GATE CS 2015 Set 3 Q.54 (CS Q.44): Boolean Algebra and Logic Minimization

Given the function $F = P' + QR$ , where $F$ is a function in three Boolean variables $P, Q$ and $R$ and $P' = \neg P$ , consider the following statements.
(S1) $F = \sum(4, 5, 6)$
(S2) $F = \sum(0, 1, 2, 3, 7)$
(S3) $F = \prod(4, 5, 6)$
(S4) $F = \prod(0, 1, 2, 3, 7)$

Which of the following is true?

(S1)- False, (S2)- True, (S3)- True, (S4)- False

(S1)- True, (S2)- False, (S3)- False, (S4)- True

(S1)- False, (S2)- False, (S3)- True, (S4)- True

(S1)- True, (S2)- True, (S3)- False, (S4)- False

Official answer and solution

Official answer: A
From: Official GATE 2015 answer key

Solution

Find the canonical sum-of-minterms and product-of-maxterms representations of F = P' + QR.

Given $F(P, Q, R) = P' + QR$ .
Expand $F$ into canonical sum of minterms (SOP):

$P' = P' (Q + Q') (R + R') = P'Q'R' + P'Q'R + P'QR' + P'QR = m_0 + m_1 + m_2 + m_3$
$QR = (P + P') QR = PQR + P'QR = m_7 + m_3$

Combining distinct minterms:

F = \sum m(0, 1, 2, 3, 7)

The complementary maxterms (POS) are the remaining indices from $\{0, 1, \dots, 7\}$ :

F = \prod M(4, 5, 6)

Evaluate the statements:

(S1) $F = \sum(4, 5, 6)$ is False.
(S2) $F = \sum(0, 1, 2, 3, 7)$ is True.
(S3) $F = \prod(4, 5, 6)$ is True.
(S4) $F = \prod(0, 1, 2, 3, 7)$ is False.

This matches: (S1) False, (S2) True, (S3) True, (S4) False (Option A).

Final answer

Common trap

Confusing minterm sum notation (Sigma) with maxterm product notation (Pi).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 3
Question: 54 of 65, printed as CS Q.44
Section: Computer Science and Information Technology

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