How to determine the scope of a proof?

Question

When given exercises for my coursework, I often encounter the problem of not knowing how pedantic to be in my proofs. Some seem to be statements so trivial that I'm forced to question whether or not the exercise is allowing me to use basic operations that I take for granted. For instance, I have the following exercise tonight:

Given $1 < b\in\mathbb{R}$, and $r = \frac{m}{n} = \frac{p}{q}$ prove that: $$(b^m)^\frac{1}{n} = (b^p)^\frac{1}{q}$$

I dont' know if I'm allowed to use a statement like $(x^a)^b = (x^{ab})$ in this context.

I've had this situation go both ways. I've taken the option of writing a page long proof, and had the professor tell me that I expanded on it too much. I've also gone the route of using statements like the above to make it a few scant lines, only to have the professor tell me I wasn't allowed to use a statement like that.

To be clear, I don't want help with this particular exercise. I'd like to know if anybody has general guidelines for knowing which theorems one is allowed to use for a given exercise.

Answer 1

This is a very interesting question and this answer might be a surprise to you: it depends on your other exercises. That is, if you answered correctly other exercises, the teacher will be inclined to accept a short proof, even if some intermediate steps are missing, as long as he or she is convinced that you master these intermediate steps.

Let me explain this idea in the context of your example (in which I suppose that $m$ and $p$ are nonnegative integers and $n$ and $q$ are positive integers). Suppose you start your proof by writing

It is easy to prove by induction on $b \in \mathbb{N}$ that, for all $a \in \mathbb{N}$, $(x^a)^b= x^{ab}$.

If you just did correctly a few exercises involving induction, and if the rest of the argument is correct, this is likely to be accepted. However, if you previously did major mistakes on some induction argument, you may well receive a nasty comment like "if this is easy to prove, why don't you do it?"

That being said, when you detect that an exercise is an application of a theorem of the previous lesson, it is best to focus on this and explain how you want to apply the theorem. This involves carefully justify that the hypotheses are filled. Short proofs are usually better, but still need to be fully justified. Moreover, you have to understand the purpose of the exercise. For your example, it is likely that you are not allowed to use the formula $(x^a)^b= x^{ab}$, with $a, b \in \mathbb{Q}$, and that a more detailed proof is needed. But you can still write

The result would immediately follow from the fact that, for all $a, b \in \mathbb{Q}$, $(x^a)^b= x^{ab}$.

and then proceed to the longer proof.

Answer 2

You have to know the following:

1. Your audience. Proofs are almost never machine-like automatons in two-column format. They're written by people for people, and as such are pieces of rhetoric. Therefore, you must know your audience.

2. Your math. Obviously, the proof needs to start from the starting-point and get to the finish line, and the steps in-between need to be valid.

3. What you are allowed to assume, and the rules of inference available to you. Different systems of logic, for example Copi's 19 Rules versus Natural Deduction, will have different ways you can get from one line to the next. As for what you're allowed to assume, that is universally held to be only those theorems that have already been proven in your course of study. Otherwise, you are guilty of circularity (a logical fallacy).