Q. 4 D5.0( 1 Vote )
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’.
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