Question

The Boolean expression \( AB+ A\bar{B} + \bar{A}C + AC \) is unaffected by the value of the Boolean variable

• A. A
• B. None of these
• C. C
• D. B

Hint:

To see which variable an expression is independent of, simplify it completely using Boolean algebra rules. If a variable disappears from the final simplified form, the expression is unaffected by that variable. Look for pairs like \( XY + X\bar{Y} \) which simplify to \( X \).

Answer 1

The Correct Option is D

To determine if the expression is unaffected by a variable, we must first simplify the expression as much as possible.
Given expression: \( Y = AB + A\bar{B} + \bar{A}C + AC \) Let's group the terms:
\[ Y = (AB + A\bar{B}) + (\bar{A}C + AC) \] Simplify the first group: Using the distributive law, we can factor out A: \[ AB + A\bar{B} = A(B + \bar{B}) \]
By the law of complementation, \( B + \bar{B} = 1 \). So, \( A(1) = A \).
Simplify the second group: Using the distributive law, we can factor out C: \[ \bar{A}C + AC = C(\bar{A} + A) \]
By the law of complementation, \( \bar{A} + A = 1 \). So, \( C(1) = C \).
Combine the simplified parts: The entire expression simplifies to: \[ Y = A + C \] The final simplified expression is \( A + C \). This expression depends on the values of A and C, but it does not contain the variable B. This means the value of the original expression is entirely independent of, or unaffected by, the value of the Boolean variable B.