I'm trying to understand how to write a proof showing that something is an equivalence relation. I am pretty terrible at this stuff but I am trying to understand and get better. Here is an exercise problem from one of my course books. I'm trying to understand how to write the solutions/proofs.
Problem:
Consider the relation between two sets defined by $S_{1}$ $\equiv$ $S_{2}$ if and only if |$S_1$| = |$S_2$|. Show that this is an equivalence relation.
Solution:
(a) reflexivity: Since |$S_1$| = |$S_1$|, so reflexivity holds.
(b) symmetry: if |$S_1$| =|$S_2$| then |$S_2$| = |$S_1$| so symmetry is satisfied.
(c) transitivity: Since |$S_1$| = |$S_3$| = |$S_2$|, transitivity holds, and the relation is an equivalence.
So, to show reflexivity I can say that something is equivalent to itself? (In this case the magnitude of $S_1$ is equivalent to itself)
With symmetry it can be flipped around right? (a=b then b=a)
With the transitive where does the |$S_3$| come from? and what does |$S_1$| = |$S_3$| = |$S_2$| say exactly? Like I know it's saying the magnitude of S1 is equivalent to S3 is equivalent to S2 but how does that show that this is an transitive?
 A: 
So, to show reflexivity I can say that something is equivalent to itself? (In this case the magnitude of $S_1$ is equivalent to itself)

Specifically: You show that anything in the domain will be related to itself.
$$\forall S~(\lvert S\rvert =\lvert S\rvert)$$

With symmetry it can be flipped around right? (a=b then b=a)

That if any two things are related in one direction, then they will be related in the inverse direction too.
$$\forall S~\forall T~\big(\lvert S\rvert =\lvert T\rvert~\to~ \lvert T\rvert =\lvert S\rvert\big)$$

With the transitive where does the |$S_3$| come from? and what does |$S_1$| = |$S_3$| = |$S_2$| say exactly? Like I know it's saying the magnitude of S1 is equivalent to S3 is equivalent to S2 but how does that show that this is an transitive?

That is not a good solution.  They are (poorly) abbreviating the following:
Transitivity: For any three sets, if the first is related to the second, and the second to the third, then the first will be related to the third.  We can see that this holds when the relation is equality of set cardinality.
$$\forall S_1\forall S_2\forall S_3: \big(\lvert S_1\rvert=\lvert S_2\rvert~\wedge~\lvert S_2\rvert=\lvert S_3\rvert~\to~\lvert S_1\rvert=\lvert S_3\rvert\big)$$
