GATE CS 2015 Set 3 Q44 — Boolean Algebra

To determine which statements are true, we need to find both the canonical Sum of Products (SOP) and Product of Sums (POS) representations of the Boolean function $F(P, Q, R) = P' + QR$.

1. Finding the Minterms (SOP Form)

We expand each term of $F$ to include all three variables $P, Q,$ and $R$:

• Term $P'$: $$\begin{aligned} P' &= P'(Q + Q')(R + R') \\ &= P'(QR + QR' + Q'R + Q'R') \\ &= P'QR + P'QR' + P'Q'R + P'Q'R' \end{aligned}$$ These correspond to minterms $m_3\ (011_2)$, $m_2\ (010_2)$, $m_1\ (001_2)$, and $m_0\ (000_2)$.
• Term $QR$: $$\begin{aligned} QR &= (P + P')QR \\ &= PQR + P'QR \end{aligned}$$ These correspond to minterms $m_7\ (111_2)$ and $m_3\ (011_2)$.

Combining the unique minterms, we get:

This tells us:

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

2. Finding the Maxterms (POS Form)

The maxterms of a function are the indices that are not present in its minterm list. Since the minterms are $\{0, 1, 2, 3, 7\}$, the remaining indices from the set $\{0, 1, 2, 3, 4, 5, 6, 7\}$ are $\{4, 5, 6\}$.

Thus, the POS representation is:

This tells us:

• (S3) $F = \Pi(4, 5, 6)$ is True.
• (S4) $F = \Pi(0, 1, 2, 3, 7)$ is False.

Conclusion

The statements evaluate to:

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

This matches Option A.