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Consider the group (G,*) where G is real number system except 1 and * is defined as a*b=a+b-ab then the inverse of -2 in this group is ________.

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It is given that G is a group. Therefore the inverse must exist. Let us find the identity element first to find the inverse.

We know,

a*e=e*a=a (where e is the identity element)

we know a*e = a+e-ae

=> a+e-ae=a

=> e=0

since identity is 0 ,we can say that (by group property)

a*b=b*a=0 (i.e identity element)

a+e-ae=0

=>e=$\frac{a}{a-1}$ and since a!=0 , this statifies

so inverse of -2 should be

e=$\frac{-2}{-2-1}$ = $\frac{2}{3}$

 

Just to check put a=-2 and e=2/3 in a*b and you should be getting a 0.

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