# GATE | GATE-CS-2015 (Set 3) | Question 65

• Difficulty Level : Easy
• Last Updated : 28 Jun, 2021

Let # be a binary operator defined as X # Y = X′ + Y′ where X and Y are Boolean variables. Consider the following two statements.

```S1: (P # Q) # R = P # (Q # R)
S2: Q # R = R # Q ```

Which of the following is/are true for the Boolean variables P, Q and R?

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(A) Only S1 is True
(B) Only S2 is True
(C) Both S1 and S2 are True
(D) Neither S1 nor S2 are True

Explanation:

```S2 is true, as X' + Y' = Y' + X'
S1 is false.
Let P = 1, Q = 1, R = 0, we get different results
(P # Q) # R = (P' + Q')' + R' = (0 + 0)' + 1 = 1 + 1 = 1
P # (Q # R) = P' + (Q' + R')' = 0 + (0 + 1)' = 0 + 0 = 0
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