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:
We are asked to simplify the Boolean expression \(A + \bar{A}B\) using the rules of Boolean algebra.


Step 2: Key Formula or Approach:
We will use the Distributive Law of Boolean algebra, which states:
\[ X + YZ = (X + Y)(X + Z) \] Additionally, we use the rule of complements: \(X + \bar{X} = 1\), and the identity rule: \(1 \cdot Y = Y\).


Step 3: Detailed Explanation:
Given the expression:
\[ A + \bar{A}B \] Apply the distributive law, treating \(A\) as \(X\), \(\bar{A}\) as \(Y\), and \(B\) as \(Z\):
\[ A + \bar{A}B = (A + \bar{A})(A + B) \] Since a variable ORed with its complement is always 1 (\(A + \bar{A} = 1\)), we can substitute this in:
\[ (A + \bar{A})(A + B) = (1)(A + B) \] Since ANDing any expression with 1 leaves the expression unchanged:
\[ 1 \cdot (A + B) = A + B \] Thus, the simplified expression is \(A + B\).


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