Question

Consider a Boolean expression given by \( F(X, Y, Z) = \Sigma(3, 5, 6, 7) \). Which of the following statements is/are CORRECT?

• A. \( F(X, Y, Z) = \Pi(0, 1, 2, 4) \)
• B. \( F(X, Y, Z) = XY + YZ + XZ \)
• C. \( F(X, Y, Z) \) is independent of input \( Y \)
• D. \( F(X, Y, Z) \) is independent of input \( X \)

Hint:

For Boolean expressions, convert between minterm and maxterm forms to verify equivalences. Use truth tables or Karnaugh maps for simplification.

Answer 1

The Correct Option is A

Step 1: Derive the truth table for \( F(X, Y, Z) \).
The given \( F(X, Y, Z) = \Sigma(3, 5, 6, 7) \) corresponds to minterms where the function evaluates to 1. These minterms are: \[ 3 (011), 5 (101), 6 (110), 7 (111). \] Step 2: Convert to product-of-maxterms.
Maxterms where \( F(X, Y, Z) = 0 \) are \( 0 (000), 1 (001), 2 (010), 4 (100) \). Thus, the complement of \( F(X, Y, Z) \) is given by \( \Pi(0, 1, 2, 4) \). This makes Option (1) TRUE. Step 3: Simplify the Boolean expression.
Using minterms, the Boolean expression can be simplified as: \[ F(X, Y, Z) = XY + YZ + XZ. \] This makes Option (2) TRUE. Step 4: Check independence of \( X \) or \( Y \).
The function depends on all three variables (\( X, Y, Z \)), as flipping any variable changes the output. Thus, \( F(X, Y, Z) \) is NOT independent of \( X \) or \( Y \), making Options (3) and (4) FALSE. Final Answer: \[ \boxed{(1), (2)} \]