Binary operation

**lehder** · March 24th 2010, 06:30 AM

Hi everybody,

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

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

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

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

How to do it??
Thanks.

**Haven** · March 24th 2010, 09:32 AM

For the identity (neutral) element.

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

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

**tonio** · March 24th 2010, 09:33 AM

Originally Posted by lehder

Hi everybody,

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

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

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

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

How to do it??
Thanks.

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

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

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

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

Tonio

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