Answer :

According to the question ,


R = {(a, b) : a * b = a + b + ab }


Formula


* is commutative if a * b = b * a


* is associative if (a * b) * c = a * (b * c)


Check for commutative


Consider , a * b = a + b + ab


And , b * a = b + a + ba


Both equations are same and will always be true .


Therefore , * is commutative ……. (1)


Check for associative


Consider , (a * b) * c = (a + b + ab) * c


= a + b + ab + c + (a + b + ab)c


=a + b + c + ab + ac + bc + abc


And , a * (b * c) = a * (b + c + bc)


= a + b + c + bc + a(b + c + bc)


=a +b + c + ab + bc + ac + abc


Both the equation are same and therefore will always be true.


Therefore , * is associative ……. (2)


Now , according to the equations (1) , (2)


Correct option will be (D)

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