Question

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 1

The correct option is A Commutative but not associative

$x\oplus y=x^2+y^2$
$y\oplus x=y^2+x^2$
As ${}^{\prime }{+}^{\prime }$ sign in commutative so $x^2+y^2$ is equal to $y^2+x^2$ so $x\oplus y$ is commutative.
Now check associativity
$x\oplus (y\oplus z)=x\oplus (y^2+z^2)$
$=x^2+(y^2+z^2{)}^2$
$=x^2+y^4+z^4+2y^2{z}^2$
$(x\oplus y)\oplus z=(x^2+y^2)\oplus z$
$=(x^2+y^2{)}^2+z^2$
$=x^4+y^4+2x^2{y}^2+z^2$
$x\oplus (y\oplus z)\ne (x\oplus y)\oplus z$
So not associative
Option (a) is correct.