Boolean Algebra and Simplification Techniques
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In the above expression, variables B, C and D are present in all eight possible combinations, and
variable A is the common factor in all eight product terms. With the application of theorem 8(a),
this expression reduces to A. Similarly, with the application of theorem 8(b), A + B + CCCCA + B +
CCCCA + B + CCCCA + B + CC also reduces to A as the variables B and C are present in all four possible
combinations in sum terms and variable A is the common factor in all the terms.
6.3.9 Theorem 9
(a) X+Y YYY = XXY and b XY + Y = X + Y
(6.18)
X + Y YYY = XXY + Y YY = XXY
Theorem 9(b) is the dual of theorem 9(a) and hence stands proved.
6.3.10 Theorem 10 (Absorption Law or Redundancy Law)
(a) X + XXY = X and b XXXX + YY = X
(6.19)
The proof of absorption law is straightforward:
X + XXY = XXX1 + YY = XX1 = X
Theorem 10(b) is the dual of theorem 10(a) and hence stands proved.
The crux of this simplification theorem is that, if a smaller term appears in a larger term, then the
larger term is redundant. The following examples further illustrate the underlying concept:
A + AAB + AABBC + AABBC + CCBBA = A
and
A + B + CCCCA + BBBBC + B + AA = A + B
6.3.11 Theorem 11
(a) ZZX + ZZXXY = ZZX + ZZY
and
(b) Z + XXXXZ + X + YY = Z + XXXXZ + YY
(6.20)
Table 6.2 gives the proof of theorem 11(a) using the method of perfect induction. Theorem 11(b) is the
dual of theorem 11(a) and hence stands proved. A useful interpretation of this theorem is that, when