Question

Let \(X\) be a 3-variable Boolean function that produces output as ‘1’ when at least two of the input variables are ‘1’. Which of the following statement(s) is/are CORRECT, where \(a, b, c, d, e\) are Boolean variables?

• A. \( X(a, b, X(c, d, e)) = X(X(a, b, c), d, e) \)
• B. \( X(a, b, X(a, b, c)) = X(a, b, c) \)
• C. \( X(a, b, X(a, c, d)) = (X(a, b, a) { AND } X(c, d, c)) \)
• D. \( X(a, b, c) = X(a, X(a, b, c), X(a, c, c)) \)

Hint:

In Boolean functions, simplify the expression using properties of logical functions like AND, OR, and NOT.

Answer 1

The Correct Option is B, D

Option (A): Incorrect. 
Option (B): Correct. The composition \( X(a, b, X(a, b, c)) \) 
simplifies to \( X(a, b, c) \) due to the function structure. 
Option (C): Incorrect. 
Option (D): Correct. The Boolean function \( X(a, b, c) \) can be rewritten as \( X(a, X(a, b, c), X(a, c, c)) \). 

Thus, the correct answers are \( \boxed{B} \) & \( \boxed{D} \).