Trouble understanding equivalence relations and equivalence classes I've read up a bit on equivalence relations and equivalence classes, but I'm a bit unsure on the whole concept.
From what I've read and equivalence relation, ~, between two mathematical objects $a$ and $b$ is a binary operation satisfying reflexivity, symmetry and transitivity.
Does this mean, that if two objects satisfy a given relation that satisfies the above properties, then these two objects can be considered equivalent (with respect to this relation)? For example, in physics if I have two Lagrangians that differ by a total derivative then they both lead to the same equations of motion, and thus describe the dynamics of the same physical system (correct?!), so would it be correct to say that these to Lagrangians are equivalent under the relation that they lead to the same equations of motion, i.e. $$\mathcal{L}:=\mathcal{L}+\frac{df}{dt} \quad\iff\quad \delta S_{1}= \delta S_{2}$$ where $S_{1}=\int \mathcal{L}\;dt$ and $S_{2}=\int [\mathcal{L}+\frac{df}{dt}]\;dt$.
Using the same example, could one then define a Lagrangian for a particular theory as the equivalence class of Lagrangians that lead to the same equations of motion?
I guess my real issue is that the definition that I've read seems very abstract and I'm unsure how to relate it to specific examples?! I get that two objects that satisfy a particular equivalence relation aren't actually equal, but is the idea that if we define a particular equivalence relation, we can then form a set of objects that satisfy this relation, which we call an equivalence class, and then if we have a particular problem that utilises this relation, then we can use any element of the equivalence class as they all satisfy this particular relation, so relative to a problem involving this particular relation, they are equivalent. (Is it kind of like a restricted equality relation? For example, saying that $y=x^{2}$ for $0<x<1$)?!
Sorry if this isn't worded very well, just trying to explain what my thoughts are on the subject so far.

Here are two examples :
$1 - $ Consider the relation $\equiv$ ( an equivalent relation), then
$$a \sim b \Leftrightarrow a\equiv b \mod 2 $$
That is, $a$ and $b$ will be in the same class $\overline{a}$ if their remainders of the division by $2$ are the same. For example $4$ and $6$ belong to the same class, which we are going to choose a representant $0$, because
$$6 = 3 \cdot 2 + \color{red}{0} \ \ \text{and} \ \ 4 = 2 \cdot 2 + \color{red}{0}$$
then we say $\overline{4} = \overline{6} = \overline{0}$. If we think, there are two distinct classes: $$\overline{0} = \{x \in \mathbb Z ; x \equiv 0 \mod 2, \text{$x$ is even}\}\ \  \text{and}\ \  \overline{1} = \{x \in \mathbb Z ; x \equiv 1 \mod 2, \text{$x$ is odd}\}$$
The set of all classes is
$$\mathbb Z_2 = \{\overline{0}, \overline{1}\}$$
$2-$ Consider the relation
$$(a,b) \sim (c,d) \Leftrightarrow ac = bd $$
This equivalent relation gives us the fractions, that is the filed of fractions of $\mathbb Z$. Similarly we choose a class representant for example,
$$\frac{1}{2} = \frac{2}{4} = \frac{3}{6 } = \cdots$$
we choose $\frac{1}{2}$ to be the class representant. Notice that $\mathbb Q = \{ \frac{a}{b} ; a,b \in \mathbb Z, \ \ \text{where}\ \  b \neq 0\}$ is the set of all classes.
