GATE CS 2024 Set 1 Q37 — Minimization (K-Maps etc.)

We are given the Boolean function of three variables: $F(X, Y, Z) = \sum(3, 5, 6, 7)$.

Checking Option A:
The product-of-maxterms ($\Pi$) representation consists of the indices not present in the sum-of-minterms ($\sum$) representation. For three variables, the complete set of indices is $\{0, 1, 2, 3, 4, 5, 6, 7\}$.
Thus, $F(X, Y, Z) = \Pi(0, 1, 2, 4)$. Hence, Option A is correct.

Checking Option B:
We simplify the minterms $\{3, 5, 6, 7\}$ using a 3-variable Karnaugh Map (K-map):

• Grouping minterms $3$ ($011$) and $7$ ($111$) yields $YZ$.
• Grouping minterms $5$ ($101$) and $7$ ($111$) yields $XZ$.
• Grouping minterms $6$ ($110$) and $7$ ($111$) yields $XY$.

Checking Options C and D:
Since the simplified expression $F(X, Y, Z) = XY + YZ + XZ$ depends on all three variables $X$, $Y$, and $Z$, the function is not independent of any input. Thus, Options C and D are incorrect.