Question

Which of the following is equivalent to the Boolean expression: \( (X+Y)(\bar{X}+Y)(X+\bar{Y}) \)?

• A. \( XY \)
• B. \( X\bar{Y} \)
• C. \( \bar{X}Y \)
• D. \( \bar{X}\bar{Y} \)

Hint:

When simplifying Boolean expressions, look for opportunities to apply key identities. The distributive law \( (A+B)(A+C) = A+BC \) is very useful for combining terms. Also, always remember the basic rules: \( A\bar{A}=0 \) and \( A+\bar{A}=1 \).

Answer 1

The Correct Option is A

Let's simplify the given Boolean expression step-by-step.
Expression: \( (X+Y)(\bar{X}+Y)(X+\bar{Y}) \) First, let's combine the first two terms using the distributive law of Boolean algebra: \( (A+B)(A+C) = A+BC \).
Here, we can see that \( Y \) is the common term, so \( A=Y, B=X, C=\bar{X} \). \[ (Y+X)(Y+\bar{X}) = Y + X\bar{X} \] Since \( X\bar{X} = 0 \) (Law of Complementation), this simplifies to: \[ Y + 0 = Y \] Now, the original expression becomes: \[ Y \cdot (X+\bar{Y}) \] Using the distributive law \( A(B+C) = AB+AC \): \[ YX + Y\bar{Y} \] Again, using the Law of Complementation, \( Y\bar{Y} = 0 \). \[ YX + 0 = YX = XY \] The simplified expression is \( XY \).