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Section5.4Exercise

1

Show that Equation (5.3.1) and Equation (5.3.2) represent the same functionality. This shows that the sum of minterms and product of maxterms are complementary.

Solution

The \(\sum\) column shows where the sum of minterms evaluates to \(1\text{,}\) and the \(\prod\) column shows where the product of maxterms evaluates to \(0\text{.}\) Of course, in the blank cells the functions evaluate to the complementary value.

\(x\) \(y\) \(z\) \(F(x,y,z) = \sum(0,1,5,6)\) \(F(x,y,z) = \prod(2,3,4,7)\)
\(\binary{0}\) \(\binary{0}\) \(\binary{0}\) \(\binary{1}\)
\(\binary{0}\) \(\binary{0}\) \(\binary{1}\) \(\binary{1}\)
\(\binary{0}\) \(\binary{1}\) \(\binary{0}\) \(\binary{0}\)
\(\binary{0}\) \(\binary{1}\) \(\binary{1}\) \(\binary{0}\)
\(\binary{1}\) \(\binary{0}\) \(\binary{0}\) \(\binary{0}\)
\(\binary{1}\) \(\binary{0}\) \(\binary{1}\) \(\binary{1}\)
\(\binary{1}\) \(\binary{1}\) \(\binary{0}\) \(\binary{1}\)
\(\binary{1}\) \(\binary{1}\) \(\binary{1}\) \(\binary{0}\)