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Im reading "How to prove it" by velleman, and in a lot of the exercises you need to prove some theorem is true. How well should I understand why the theorem is actually true before proving it? For me it's pretty easy to prove something just by following the rules, but idk if that's smart.

For example:
Suppose $R$ and $S$ are symmetric on $A$.
Prove that $R\circ S$ is symmetric iff $R\circ S=S\circ R$

Here is my answer: enter image description here

Before writing the proof, I didn't take the time to fully understand why its true, but I could still write the proof by following the rules.
What I mean by not fully understanding it, is that if someone asked me:
"If you have two relations on a set, why is it true that their composition is symmetric if and only if their composition is commutative?"
I could only explain it by doing a step by step proof.

I feel like that kind of says I don't understand relations, compositions and symmetricity. Because with something like 2+2=4 I obviously understand why its true without writing a proof.

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    $\begingroup$ But can you prove that 2+2=4? $\endgroup$
    – Asaf Karagila ♦
    May 28 at 16:48
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    $\begingroup$ Your question is clear and imo very interesting, but too general to get an appropriate answer, I think. As for you example, the "reason why" is that a relation $R$ is symmetric iff it is equal to its opposite $R^{op},$ and that $(R\circ S)^{op}=S^{op}\circ R^{op}.$ $\endgroup$
    – Anne Bauval
    May 28 at 16:53
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    $\begingroup$ Sometimes it's probably not possible to recast the result to something that allows you to understand why it is correct, such as the product of $3384$ and $2187$ is equal to $7400808.$ Other times such a reduction may not occur until you learn more, such as what @Anne Bauval said in a comment. Indeed, a large part of the "theory-building" part of mathematics is seeking such reductions (see The Two Cultures of Mathematics by Gowers and this search). $\endgroup$
    – Dave L. Renfro
    May 28 at 18:15
  • $\begingroup$ You asked: "How well should I understand why the theorem is actually true before proving it?" I would say there is no need to understand why the theorem is true before proving it. Of course, it is nice to have an intuitive understanding of why a theorem is true. But if you insist on achieving that understanding before you write your proof, then the only theorems you will ever be able to prove are those that are so simple that you can see intuitively why they are true before working out the proof. $\endgroup$
    – Dan Velleman
    May 30 at 15:39
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    $\begingroup$ By the way: Nice proof. $\endgroup$
    – Dan Velleman
    May 30 at 15:39

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Some theorems have profound proofs, and require developing a clever idea. And some theorems follow very directly from definitions, and their proofs are basically bookkeeping*. Your example seems to be of the latter type, and I wouldn't worry about the way you're approaching them.

Of course, even trivial facts can be seen in a more general setting, and one can get a bit more insight viewing them that way. That's what AnneBauval's comment does. But that usually comes later, and you're not expected to come up with that insight the first time you encounter an idea.

*In algebraic topology there are some theorems whose proof just say "diagram chase", which means "At each step there is only one obvious thing to do. If you do that, everything does in fact work out.".

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