The author of my book simplifies his solutions to an extent that I am uncomfortable with, so are my solutions to homework over doing it?

Question

This question can be summarized as: How explicit does one need to be when writing proofs? To what extent can one implicity write a proof safely?

The first chapter of our text in elementary discrete math is very brief in the solutions. As an example from the text:

If

$C = \{1, 3\}$ and $A = \{1, 2, 3, 4\}$

by inspection, every element of $C$ is an element of $A$. Therefore, $C$ is a subset of $A$ and we write $C \subseteq A$

So when I get to homework that asks: Show, as in the previous example, that $A \subseteq B$.

$A = \{1, 2\}$, $B = \{3, 2, 1\}$

I solved this myself but, first, here is the text's solution:

Let $x \in A$. Then $x = 1$ or $x = 2$. In either case, $x \in B$. Therefore $A \subseteq B$

Given that sets are the first section of the first chapter, and in fact this is all covered in the first dozen pages, we obviously have not talked about proofs, or what flavor of statements take the form of expressions like $x = 1$ or $x = 2$. So I am unable myself at this point to critique the author on his proofs at such an early stage.

My question, is the author simplifying this process dramatically and therefore showing a thorough proof is undesirable at this stage, and in practice unexpected? Or has the author shown a completely valid proof given that the question itself is very much simple (perhaps I should be as explicit as the author, or the question asked)?

Consider my eccentric proof to the same problem:

$\forall x(x \in B \rightarrow (0 < x \leq 3 \land x \in \mathbb{Z}^+))$

Assume $x \in A$, then $x \in \mathbb{Z}^+$ and $0 < x \leq 2 < 3$

Therefore $\forall x(x \in A \rightarrow x \in B)$

Therefore $A \subseteq B$

Is this over doing things, as the author's solutions suggest? I really need to ask my teacher tomorrow to know in terms of the class assignments, but I would really like to know if the community agrees or disagrees and to what extent when it comes to being abrupt in the proof writing.

Often in basic algebra classes, for example, the author will simplify evaluating or solving larger expressions, at many times skipping two or more steps that could have been written explicitly, but it is assumed that one could follow logically without seeing those steps. So another way of asking my question is am I myself skipping such steps, despite my attempts to be explicit, and is it, synonymously with many basic algebra solutions, a waste of time to be overly explicit?

Answer 1

These are very straightforward solutions. When showing one set is a subset of another, all you need to show is that $x\in A$ implies $x\in B$. So in your exercise, $x\in A$ implies $x=1$ or $x=2$. The fact that $1$ and $2$ are both in $B$ implies that regardless of what $x$ is, it necessarily must be in $B$.

Later in the textbook, you may see much more implicit "proofs" where the author gives you a sketch of what to do while leaving the tedious details to you, the student.

When it comes to homework, you should certainly have your specific teacher/professor critique your work. What may be blatantly obvious to you may not be so obvious to another person, or your professor could want you to fully justify every detail for the mental exercise.