# Thread: Operation

1. ## Operation

Hi,

$\displaystyle _* \$is a binary operation in E,

C is a set defined as:

$\displaystyle C={ y \in E/ (\forall x \in E): y_*x=x_*y }$

I must determine the specifications of $\displaystyle (C,_*)$: associativity, commutativity, neutral element, inverse element...

I found that $\displaystyle _*$ is commutative but for the others i don't know how to do it.

2. Originally Posted by bhitroofen01
Hi,

$\displaystyle _* \$is a binary operation in E,

C is a set defined as:

$\displaystyle C={ y \in E/ (\forall x \in E): y_*x=x_*y }$

I must determine the specifications of $\displaystyle (C,_*)$: associativity, commutativity, neutral element, inverse element...

I found that $\displaystyle _*$ is commutative but for the others i don't know how to do it.
If E has an identity element (is that what you mean by "neutral element"?) then that identity element commutes with everything in E.

If * is associative on E, then it's clearly associative on any subset of E.

If y is in C and x is in E, then $\displaystyle y^{-1}x=(x^{-1}y)^{-1}=(yx^{-1})^{-1}=xy^{-1}$.