GATE CS 2024 Set 1 | Question 47

MSQ · 2 marks

Consider a Boolean expression given by \(F(X, Y, Z) = \sum(3,5,6,7)\).

Which of the following statements is/are CORRECT?

  1. Option A: \(F(X, Y, Z) = \Pi(0,1,2,4)\) (correct)
  2. Option B: \(F(X, Y, Z) = XY + YZ + XZ\) (correct)
  3. Option C: \(F(X, Y, Z)\) is independent of input \(Y\)
  4. Option D: \(F(X, Y, Z)\) is independent of input \(X\)

Answer

(A), (B)

Explanation

  • A is true. The maxterms of a function are exactly the minterms it does not contain; out of 0 to 7 these are 0, 1, 2 and 4.
  • B is true. \(XY + YZ + XZ\) is 1 exactly when at least two inputs are 1, that is for \(XYZ\) = 011, 101, 110 and 111, which are minterms 3, 5, 6 and 7. \(F\) is the 3-input majority function.
  • C and D are false. \(F(X, 0, Z) = XZ\) while \(F(X, 1, Z) = X + Z\), so \(F\) depends on \(Y\). Likewise \(F(0, Y, Z) = YZ\) and \(F(1, Y, Z) = Y + Z\) differ, so \(F\) depends on \(X\).

Answer: A, B