The GATE Grind

GATE 2025 CS (CS2) – Question 50

Digital Logic · Boolean Algebra and Minimization · 2 marks · Multiple select

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

  1. $\bar{A}BC + A\bar{B}\bar{C} + \bar{A}\bar{B}\bar{C} + A\bar{B}C + ABC = BC + \bar{B}\bar{C} + \bar{A}\bar{B}$
  2. $AB + \bar{A}C + BC = AB + \bar{A}C$
  3. $(A + C)(\bar{A} + B) = AB + \bar{A}C$
  4. $\overline{(A + \bar{B} + \bar{D})(C + D)(\bar{A} + C + D)(A + B + \bar{D})} = \bar{A}D + \bar{C}\bar{D}$

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Correct answer: (B) $AB + \bar{A}C + BC = AB + \bar{A}C$; (C) $(A + C)(\bar{A} + B) = AB + \bar{A}C$; (D) $\overline{(A + \bar{B} + \bar{D})(C + D)(\bar{A} + C + D)(A + B + \bar{D})} = \bar{A}D + \bar{C}\bar{D}$

Explanation

A fails because the LHS has minterm 5 but not minterm 1, while the RHS has minterm 1 but not 5. B is the consensus theorem. C expands to $AB + \bar{A}C + BC = AB + \bar{A}C$. For D, the complement is $\bar{A}BD + \bar{C}\bar{D} + A\bar{C}\bar{D} + \bar{A}\bar{B}D = \bar{A}D + \bar{C}\bar{D}$.