Digital Electronics: Principles, Devices and Applications

206 Digital ElectronicsThe dual of AB + AB = A + BA + B. ThereforeAB + AB = A + BA + B6.4.3 Expanded Forms of Boolean ExpressionsExpanded sum-of-products and product-of-sums forms of Boolean expressions are useful not onlyin analysing these expressions but also in the application of minimization techniques such as theQuine–McCluskey tabular method and the Karnaugh mapping method for simplifying given Booleanexpressions. The expanded form, sum-of-products or product-of-sums, is obtained by including allpossible combinations of missing variables.As an illustration, consider the following sum-of-products expression:AB + BC + ABC + ACIt is a three-variable expression. Expanded versions of different minterms can be written as follows:• AB = ABC + C = ABC + ABC• BC = BCA + A = BCA+ BCA• ABC is a complete term and has no missing variable.• AC = ACB + B = ACB + ACB.The expanded sum-of-products expression is therefore given byABC + ABC + ABC + ABC + ABC + ABC + ABC = ABC + ABC+ ABC + ABC + ABC + ABCAs another illustration, consider the product-of-sums expressionA + BA + B + C + DIt is four-variable expression with A, B, C and D being the four variables. A + B in this case expandsto A + B + C + DA + B + C + DA + B + C + DA + B + C + D.The expanded product-of-sums expression is therefore given byA + B + C + DA + B + C + DA + B + C + DA + B + C + DA + B + C + D= A + B + C + DA + B + C + DA + B + C + DA + B + C + D6.4.4 Canonical Form of Boolean ExpressionsAn expanded form of Boolean expression, where each term contains all Boolean variables in their trueor complemented form, is also known as the canonical form of the expression.As an illustration, fAB C = ABC + ABC + ABC is a Boolean function of three variablesexpressed in canonical form. This function after simplification reduces to AB + ABC and loses itscanonical form.