a∈Q/{−1}
b∈Q/{−1}
a∗b=a+b+ab∈Q/{−1}
∴∗,is a binary operation in(Q/{−1},∗)
Associativity: Suppose a,b,c∈Q/{−1}
(a∗b)∗c=(a+b+ab)∗c
=(a+b+ab)+c+(a+b+ab)c
=a+b+c+ab+bc+ca+abc
So,
a∗(b∗c)=a∗(b+c+bc)
=a+(b+c+bc)+a(b+c+bc)
=a+(b+c+bc)+a(b+c+bc)
=a+b+c+ab+bc+ca+abc
(a∗b)∗c=a∗(b∗c)
* is associative.
Existence of Identity:
0∈Q/{−1}
is the identity element.
Because a∈Q/{−1}
0∗a=0+a+0a=a
a∗0=a+0+a0=a
Existence of Inverse :
Every a∈ Q/{−1}
For -a/a+1 ∈ Q/{-1}
a=−1, which is inverse of a
Because
(−a/(a+1))∗a =
=(−a/(a+1))+a+(−a/(a+1))∗a=0
a∗(−a/(a+1))=
=a+(−a)/(a+1)+a(−a/(a+1))=0
Commutativity:− a,b∈Q/{−1}
a∗b=a+b+ab
b∗a=b+a+ba
So,
a∗b=b∗a
Operation * is commutative.
The above discussion proves that the group is abelian.
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