As a pos expression it is a single sum term

As a
POS expression, it is a single sum term comprising three literals. As an SOP expression, it is
the logical OR of three product terms, each of which comprising a single literal. For cases
such as these the only rule is to have a consistent interpretation.
Page 142 CPSC 5155 Last Revised On July 2, 2011
Copyright © 2011 by Ed Bosworth

Chapter 4 Boz–7 Boolean Algebra and Combinational Logic
We now consider the concept of normal and canonical forms. These forms apply to both Sum
of Products and Produce of Sums expressions, so we may have quite a variety of expression
types including the following.
1. Not in any form at all.
2. Sum of Products, but not normal.
3. Product of Sums, but not normal.
4. Normal Sum of Products.
5. Normal Product of Sums.
6. Canonical Sum of Products.
7. Canonical Product of Sums.
In order to define the concept of a normal form, we must consider the idea of inclusion.
A product term X is included in another product term Y if every literal that is in X is also in
Y. A sum term X is included in another sum term Y if every literal that is in X is also in Y.
For inclusion, both terms must be product terms or both must be sum terms. Consider the
SOP formula A  B + A  C + A  B  C. Note that the first term is included in the third term as is
the second term. The third term A  B  C contains the first term A  B, etc.
Consider the POS formula (A + B)  (A + C)  (A + B + C). Again, the first term (A + B) is
included in the third term (A + B + C), as is the second term.
An extreme form of inclusion is observed when the expression has identical terms.
Examples of this would be the SOP expression A  B + A  C + A  B and the POS expression (A
+ B)  (A + C)  (A + C). Each of these has duplicate terms, so that the inclusion is 2-way. The
basic idea is that an expression with included (or duplicate) terms is not written in the
simplest possible form. The idea of simplifying such expressions arises from the theorems of
Boolean algebra, specifically the following two.
T1 Idempotency a) X + X = X, for all X
b) X  X = X, for all X
T3 Absorption a) X + (X  Y) = X, for all X and Y
b) X  (X + Y) = X, for all X and Y
As a direct consequence of these theorems, we can perform the following simplifications.
A  B + A  C + A  B = A  B + A  C
(A + B)  (A + C)  (A + C) = (A + B)  (A + C)
A  B + A  C + A  B  C = A  B + A  C
(A + B)  (A + C)  (A + B + C) = (A + B)  (A + C)
Page 143 CPSC 5155 Last Revised On July 2, 2011
Copyright © 2011 by Ed Bosworth

Chapter 4 Boz–7 Boolean Algebra and Combinational Logic