Mark the tick against the correct answer in the following:
Let Z be the set of all integers. Then, the operation * on Z defined by
a * b = a + b - ab is
According to the question ,
Q = { All integers }
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 the 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 – ab + c – ac – bc + abc
And , a * (b * c) = a * (b + c - bc)
= a + (b + c - bc) - a(b + c - bc)
=a + b + c – bc - ab – ac + abc
Both the equation are the same and therefore will always be true.
Therefore , * is associative ……. (2)
Now , according to the equations (1) , (2)
Correct option will be (D)