Question

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

• A. \( A B C + A \bar{B} C + A \bar{B} C + A \bar{B} C + A B C = B C + \bar{B} C + A \bar{B} \)
• B. \( A B + A C + B C = A B + A C \)
• C. \( (A + C)(A + B) = A B + A C \)
• D. \( (A + \bar{B} + \bar{D})(C + D)(A + C + D)(A + B + \bar{D}) = A D + C \bar{D} \)

Hint:

{When simplifying Boolean expressions, use absorption, distribution, and complement rules to reduce terms.}

Answer 1

The Correct Option is B, C, D

1. Option (A) is incorrect: - The left-hand side simplifies to \( A B C + A \bar{B} C + A B C \), which does not simplify to the right-hand side. The terms do not cancel out or combine in a way that would make the left-hand side equal to the right-hand side.

2. Option (B) is correct: - Using absorption: \[ A B + A C + B C = A B + A C \quad \text{(since \( B C \) is redundant in the presence of \( A B \) and \( A C \))} \] This simplification follows from the Boolean absorption law, where \( A B \) and \( A C \) already cover the possibilities where \( B C \) would be true.

3. Option (C) is correct: - Expanding using the distributive property: \[ (A + C)(A + B) = A A + A B + A C + B C = A + A B + A C + B C = A B + A C \] This simplification follows from the fact that \( A A = A \) and the expression reduces to \( A B + A C \).

4. Option (D) is correct: - Simplifying step by step using Boolean rules, we get: \[ (A + \bar{B} + \bar{D})(C + D)(A + C + D)(A + B + \bar{D}) = A D + C \bar{D} \] This simplification uses a combination of distribution and the elimination of redundant terms.

Thus, the correct options are (B), (C), (D).