Question

Which of the following is the simplified equivalent of the Boolean expression \(A + \bar{A}B\)?

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

Hint:

The expression \(A + \bar{A}B = A + B\) is a standard Boolean algebra rule often called the "Redundancy Law" or "Absorption Law variant". Memorizing it saves time.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The task is to simplify the given Boolean expression \(A + \bar{A}B\) by applying standard Boolean algebra laws.


Step 2: Key Formula or Approach:
We will apply the Distributive Law:
\[ X + YZ = (X + Y)(X + Z) \] Along with the Complement Law (\(X + \bar{X} = 1\)) and the Identity Law (\(1 \cdot Y = Y\)).


Step 3: Detailed Explanation:
Start with the original expression:
\[ A + \bar{A}B \] Using the Distributive Law, we can expand this as:
\[ A + \bar{A}B = (A + \bar{A})(A + B) \] According to the Complement Law, a variable ORed with its inverse equals 1 (\(A + \bar{A} = 1\)). Substituting this yields:
\[ (1)(A + B) \] By the Identity Law, multiplying by 1 does not change the expression, so we are left with:
\[ A + B \]

Step 4: Final Answer:
The correct choice is (B).