GATE CS 2024 Set 1 Q.47: Boolean Algebra and Logic Minimization

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

Which of the following statements is/are CORRECT?

$F(X, Y, Z) = \prod(0, 1, 2, 4)$

$F(X, Y, Z) = XY + YZ + XZ$

$F(X, Y, Z)$ is independent of input $Y$

$F(X, Y, Z)$ is independent of input $X$

Official answer and solution

Solution

Convert the sum-of-minterms expression to product-of-maxterms and minimize using Boolean algebra or a 3-variable K-map.

(A) is CORRECT: For 3 variables $(X, Y, Z)$ , the set of all indices is $\{0, 1, 2, 3, 4, 5, 6, 7\}$ . Since $F = \sum(3, 5, 6, 7)$ , the complementary indices where $F = 0$ give the maxterm expansion $F(X, Y, Z) = \prod(0, 1, 2, 4)$ .
(B) is CORRECT: Writing out the minterms (the 3-input majority function): $F(X, Y, Z) = \overline{X}YZ + X\overline{Y}Z + XY\overline{Z} + XYZ = YZ(\overline{X}+X) + XZ(\overline{Y}+Y) + XY(\overline{Z}+Z) = XY + YZ + XZ$
(C) and (D) are FALSE: $F$ depends symmetrically on all three inputs $X, Y,$ and $Z$ .

Final answer

Options (A), (B)

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

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