Question 1

The binary operation is defined as $x \oplus y = x^2 + y^2$.

1. Commutativity:

   $$y \oplus x = y^2 + x^2 = x^2 + y^2 = x \oplus y$$

   Since addition on integers is commutative, $x \oplus y = y \oplus x$ for all integers $x, y$. Thus, $\oplus$ is commutative.

2. Associativity:

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

   Testing with concrete values, say $x = 1, y = 2, z = 3$:

   $$(1 \oplus 2) \oplus 3 = (1^2 + 2^2)^2 + 3^2 = 5^2 + 9 = 25 + 9 = 34$$
   $$1 \oplus (2 \oplus 3) = 1^2 + (2^2 + 3^2)^2 = 1 + 13^2 = 1 + 169 = 170$$

   Since $34 \neq 170$, $(x \oplus y) \oplus z \neq x \oplus (y \oplus z)$ in general.

Therefore, $\oplus$ is commutative but not associative.