Answer :

Given that,


‘*’ be binary operation defined on R by a * b = 1 + ab, a, b R


A binary operation ‘*’ is commutative if a*b = b*a a, b R


Now,


a*b = 1+ab = 1+ba [ ab=ba since multiplication is commutative


on R]


1+ba = b*a


a*b = b*a a, b R


So, ‘*’ is commutative on R.


A binary operation ‘*’ is associative if (a*b)*c = a*(b*c) a, b,c R


Now,


(a*b)*c = (1+ab)*c = 1+(1+ab)c = 1+c+abc


a*(b*c) = a*(1+bc) = 1+a(1+bc) = 1+a+abc


(a*b)*c ≠ a*(b*c)


So, ‘*’ is not associative on R.


Hence, ‘*’ is commutative but not associative on R.


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