Q: $S$ is the set of complex numbers $x+iy$ with real part $x \geq 0$, the binary operation is the addition.
My method:
(1) To Check its Associative Property
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For $a,b,c \in \mathbb{C}$, $(a \ast b)\ast c$ is in fact the same as $a \ast (b\ast c)$ since it is addictive. LHS should equal to RHS.
(2) Exist an identity element $e$
$a\ast e=a$
It is obvious that $0$ is the identity element.
(3) Exist an inverse
Let $a=x_1+y_1i$, $a'=x_2+y_2i$
then,
$a\ast a'=(x_1+y_1i) + (x_2+y_2i)=0$
It is also obvious that there exist an inverse if $x_1=-x_2$ and $y_1=-y_2$.
Is there any problems with it?? In fact, I still not understand 100% about "group". What I know are the 3 rules such that it is classified as a group. But it seems every time I try to prove, it satisfies the conditions. So I am a little bit afraid that I am in fact making the wrong things.
Best Answer
No, it is not obvious that an inverse exists. For instance, $1+i$ has no inverse. From you you wrote, the inverse would be $-1-i$. Problem: $-1-i\notin S$.
You are right about the other two properties, though.