GATE CS 2024 Set 2 Q.50: Combinational Circuits

Consider 4-variable functions $f1, f2, f3, f4$ expressed in sum-of-minterms form as given below.

f1 = \sum(0,2,3,5,7,8,11,13)

f2 = \sum(1,3,5,7,11,13,15)

f3 = \sum(0,1,4,11)

f4 = \sum(0,2,6,13)

With respect to the circuit given above, which of the following options is/are CORRECT?

$Y = \sum(0,1,2,11,13)$

$Y = \prod(3,4,5,6,7,8,9,10,12,14,15)$

$Y = \sum(0,1,2,3,4,5,6,7)$

$Y = \prod(8,9,10,11,12,13,14,15)$

Official answer and solution

Solution

Compute the minterm sets of $f_1 \cdot f_2$ (intersection), $f_3 + f_4$ (union), and $Y = (f_1 \cdot f_2) \oplus (f_3 + f_4)$ (symmetric difference).

Output of the AND gate ( $f_1 \cdot f_2$ ): Taking the intersection of the minterm lists of $f_1$ and $f_2$ : $f_1 \cdot f_2 = \sum(3, 5, 7, 11, 13)$
Output of the OR gate ( $f_3 + f_4$ ): Taking the union of the minterm lists of $f_3$ and $f_4$ : $f_3 + f_4 = \sum(0, 1, 2, 4, 6, 11, 13)$
Output of the XOR gate ( $Y = (f_1 \cdot f_2) \oplus (f_3 + f_4)$ ): Taking the symmetric difference (removing common minterms $11$ and $13$ ): $Y = \sum(0, 1, 2, 3, 4, 5, 6, 7)$
Maxterm (Product-of-Maxterms) form of $Y$ : The complement within $\{0, 1, \dots, 15\}$ is: $Y = \prod(8, 9, 10, 11, 12, 13, 14, 15)$

Therefore, (C) and (D) are both correct.

Final answer

Options (C), (D)

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

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