I am familiar with all 3 of the entities I have listed in my question. I know the definitions of "reflexive", "symmetric", and "transitive". However, I am afraid I do not mechanistically understand the "flow" of how we ultimately generate equivalence classes from a particular relation that exhibits the 3 properties of equivalence.
To help illustrate my confusion, consider the following example:
$S=\{1,2,3,4,5,6\}$
Let $R_1$ be a relation on $S$ such that $x-y$ is divisible by $3$
So, firstly, from what I understand about relations, I am going to find all of the order pairs that satisfy this (these ordered pairs are a subset of the cartesian product $S$ x $S$).
$R_1 = \{(1,4) (2,5) (3,6) (4,1) (5, 2) (6, 3)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)\}$
Ok cool. These are all of the ordered pairs that "satisfy" or "make the $R_1$ relation true".
For this given relation, I can observe the following:
1 - The reflexive property is satisfied because of the presence of $(1,1), (2,2),\ etc$
1st question: If, for example (6,6) was not in this set, $R_1$ could not be deemed reflexive because $(3,6)$ and $(6,3)$ are present, correct? (i.e. because the element "6" shows up as as an ordered pair, $(6,6)$ MUST show up as well in order to declare this relation reflexive)
2 - The symmetric property is satisfied because of the presence of $(1,4) \& (4,1)$, $(2,5)\&(5,2),\ etc$
3 - The transitive property is satisfied because...
2nd question: I actually do not immediately see why the transitive property is satisfied (I believe that the transitive property should be satisfied because the "congruence modulo n" relation is an equivalence relation...and I'm fairly certain that the relation $R_1$ that I described is of that form). Is it just because my set is too small to see the transitive property in its stereotypical form?
So, assuming that this relation IS an equivalence relation (I believe that it is...for the reason mentioned above), I really do not understand how we go from this single set of ordered pairs to equivalence classes. From example videos I have seen, I know that a set of integers mod 3 will create three equivalence classes...namely, the integers with remainder 0, 1, and 2 when divided by 3.
3rd question: However, I do not really understand, mechanistically, how we "separate" these ordered pairs. All of the ordered pairs are initially grouped together. How do we decide, from this initial $R_1$ set, which ordered pairs belong to which equivalence class? Obviously, if you know how mod 3 works, you could sort of intuit that 1 and 4 go together because
$1 mod (3) = 4 mod (3)$
...however, if I knew nothing about how $mod (3)$ worked, how would I know how to make the appropriate partitions?