Let $\#$ be a binary operator defined as $$X \# Y = X^{\prime} + Y^{\prime} \text{ where } X \text{ and } Y \text{ are Boolean variables.}$$ Consider the following two statements. $(S1) \quad (P \# Q)\# R = P\#(Q \# R)$ $(S2) \quad Q\# R = R\# Q$ Which of the following is/are true for the Boolean variables $P$, $Q$ and $R$?

The given operation is $X \# Y = X^{\prime} + Y^{\prime} = (XY)'$, which represents the standard NAND operation.

1. Statement (S1) - Associativity:

   • Left-hand side (LHS):

     $$(P \# Q) \# R = (P \# Q)' + R^{\prime} = (P^{\prime} + Q^{\prime})' + R^{\prime} = PQ + R^{\prime}$$

   • Right-hand side (RHS):

     $$P \# (Q \# R) = P^{\prime} + (Q \# R)' = P^{\prime} + (Q^{\prime} + R^{\prime})' = P^{\prime} + QR$$

   • Testing with $P = 0, Q = 0, R = 1$:

     $$\text{LHS} = (0)(0) + 1' = 0$$

     $$\text{RHS} = 0' + (0)(1) = 1$$

   • Since $\text{LHS} \neq \text{RHS}$, the NAND operation is not associative. Thus, S1 is false.

2. Statement (S2) - Commutativity:

   $$Q \# R = Q^{\prime} + R^{\prime} = R^{\prime} + Q^{\prime} = R \# Q$$

   • Since Boolean OR is commutative, S2 is true.

Therefore, Only S2 is true (Option B).