Problem: Determine a minimum sum-of-products expression

Chapter 5, Problem 5.2

(choose chapter or problem)

Determine a minimum sum-of-products expression for

\(f(a, b, c, d, e)=\left(a^{\prime}+c+d\right)\left(a^{\prime}+b+e\right)\left(a+c^{\prime}+e^{\prime}\right)\left(c+d+e^{\prime}\right) \left(b+c+d^{\prime}+e\right)\left(a^{\prime}+b^{\prime}+c+e^{\prime}\right)\)

The first step in the solution is to plot a map for f. Because f is given in product-of-sums form, it is easier to first plot the map for \(f^\prime\) and then complement the map. Write \(f^\prime\) as a sum of products:

\(f^\prime\) = _________________________________________

Now plot the map for \(f^\prime\). (Note that there are three terms in the upper layer, one term in the lower layer, and two terms which span the two layers.)

Next, convert your map for \(f^\prime\) to a map for f.

The next step is to determine the essential prime implicants of f.

(a) Why is \(a^\prime d^\prime e^\prime\) an essential prime implicant?

(b) Which minterms are adjacent to \(m_3\)? ___________ To \(m_{19}\)? ___________

(c) Is there an essential prime implicant which covers \(m_{3}\) and \(m_{19}\)?

(d) Is there an essential prime implicant which covers \(m_{21}\)?

(e) Loop the essential prime implicants which you have found. Then, find two more essential prime implicants and loop them.

(a) Why is there no essential prime implicant which covers \(m_{11}\)?

(b) Why is there no essential prime implicant which covers \(m_{28}\)?

Because there are no more essential prime implicants, loop a minimum number of terms which cover the remaining 1’s.

Write down two different minimum sum-of-products expressions for f.

f = ___________________________________

f = ___________________________________