On the set Q of all rational numbers the operation * which is both associative and commutative is given by a * b=

Step-by-step solution

If a * b=b * a , then the operation is commutative. If (a * b) * c=a *(b * c) , then the operation is associative. By option (d), aligned &a * b=a+b+a b &b * a=b+a+b a=a+b+a b aligned Here, a * b=b * a , so the operation is commutative. aligned (a * b) * c &=(a+b+a b) * c &=(a+b+a b)+c+(a+b+a b) c &=a+b+a b+c+a c+b c+a b c &=a+b+c+a b+a c+b c+a b c a *(b * c) &=a *(b+c+b c) &=a+b+c+b c+a(b+c+b c) &=a+b+c+b c+a b+a c+a b c Here, (a * b) & * c=a *(b * c) . aligned So, the operation is associative.