Check the commutativity and associativity of each of the following binary operations:
‘o’ on Q defined by a o b = ab/2 for all a,b Q.
Given that ο is a binary operation on Q defined by 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:
⇒
⇒
⇒ b*a = a*b
∴ The 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:
⇒
⇒
⇒ ...... (1)
⇒
⇒
⇒ ...... (2)
From (1) and (2) we can clearly say that associativity hold for the binary operation ‘ο’ on ‘Q’.