Working with Equivalence Classes and Quotient Sets

I have a doubt about working with equivalence classes and quotient sets. The definition that I know, is that given an equivalence relation $\sim$ on a set $A$, the set of all elements of $A$ equivalent to a certain element $x$ is the set $\left[x\right]\subset A$ which is called the equivalence class of $x$. The set of all equivalence classes is then the set $A/\sim$ called quotient set.

Well, I understand this. However I’m a bit confused when working with this, because the set $A/\sim$ is a set of sets right? So if $x \in A/\sim$, $x$ is not an element of $A$ but one entire collection of such elements. However, there’re texts that it seems that the author is thinking of $x \in A/\sim$ as just and element of $A$.

For instance: in the context of manifolds, when defining tangent vector as equivalence classes of curves, one doesn’t think of an element of the tangent space as a set of curves. Or on the context of multilinear algebra, we define the tensor product of vector spaces using one quotient, but we don’t think of the elements as sets of vectors.

I’ve heard that “we think of an element $x \in A/\sim$ as an element subjected to the relation $\sim$”, however, we are thinking of $x$ related to what element of $A$ ? I really didn’t get the way that we should work/think about equivalence classes and quotient sets.

Thanks in advance for your help, and sorry if the question is to silly.

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Usually when you talk about an equivalence class in a quotient set, you refer to it by using a representative, i.e. an element of that class. For example, consider $\Bbb Z/n\Bbb Z$, the integers modulo $n$. This is defined as a set of sets, but usually you just identify it with $\{0,\dotsc,n-1\}$.

It might be (and usually this is the case) that all elementary mathematical objects are, in fact, sets. For example take the real line $\mathbb R$. You know that $\pi$ is an element of $\mathbb R$. But in many assiomatization $\pi$ is itself an equivalence class (namely the set of all sequences of rational numbers converging to it). This might sound strange but doesn’t change our imagination of $\pi$ being a number or a point in the real line.

Actually it might be that all mathematical objects are sets. For example one can define natural numbers as $0=\{\}, 1=\{0\}, 2=\{0,1\}\dots$