Consider the following minterm expression for \(F(P,Q,R,S) = \sum…
2014
Consider the following minterm expression for
\(F(P,Q,R,S) = \sum 0,2,5,7,8,10,13,15\)
The minterms 2, 7, 8 and 13 are ‘do not care’ terms. The minimal sum-of-products form for \(F\) is
- A.
\(Q \bar S+ \bar QS\) - B.
\(\bar Q \bar S+ QS\) - C.
\(\bar Q \bar R \bar S+ \bar QR \bar S+Q \bar R S+QRS\) - D.
\(\bar P \bar Q \bar S+ \bar P QS+PQS+P \bar Q \bar S\)
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Correct answer: B
Key insight: use the don't-care minterms to make the largest groups on a Karnaugh map.
Minterms required: 0, 5, 10, 15. Don't-cares: 2, 7, 8, 13.
Form a 4-cell group covering indices 0,2*,8*,10 (rows where Q=0 and S=0). This yields the implicant \bar Q \bar S
Form a 4-cell group covering indices 5,7*,13*,15 (rows where Q=1 and S=1). This yields the implicant Q S
Combine the implicants to get the minimal SOP:
Q S + \bar Q \bar S
Notes: The expression Q \bar S + \bar Q S is the exclusive-OR of Q and S and does not match F (it is 0 at minterm 0 while F includes minterm 0). The four-term expressions that include R or P are algebraically equivalent to Q S + \bar Q \bar S (they contain redundant R or P factors) and should be simplified to the two-product minimal form above.
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