Question

Dual of \( (x + y)(x + 1) = x + x \cdot y + y \) is:

• A. \( (x \cdot y) + (x \cdot 0) = (x + y) \cdot y \)
• B. \( (x + y) + (x \cdot 1) = x \cdot (x + y) \cdot y \)
• C. \( (x \cdot y) \cdot (x \cdot 0) = (x + y) \cdot y \)
• D. None of the above

Hint:

To find the dual of a Boolean expression, swap AND and OR operations and replace 1 by 0 and 0 by 1.

Answer 1

The Correct Option is A

Step 1: Understanding the dual.
The dual of a Boolean expression is obtained by interchanging the AND (\( \cdot \)) and OR (\( + \)) operations and replacing 1 by 0 and 0 by 1. Given the expression: \[ (x + y)(x + 1) = x + x \cdot y + y, \] to find its dual, we follow the rule for duality.
Step 2: Applying duality.
- Replace all \( + \) (OR) with \( \cdot \) (AND). - Replace all \( \cdot \) (AND) with \( + \) (OR). - Replace 1 with 0 and 0 with 1. The dual of the given expression is: \[ (x \cdot y) + (x \cdot 0) = (x + y) \cdot y. \]
Step 3: Conclusion.
Thus, the correct dual is \( (x \cdot y) + (x \cdot 0) = (x + y) \cdot y \), and the correct answer is (a).