GATE CS 2025 Set 2 Q.50: Boolean Algebra and Logic Minimization

Which of the following Boolean algebraic equation(s) is/are CORRECT?

$\bar{A}BC + A\bar{B}\bar{C} + \bar{A}\bar{B}\bar{C} + A\bar{B}C + ABC = BC + \bar{B}\bar{C} + \bar{A}\bar{B}$

$AB + \bar{A}C + BC = AB + \bar{A}C$

$(A + C)(\bar{A} + B) = AB + \bar{A}C$

$\overline{(A + \bar{B} + \bar{D})(C + D)(\bar{A} + C + D)(A + B + \bar{D})} = \bar{A}D + \bar{C}\bar{D}$

Official answer and solution

Solution

Simplify the LHS of each Boolean equation using consensus, absorption, and De Morgan's laws.

(A) is FALSE: Grouping terms on the LHS gives: $(\bar{A}BC + ABC) + (A\bar{B}\bar{C} + \bar{A}\bar{B}\bar{C}) + A\bar{B}C = BC + \bar{B}\bar{C} + A\bar{B}C = BC + \bar{B}\bar{C} + A\bar{B}$ whereas the RHS has $\bar{A}\bar{B}$ instead of $A\bar{B}$ (at $A=0, B=0, C=1$ , LHS $= 0$ while RHS $= 1$ ).
(B) is TRUE: By the Consensus Theorem, the consensus term $BC$ of $AB$ and $\bar{A}C$ is redundant, so $AB + \bar{A}C + BC = AB + \bar{A}C$ .
(C) is TRUE: Expanding $(A + C)(\bar{A} + B) = A\bar{A} + AB + \bar{A}C + BC = AB + \bar{A}C + BC = AB + \bar{A}C$ .
(D) is TRUE: Inside the overbar, combine $(A + \bar{B} + \bar{D})(A + B + \bar{D}) = A + \bar{D}$ and $(C + D)(\bar{A} + C + D) = C + D$ . Applying De Morgan's Law to $\overline{(A + \bar{D})(C + D)}$ yields: $\overline{(A + \bar{D})} + \overline{(C + D)} = \bar{A}D + \bar{C}\bar{D}$

Final answer

Options (B), (C), (D)

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

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 50 of 65
Section: Computer Science and Information Technology

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