The difference between Quotient Set and other definition it a new course and material that I learn, we defined 2 pretty similiar definitions and I didnt understand what is the difference between the definitions. 
Definition 1: A subset $T\subseteq X$ is called "slice" (dont know how to transkate it correctly) if $\forall x\in X\:\exists! t\in T\colon x\sim t$.
Then we defined the Qutient Set: $X/{\sim}:=\left\{\,\left[x\right]:\:x\in T\,\right\}$
Now an example: 
$$X=\mathbb R^2$$
$$\left(x_1,y_1\right)\sim\left(x_2,y_2\right):\:x_1^2+y_1^2=x_2^2+y_2^2$$
So by our definition the slice is: $T=\left\{\,\left(a,0\right):a\ge 0\,\right\}$
and the quotient set is be like: $\mathbb R^2/{\sim}=\left\{\,\left(r,0\right):r\ge 0\,\right\}$
Does any one understand what did I mean in the "slice ($T$) definition" and can somebody explain me the difference between them both? 
I just copied it as I wrote it in class, and I dont understand the "slice" terminology or definition. Maybe by the definition somebody could translate it correctly? 
 A: First note that while you wrote $X/{\sim}=\{\,[x]:x\in T\,\}$ you might just as well write $X/{\sim}=\{\,[x]:x\in X\,\}$. The fact that classes are "enumerated" repeatedly this way does note matter, after all you may recall that e.g. $\{1,2,3,2,1,1,4\}=\{1,2,3,4\}$. So, we do not need the slice (or system of representatives)  $T$ to define the quotient space.
However, by the definitng property of a slice, we have that the map $T\to X/{\sim}$, $x\mapsto [x]$ is a bijection. Therefore we can identify (or at least nicely visualize) $X/{\sim}$ via $T$. And using this bijection as a means of identificatin, we see that - in a way - there is no difference between the two concepts. At least this may simplify or support imagination because after all the elements of $T$ are just normal "points" (or whatever the elements of $X$ look like) whereas the elements of $X/{\sim}$ are sets of points.
On the other hand, it is often more elegant (and I prefer) to work with $X/{\sim}$ directly instead of $T$; for example, defining addition on $\mathbb Z/n\mathbb Z$ (which is $\mathbb Z$ modulo the equivalence relation $x\sim y\iff n\mid (x-y)$)  requires case distinction if one works with a system of representatives such as $\{0,1,\ldots,n-1\}$, but it can be formulated directly when working with the equivalence classes themselves.
Also note, that finding a nice slice explicitly may not always be straightforward and is certainly not unique (except in trivial cases).
