Define the connective * for Boolean variables X and Y as: X * Y = XY' + X'Y Let Z = X * Y. Consider the following expressions P, Q and R: P: X = Y + Z, Q: Y = X * Z, R: X * Y + Z' = 1 Which of the following is TRUE? A. Only P and Q are valid. B. Only Q and R are valid. C. Only P and R are valid. D. All P, Q, R are valid.

GATE 2007 · Digital Logic · Boolean Algebra · medium

Answer: Only Q and R are valid identities. Answer: B.

1. Check P: X = Y + Z (Boolean OR): Case X=0,Y=0: Z=0, Y+Z=0+0=0, X=0 ✓ Case X=0,Y=1: Z=1, Y+Z=1+1=1, but X=0 ✗ P fails at (X=0, Y=1), so P is NOT a valid identity.
2. Check Q: Y = X * Z: Using XOR associativity: X XOR X XOR Y = 0 XOR Y = Y. So X * Z = Y for all X, Y. Q is always true. ✓
3. Check R: X * Y + Z' = 1: Z + Z' = 1 is the complement law, which holds for any Boolean variable Z. So R is always true. ✓