GATE CS 2013 Q1 — Sets, Relations and Functions

To determine the properties of the binary operation $x \oplus y = x^2 + y^2$ on the set of integers, we check for commutativity and associativity:

• Commutativity: An operation $\oplus$ is commutative if $x \oplus y = y \oplus x$ for all integers $x$ and $y$.
  $$x \oplus y = x^2 + y^2$$$$y \oplus x = y^2 + x^2 = x^2 + y^2$$Since addition of integers is commutative, $x \oplus y = y \oplus x$. Thus, the operation is commutative.
• Associativity: An operation $\oplus$ is associative if $(x \oplus y) \oplus z = x \oplus (y \oplus z)$ for all integers $x, y,$ and $z$.
  Let us test this with a simple counterexample using $x = 1, y = 2,$ and $z = 3$:
  $$(1 \oplus 2) \oplus 3 = (1^2 + 2^2) \oplus 3 = 5 \oplus 3 = 5^2 + 3^2 = 25 + 9 = 34$$$$1 \oplus (2 \oplus 3) = 1 \oplus (2^2 + 3^2) = 1 \oplus 13 = 1^2 + 13^2 = 1 + 169 = 170$$Since $34 \neq 170$, the operation is not associative.

Hence, the operation is commutative but not associative.