA + BBBBA + BB as follows:
A + BBBBA + BBBBA + BBBBA + BB = A + BBBBBBA + BBBB = A + 0A + 0 = AAA = 0
6.3.8 Theorem 8
(a) XXY + XXY = X and (b) X + YYYYX + Y Y = X
This is a special case of theorem 7 as
XXY + XXY = XXXY + Y Y = XX1 = X and X + YYYYX + Y Y = X + YYY = X + 0 = X
This theorem, however, has another very interesting interpretation. Referring to theorem 8(a), there
are two two-variable terms in the LHS expression. One of the variables, Y , is present in all possible
combinations in this expression, while the other variable, X, is a common factor. The expression
then reduces to this common factor. This interpretation can be usefully employed to simplify many a
complex Boolean expression.
As an illustration, let us consider the following Boolean expression:
AABBCCD + AABBCCD + AABBCCD + AABBCCD + AABBCCD + AABBCCD + AABBCCD + AABBCCD