Consider the minterm list form of a Boolean function πΉ given below. πΉ(π,π,β¦
2018
Consider the minterm list form of a Boolean function πΉ given below.
πΉ(π,π, π , π) = βπ(0, 2, 5, 7, 9, 11) + π(3, 8, 10, 12, 14)
Here, π denotes a minterm and π denotes a donβt care term. The number of essential prime implicants of the function πΉ is ______.
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Correct answer: 3
Answer: 3 essential prime implicants.
Work through:
Combine minterms and donβt-cares to form the one-set: 0, 2, 3, 5, 7, 8, 9, 10, 11, 12, 14
Identify prime implicants (groups of adjacent ones/donβt-cares):
P AND NOT Q (P Q'): covers indices 8, 9, 10, 11 β therefore covers minterms 9 and 11.
NOT Q AND NOT S (Q' S'): covers indices 0, 2, 8, 10 β therefore covers minterms 0 and 2.
NOT P AND Q AND S (P' Q S): covers indices 5 and 7 β therefore covers minterms 5 and 7.
Check essentiality: each minterm is covered by exactly one of the listed prime implicants:
Minterms 0 and 2 are covered only by NOT Q AND NOT S (Q' S').
Minterms 9 and 11 are covered only by P AND NOT Q (P Q').
Minterms 5 and 7 are covered only by NOT P AND Q AND S (P' Q S).
Since each of the three prime implicants uniquely covers (at least) one minterm, all three are essential. Therefore the number of essential prime implicants is 3.
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