# Operation

• Mar 24th 2010, 12:59 PM
bhitroofen01
Operation
Hi,

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

C is a set defined as:

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

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

I found that $_*$ is commutative but for the others i don't know how to do it.
• Mar 24th 2010, 07:20 PM
Tinyboss
Quote:

Originally Posted by bhitroofen01
Hi,

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

C is a set defined as:

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

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

I found that $_*$ 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 $y^{-1}x=(x^{-1}y)^{-1}=(yx^{-1})^{-1}=xy^{-1}$.