How to express uniqueness of the identity element

[Jbreezy]

Homework Statement

Hi All,
I was trying to write that $I$ is unique. $I$ is my identity element.

Homework Equations

The Attempt at a Solution

Maybe I can write

$\exists I | I = I$
$\forall I \in G$
Except with spaces. Is this not a way?


Just trying to specify that this thing is unique.

 

[lurflurf]

That is very confusing. It could be made more clear, but ! means unique, You could just write

$$\exists ! I$$

Uniqueness quantification - Wikipedia, the free encyclopedia

 

[Jbreezy]

Could you say that and then just start with this assumption I = I and proceed with a proof from there?Thanks

 

[lurflurf]

What does I=I mean? It is always true. You want to say something like there exist a unique identity or if I and I' are any two identities then I=I'.

 

[Jbreezy]

The question tells "Given there exits a unique identity element." So I'm just trying to use this. But having a hard time saying what I want. What I was trying to say before this was $\exists ! I$ such that I = I.
But I guess...not sure how I'm trying to say this.

 

[lurflurf]

I would say something like
$$\exists ! I \in G |(xI=Ix=x) \forall x \in G$$
or
$$(xy=yx=x) \forall x \in G \implies y=I$$

 

[HallsofIvy]

"I= I" is always true whether I is denoting something that is "unique" or not. For example $\frac{1}{2}= \frac{1}{2}$ even though $\frac{1}{2}= \frac{2}{4}$ as well.