Correct Answer - Option 1 : Commutative but not associative
A binary operation ∗ on a set S is said to be associative if it satisfies the associative law:
a ∗ (b ∗c) = (a ∗ b) ∗ c for all a, b, c ∈S.
A binary operation ∗ on a set S is said to be commutative if it satisfies the condition:
a ∗b=b ∗a for all a, b, ∈S.
First started with checking for associative, x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z.
L.H.S: x ⊕ (y ⊕ z) = x ⊕ (y2 + z2) = x2 + (x2 + z2)2 .
R.H.S: (x ⊕ y) ⊕ z = (x2 + y2) ⊕ z = (x2 + y2)2 + z2 .
L.H.S ≠ R.H.S then the operator is not associative.
For commutative, x ⊕ y = y ⊕ x
L.H.S: x ⊕ y = x2 + y2
R.H.S: y ⊕ x = y2 + x2
L.H.S = R.H.S then the operator is Commutative.