GATE CSE 2013 | Question: 1

Question

A binary operation $\oplus$ on a set of integers is defined as $x \oplus y = x^{2}+y^{2}$. Which one of the following statements is TRUE about $\oplus$?

• A. Commutative but not associative
• B. Both commutative and associative
• C. Associative but not commutative
• D. Neither commutative nor associative

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Complete Analysis Here

Answer 1

Answer is (A) Commutative but not associative.

$y \oplus x = y^2 + x^2 = x \oplus y$. Hence, commutative.

$ (x \oplus y) \oplus z = (x^2 + y^2) \oplus z = (x^2 + y^2)^2 + z^2$
$ x \oplus (y \oplus z) = x \oplus (y^2 + z^2) = x^2 + (y^2 + z^2)^2$

So, $( (x \oplus y) \oplus z) \neq (x \oplus (y \oplus z))$, hence not associative.

Answer 2

it is commutativa but not associative

Answer 3

 

The correct option is A Commutative but not associative

x⊕y=x2+y2
y⊕x=y2+x2

As ′+′ sign in commutative so x²+y² is equal to y²+x² so x⊕y is commutative.
Now check associativity

x⊕(y⊕z)=x⊕(y²+z²)

=x²+(y²+z²)²

=x²+y^4+z^4+2y²z²

(x⊕y)⊕z

=(x²+y²)⊕z

=(x²+y²)²+z²

=x^4+y^4+2x²y²+z²

x⊕(y⊕z)≠(x⊕y)⊕z

So not associative

Option (a) is correct.