GATE CS 2025 Set 2 | Question 50

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

Option A: \(\overline{A}\,B\,C + A\,\overline{B}\,\overline{C} + \overline{A}\,\overline{B}\,\overline{C} + A\,\overline{B}\,C + A\,B\,C = B\,C + \overline{B}\,\overline{C} + \overline{A}\,\overline{B}\)
Option B: \(A\,B + \overline{A}\,C + B\,C = A\,B + \overline{A}\,C\) (correct)
Option C: \((A + C)\,(\overline{A} + B) = A\,B + \overline{A}\,C\) (correct)
Option D: \(\overline{(A + \overline{B} + \overline{D})\,(C + D)\,(\overline{A} + C + D)\,(A + B + \overline{D})} = \overline{A}\,D + \overline{C}\,\overline{D}\) (correct)

Answer

(B), (C), (D)

(A) is wrong. The left side is \(\sum m(0, 3, 4, 5, 7)\) over \(ABC\). On the right, \(B\,C\) gives \(m_3, m_7\), \(\overline{B}\,\overline{C}\) gives \(m_0, m_4\) and \(\overline{A}\,\overline{B}\) gives \(m_0, m_1\): it contains 001 and misses 101. With \(A\,\overline{B}\) as the last term it would hold.
(B) is the consensus theorem: \(BC = ABC + \overline{A}BC\), and these two terms are absorbed by \(AB\) and \(\overline{A}C\).
(C) Expanding gives \(A\overline{A} + AB + \overline{A}C + BC = AB + \overline{A}C + BC\), which reduces to \(AB + \overline{A}C\) by (B).
(D) By De Morgan, the complement of the product is the sum of the complemented factors: \(\overline{A}BD + \overline{C}\,\overline{D} + A\overline{C}\,\overline{D} + \overline{A}\,\overline{B}D\). The first and last terms combine to \(\overline{A}D\), and \(A\overline{C}\,\overline{D}\) is absorbed by \(\overline{C}\,\overline{D}\).

Answer: B, C, D