GATE CS 2024 Set 2 | Question 50

MSQ · 2 marks

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

\(f1 = \sum(0,2,3,5,7,8,11,13)\)
\(f2 = \sum(1,3,5,7,11,13,15)\)
\(f3 = \sum(0,1,4,11)\)
\(f4 = \sum(0,2,6,13)\)

Logic circuit: inputs f1 and f2 enter an AND gate, inputs f3 and f4 enter an OR gate, and the outputs of the AND gate and the OR gate are the two inputs of an XOR gate whose output is Y.

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

  1. Option A: \(\boldsymbol{Y} = \sum(0,1,2,11,13)\)
  2. Option B: \(\boldsymbol{Y} = \Pi(3,4,5,6,7,8,9,10,12,14,15)\)
  3. Option C: \(\boldsymbol{Y} = \sum(0,1,2,3,4,5,6,7)\) (correct)
  4. Option D: \(\boldsymbol{Y} = \Pi(8,9,10,11,12,13,14,15)\) (correct)

Answer

(C), (D)

Explanation

Work with minterm sets: AND keeps the minterms common to both inputs, OR takes their union, and XOR keeps the minterms that appear in exactly one input.

  • \(f1 \cdot f2\): common minterms \(\{3, 5, 7, 11, 13\}\).
  • \(f3 + f4\): \(\{0, 1, 2, 4, 6, 11, 13\}\).
  • XOR: 11 and 13 are in both sets and cancel, leaving \(\{0, 1, 2, 3, 4, 5, 6, 7\}\).

So \(Y = \sum(0,1,2,3,4,5,6,7)\), which is (C). Its maxterms are the remaining minterms 8 to 15, so (D) is the same function. Options (A) and (B) both describe \(\sum(0,1,2,11,13)\), which is not \(Y\).

Answer: C, D