Answer :

Given: binary operation * on A = R – {-1} defined as


a * b = a + b + ab for all a and b belongs to A


To prove: given operation is is commutative and associative on A and every element of A is invertible.


To find: the identify element of * in A


Commutativity:



a * b = a + b + ab and b * a = b + a + ba


Since, a + b + ab = b + a + ba


a * b = b * a


This shows that operation is commutative


Associativity:



(a * b) * c


= (a + b + ab) * c


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


= a + b + ab + c + ac + bc + abc……………(1)


and


a * (b * c)


= a * (b + c + bc)


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


= a + b + c + bc + ab + ac + abc………………(2)


Since, (1) = (2)


(a * b) * c = a * (b * c)


This shows that operation is associative


Existence of identity:


Let e be the identity element



a * e = a = e * a


a + e + ae = a and e + a + ea = a


e(1 + a) = a – a


e(1 + a) = 0


e = 0



So, 0 is the identity element of the * operation


Existence of inverse:



a * b = e = e * b


a + b + ab = 0








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