A binary operation ⊕ on the set of integers is defined as x⊕y = x2 + y2 Which…

Concept

A binary operation ⊕ on a set S is commutative if a⊕b = b⊕a for every a, b in S. It is associative if (a⊕b)⊕c = a⊕(b⊕c) for every a, b, c in S. Commutativity and associativity are independent properties — an operation can satisfy one without the other, so each must be tested separately.

Applying it to this operation

Here the operation is x⊕y = x2 + y2, defined on the integers.

Test commutativity

1. Compute x⊕y = x2 + y2.

2. Compute y⊕x = y2 + x2.

3. Ordinary addition is commutative, so x2 + y2 = y2 + x2. Hence x⊕y = y⊕x for all integers — the operation IS commutative.

Test associativity

1. Left grouping: (x⊕y)⊕z = (x2 + y2)⊕z = (x2 + y2)2 + z2.

2. Right grouping: x⊕(y⊕z) = x⊕(y2 + z2) = x2 + (y2 + z2)2.

3. These two expressions are not equal in general, so the operation is NOT associative.

Cross-check with a counterexample

Take x = 1, y = 1, z = 0:

• (1⊕1)⊕0 = (1 + 1)2 + 0 = 22 + 0 = 4.

• 1⊕(1⊕0) = 1 + (1 + 0)2 = 1 + 12 = 2.

Since 4 ≠ 2, associativity fails. The operation is commutative but not associative.