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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