Question 1

1. A binary operation ⊕ on a set of integers is defined as x ⊕ y = x^2 + y^2 . Which one of the following statements is TRUE about ⊕ ?

(A) Commutative but not associative
(B) Both commutative and associative 
(C) Associative but not commutative
(D) Neither commutative nor associative

Answer : A

Explanation :

Is this commutative ?
          x ⊕ y = x^2 + y^2 = y^2 + x^2 = y ⊕ x
         Since  x ⊕ y = y ⊕ x, it is commutative.

Is this associative ?
    That means is (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z ) ?
    Lets take an example: x = 1, y = 2, z = 3
       (x ⊕ y) ⊕ z = (1 + 4) ⊕ 3 = 5 ⊕ 3 = 25 + 9 = 34
       x ⊕ (y ⊕ z ) = 1 ⊕ (4 + 9) = 1 ⊕ 13 = 1 + 169 = 170

    So,  (x ⊕ y) ⊕ z != x ⊕ (y ⊕ z ). That means, it is not associative.

Now lets look at the answers and see which one makes sense.

(A) Commutative but not associative : This makes sense according to our analysis till now.
(B) Both commutative and associative  : This does not make sense, since it is not associative.
(C) Associative but not commutative : This does not make sense, since it is commutative.
(D) Neither commutative nor associative : This does not makes sense, since it is commutative.

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