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# 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,bQ.

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’.

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