When asked to show that $X=Y$, is it reasonable to manipulate $X$ without thinking about $Y$?

Question

When I'm asked something like "show X is equal to Y", I first try to manipulate what I know (X) into the result (Y). A lot of the time, I do not investigate the result I'm trying to conclude with. I feel like I should be approaching the problem without a goal in mind (zero knowledge about the result) and that I should try to prove the result in a way that it was mostly likely proved to begin with: in an exploratory way. What ends up happening is that I just struggle pathetically without having any sort of direction.

The opposite approach is to investigate what it is I'm trying to prove, find its consequences perhaps, and to keep that result in mind and have it guide me. But I really feel like that's cheating. My main question is: is it?

Answer 1

This is a bit long for a comment, so posting as an answer.

No, it is not "cheating" to have the goal in mind when trying to prove a result. A given set of hypotheses may have many possible consequences, and very often, simply trying to explore the consequences may lead to aimless flailing around, without any progress towards the result — in other words, what you observe is entirely natural, and you should not be worried. Having the goal in mind focuses the direction of your explorations, and is helpful. Everyone does this. If you know what you want to prove, you might as well use it — there is no need to unnecessarily constrain yourself to ignore it. Even with research mathematicians, though their explorations may induce them to guess/conjecture a certain result, when trying to prove the result, they too will use everything about the result to aid their proof.

Having said all that, there are some things you can do after being done with the proof, to maximize your effectiveness. You can take a step back, examine the structure of the proof, see what steps are specific to the goal and what are natural consequences of the hypotheses, and try to restructure it as a path from X to Y (if it makes it cleaner).

This is exactly parallel to the issue of proof by contradiction, about which I've written in this answer before. The question is whether, when trying to prove that $P \implies Q$, you should just work forwards from $P$, or use both "$P$" and "not $Q$" and work towards a contradiction. The latter is strictly easier (you have more information), and it's undesirable to cripple yourself by avoiding it: Hilbert said

Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists.

So when trying to prove the result, do use everything you have at your disposal, but when trying to write down the proof (or just explain it to yourself), take another look at it and see if restructuring it makes it cleaner / more illuminating, so that it can help you in future.

As a concrete (though perhaps trivial) example, see this question, something about simple algebra and inequalities. I wrote an answer, whose first revision was almost entirely as I thought (using both the hypotheses and desired conclusion, and working from both ends and trying to make them meet), but after writing it down, I was able to slightly see better what was actually happening, and condense the working to the second revision. Not a very good example, but I hope you get the idea.

Answer 2

If you are asked to "show that $X$ is equal to $Y$", then you are obviously in a learning situation, not doing mathematical research. In a learning situation there are clear reasons to set specific goals; asking a question like "find out something interesting about $X$" would not only be a much harder task, it would also be very hard to grade, as some will come up with nothing, some with interesting but false results, and others with things that might look interesting at first glance but which are completely trivial for a more experienced person; it is just too vague about where one puts the bar. Having a well chosen (by the instructor) concrete goal both poses a more concrete challenge and may avoid wandering down unintended paths after taking a wrong turn. It also unfortunately stimulates a form of attempt at cheating, consisting of turning a bit aimlessly around $X$, and a also bit around $Y$, and finally inserting an unmotivated equality between two randomly chosen points from either side to connect the ends, in the hope that the person grading won't notice (and indeed if carefully done, this equality won't even be false).

By the way, the formulation I cited above is considered a level$~$5 question in Concrete Mathematics, section 3.2, where previous levels are 1: "show $P(x)$ for a concrete $x$", 2: "show $P(x)$ for a all $x\in X$", 3: "prove or disprove that $P(x)$ for a all $x\in X$", and 4: "find a necessary and sufficient condition $Q(x)$ for $P(x)$".

In a research situation things are much more complicated of course, but you won't be able to tackle them without a lot of experience obtained in learning situations. What mostly happens is that one wishes to understand some interesting, nonobvious, question in terms of conditions that are more transparent to evaluate; level 4 above, with the requirement that $Q$ have some simple form. But in the end, once you've guessed what $Q$ should be, you are left with finding a proof for a concrete goal, so not so far from what you saw in the learning situation (however without the certainty that the goal you set is attainable, so one is really down to level 3 only).

Answer 3

Take a look at Pólya's classic "How to prove it", it explains clearly (in a school setting problem, mostly) how to solve problems, general strategies to use. Highly recommended.

Hammack's "The Book of Proof" might come handy. Also look for lecture notes for introductory mathematics classes, there are some on "how to prove" (or which include the matter).