Solution the function has three variables a b and c

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Unformatted text preview: be made so by first expanding the expression into a sum of AND terms. Each term is then inspected to see if it contains all the variables. If it misses one or more variables, it is ANDed with an expression of the form (x + x), where x is one of the missing variables. The following example clarifies this procedure. Example 6.4. Express the Boolean function F = A+ B-C in the sum-of-minterms (products) form. Solution: The function has three variables A, B and C. The first term A is missing two variables, therefore A = A • (B + B) = AB + AB This is still missing one variable, therefore A = A • B • C+C) + A•B• (C+C) =A•B•C•+A•B•C+A•C+A•B•C The second term B • C is missing one variable, therefore B • C = B • C • (A+A) = A • B • C+ A • B • C Hence by combining all the terms we get F = A•B•C + A•B•C +A•B • C + A•B • C +A•B • C+A •B • C But in the above expression, the term ABC appears twice and according to theorem l(a) we have x + x = x. Hence it is possible to remove one of them. Rearranging the minterms in ascending order, we finally obtain: F=A•B•C+A•B•C+A•B•C+A•B•C = m1 m4 + m5 + m6 + m7 It is sometimes convenient to express the Boolean function, when in its sum-of-minterms, in the following short notation: F(A,B,C) = Z(1,4,5,6,7) The summation symbol 'Z' stands for the ORing of terms. The numbers following it are the minterms of the function. Finally, the letters in parentheses with F form a list of the variables in the order taken when the minterm is converted to an AND term. Product-of-Sums A product-of-sums expression is a sum term (maxterm) or several sum terms (maxterms) logically multiplied (ANDed) together. For example, the expression (x +y)-(x+ y) is a product of sums expression. The following are all product-of-sums expressions: x (X+y) (x + y)-z (x+ y)-(x+y)-(x + y) (x + y)-(x+y + z) The following steps are followed to express a Boolean function in its product-of-sums form: 1. Construct a truth table for the given Boolean function. 2. Form a maxterm for each combination of the varibles which produce a 0 in the function. 3. The desired expression is the product (AND) of all the maxterms obtained in s...
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This document was uploaded on 04/07/2014.

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