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Q.

A binary operation ⊕ on a set of integers is defined as x⊕y=x2⊕y2. 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 & Explanation

Answer: Option [A]

A binary operation ⊕ on a set of integers is defined as x⊕y=x2⊕y2. Let us test first, the operation supports commutative law or not,

Given, x⊕y=x2⊕y2

y⊕x=y2⊕x2=x2⊕y2[because Addition operation is commutative]

So, ⊕ supports commutative law.

Now, we test for associative law

(x⊕y)⊕z = (x2+y2)2+z2

x⊕(y⊕z) = x2+(y2+z2)2

We get, (x2+y2)2+z2 ≠ x2+(y2+z2)2

⊕ is not associative.

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