For the Boolean function F(A,B,C,D)= m(0,2,5,7,8,10,12,13,14,15), the essential prime implicants are .

GATE 2024 EC – Question 19

Digital Circuits · Combinatorial Circuits · 1 mark · Multiple choice

For the Boolean function

$$F(A,B,C,D)=\sum m(0,2,5,7,8,10,12,13,14,15),$$

the essential prime implicants are ____________.

$BD,\ \bar B\bar D$
$BD,\ AB$
$AB,\ \bar B\bar D$
$BD,\ \bar B\bar D,\ AB$

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Show answer and explanation

Correct answer: (A) $BD,\ \bar B\bar D$

Explanation

Prime implicants from the K-map: $\bar B\bar D$ covers 0, 2, 8, 10; $BD$ covers 5, 7, 13, 15; $AB$ covers 12, 13, 14, 15; $A\bar D$ covers 8, 10, 12, 14. Minterm 0 is covered only by $\bar B\bar D$, and minterms 5 and 7 only by $BD$, so those two are essential. Minterms 12 and 14 can be covered by either $AB$ or $A\bar D$, so neither of those is essential.