 # As a pos expression it is a single sum term

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As aPOS expression, it is a single sum term comprising three literals.As an SOP expression, it isthe logical OR of three product terms, each of which comprising a single literal.For casessuch as these the only rule is to have a consistent interpretation.Page 142CPSC 5155Last Revised On July 2, 2011Copyright © 2011 by Ed Bosworth
Chapter 4Boz–7Boolean Algebra and Combinational LogicWe now consider the concept of normal and canonical forms. These forms apply to both Sumof Products and Produce of Sums expressions, so we may have quite a variety of expressiontypes 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 isincludedin another product term Y if every literal that is in X is also inY.A sum term X isincludedin 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 theSOP formula AB + AC + ABC.Note that the first term is included in the third term as isthe second term.The third term ABC contains the first term AB, etc.Consider the POS formula (A + B)(A + C)(A + B + C).Again, the first term (A + B) isincluded 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 AB + AC + AB and the POS expression (A+ B)(A + C)(A + C).Each of these has duplicate terms, so that the inclusion is 2-way.Thebasic idea is that an expression with included (or duplicate) terms is not written in thesimplest possible form.The idea of simplifying such expressions arises from the theorems ofBoolean algebra, specifically the following two.T1Idempotencya) X + X = X, for all Xb) XX = X, for all XT3Absorptiona) X + (XY) = X, for all X and Yb) X(X + Y) = X, for all X and YAs a direct consequence of these theorems, we can perform the following simplifications.AB + AC + AB= AB + AC(A + B)(A + C)(A + C)= (A + B)(A + C)AB + AC + ABC= AB + AC(A + B)(A + C)(A + B + C)= (A + B)(A + C)Page 143CPSC 5155Last Revised On July 2, 2011Copyright © 2011 by Ed Bosworth
Chapter 4Boz–7Boolean Algebra and Combinational Logic

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Boolean Algebra, Logic gate, Boolean function
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