1. ## Binary operation

Hi everybody,

f is a binary operation defined in $\displaystyle \mathbb{R}$ as follows:

$\displaystyle (\forall (x,y) \in \mathbb{R}^2) x_* y=xy-2(x+y)+6$

I find that $\displaystyle _*$ is commutative and associative.

I must show that $\displaystyle _*$ takes a neutral element; and i must find if each element of $\displaystyle \mathbb{R}$ has an inverse element?

How to do it??
Thanks.

2. For the identity (neutral) element.

You must solve $\displaystyle x * e = x$
so$\displaystyle x * e=xe-2(x+e)+6 = x$

Then for the inverses, solve the equation$\displaystyle x * x^{-1} = e$
so $\displaystyle x* x^{-1}=xx^{-1}-2(x+x^{-1})+6= e$

3. Originally Posted by lehder
Hi everybody,

f is a binary operation defined in $\displaystyle \mathbb{R}$ as follows:

$\displaystyle (\forall (x,y) \in \mathbb{R}^2) x_* y=xy-2(x+y)+6$

I find that $\displaystyle _*$ is commutative and associative.

I must show that $\displaystyle _*$ takes a neutral element; and i must find if each element of $\displaystyle \mathbb{R}$ has an inverse element?

How to do it??
Thanks.
Denote by $\displaystyle z$ the neutral element, in case it exists. Then it must fulfill $\displaystyle x=x_*z=xz-2(x+z)+6\,,\,\,\forall x\in\mathbb{R}$ .

In particular, this must be true for $\displaystyle x=0: 0=0_*z=0\cdot z-2(0+z)+6=-2z+6\Longrightarrow z=3$ , and now we check with a general element $\displaystyle x\in\mathbb{R}$ :

$\displaystyle 3_*x=3x-2(3+x)+6=3x-6-2x+6=x$ ...check!

Well, now find $\displaystyle \forall x\in\mathbb{R}$ an element $\displaystyle x'\,\,\,s.t.\,\,\,x_*x'=3$ .

Tonio