Perhaps there is not a correct way to think about it but I would want to know how others think about it. Here are my problems/questions, after my definitions:
Definition 1. Let $X$ be a set and $\sim$ be an equivalence relation on $X$. Then $[x]:=\{y \in X \mid y \sim x\}$ and $X/{\sim} := \{[x] \mid x \in X\}$.
My question could be summarized to "How should I think about $X/{\sim}$?". Consider $\mathbf{Z}/{\sim}$ with $z_1 \sim z_2$ $\iff$ $z_1-z_2$ is even. One then obtains $\mathbf{Z}/{\sim} = \{[0],[1]\}=\{\{…,-4,-2,0,2,4,…\},\{…,-5,-3,-1,1,3,5,…\}\}.$
The way I think about the set of all equivalence classes is that one collects all equivalent elements into one set for all elements and obtains the set on the very right in the example. Then one picks a "name" for each of those sets, calling it by one of its members. In the example one has the canonical choices of $[0],[1]$. If I now pick an arbitrary element $a \in \mathbf{Z}/{\sim}$, then there exists a $z \in \mathbf{Z}$ such that $a=[z]$. This is because I can simply call the set $a$ by one of its representatives, in this case $z$ or in the example above $[0]$ or $[1]$. When defining a function it then suffices to define it on all the "names" $[z]$ because I can give each object in $\mathbf{Z}/{\sim}$ one. The function being well defined then comes down to showing that it is independent of the name each object has been given.
Is this a valid way to think about this concept or are there other, perhaps better ways to do so? I am not sure if I am satisfied with the way I would explain it to myself since the "giving it a name" does not really sound that rigorous. I guess one could also view this as a sort of assignment which assigns to every set of equivalent elements a member of it (which is not well defined) and then assigns to it a value such that this process is well defined.
Edit: The following is still not entirely clear to me. When defining a function from a quotient set to another set, one usually defines this in the following way: $$f: X/{\sim} \to A, \ [x] \mapsto a(x).$$ How should I think about this? Do I first choose a (arbitrary) complete system of representatives, define this function for them and then show that it is not dependent of the choice of the complete system, or do I map all $[x]$, $ x \in X$ and then realize that the images of equivalent elements are the same, meaning that the function is well defined?
Best Answer
Frankly, the way you have worded all this seems very sound to me.
The one issue is that one cannot generally expect to be able to pick a "canonical" element. So we generally are satisfied with the ambiguity of the "name" $[x]$ for the equivalence class that contains $x$. Formally, one simply uses that the statements $[x]=[y]$ and $x \sim y$ are logically equivalent, and one remembers (just as you say) to check that all definitions based on such "names" are well-defined.
In fact, it is even somewhat problematical to assert the existence of an assignment, to each equivalence class, of a member of that class. There is a special set theory axiom devoted to the assertion that such assignments exist in complete generality: the Axiom of Choice. However, there are plenty of situations where one can construct a choice assignment by hand without that axiom, and there are plenty of situations where one does not need to bother with a choice assignment.
Regarding your latest edit, in this situation it is not necessary to choose representatives. To define a function $f : X/\!\sim \, \to A$, you can first define a function $F : X \to A$, and then you prove that $F$ has the following property:
You may then write down the formula $$f([x])=F(x) $$ and you are guaranteed that this formula gives a well-defined function $f : X / \! \sim \, \to A$.