Check the commutativity and associativity of each of the following binary operations:

‘Ο’ on Q defined by aΟb = a² + b² for all a,b∈Q

Given that Ο is a binary operation on Q defined by aΟb = a² + b² for all a,b∈Q.

We know that commutative property is pΟq = qΟp, where Ο is a binary operation.

Let’s check the commutativity of given binary operation:

⇒ aΟb = a² + b²

⇒ bΟa = b² + a² = a² + b²

⇒ bΟa = aΟb

∴ Commutative property holds for given binary operation ‘Ο’ on ‘Q’.

We know that associative property is (pΟq)Οr = pΟ(qΟr)

Let’s check the associativity of given binary operation:

⇒ (aΟb)Οc = (a² + b²)Οc

⇒

⇒ (aΟb)Οc = a⁴ + b⁴ + 2a²b² + c² ...... (1)

⇒ aΟ(bΟc) = aΟ(b² + c²)

⇒

⇒ aΟ(bΟc) = a² + b⁴ + c⁴ + 2b²c² ...... (2)

From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘Q’.