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Sorry if this isn't allowed, let me know and I'll delete, but I'm in first year algebra focused on proofs (congruences, complex numbers, etc.) And I was wondering if anyone had any tips on writing final exams on this topic. One of the things I did a lot on the midterm was finish a proof feeling satisfied but I would do something wrong at the start and lose all the marks because I based my proof off of something false.

Another thing is, if I get a true or false question, for example $\forall x \in (2^k + 1: k \in \Bbb N), \exists l \in \Bbb Z, x = 3l$, what should I do to make sure that this is true or false? Or maybe if it says "prove or disprove this statement"

This example is really specific, but for any question where it says disprove or prove, or true or false, how should I know whether to prove it (or that it's true), or to disprove it (or say false). Should I plug in odd numbers, even numbers, negative numbers, positive numbers, etc.. in all the variables? Just go with my gut? Try to prove or disprove right away and see if I get an answer? Use contradiction, and if it's false I know the statement is true and vice versa?

Thanks! (Any other tips are welcome :))

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  • $\begingroup$ This is totally unrelated: try $\ddot\smile$ $\endgroup$ – gen-ℤ ready to perish Dec 13 '18 at 5:43
  • $\begingroup$ I know the linked duplicate is written with a specific subject in mind, but the advice given in the solutions addresses your question, is good, and is not subject-based. $\endgroup$ – rschwieb Dec 13 '18 at 14:27
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This example is really specific, but for any question where it says disprove or prove, or true or false, how should I know whether to prove it (or that it's true), or to disprove it (or say false). Should I plug in odd numbers, even numbers, negative numbers, positive numbers, etc.. in all the variables? Just go with my gut? Try to prove or disprove right away and see if I get an answer? Use contradiction, and if it's false I know the statement is true and vice versa?

It just ... depends. I'm not saying that to be a jerk, it just is how it is.

After a bit of experience, you sort of get a "gut instinct" for things that might be true or false, just by looking at the statement, and you will just want to jump in there and try to work with it. Does it look too good to be true? Does something just strike you as "off," even if you can't find what exactly? These are signs that it might be best to try to disprove the theorem in question - counterexample is usually best and is a valid means of disproving something.

Remember that it only takes one counterexample to break a theorem. If it says it holds for all integers and there are no restrictions (e.g. it doesn't say "integers greater than (this)," "even integers," etc.), if it doesn't hold for any particular integer, it fails - because it doesn't hold for all the integers it claims to hold for.

In that sense, that it's so easy to break a theorem, if you're honestly unsure about the status of a theorem, consider a few quick simple examples. (What examples are nice depend on the context.) If a few simple examples work, then it might be valid. If they break it, well, good on you. It's much quicker and easier to disprove something, generally, than it is to prove it.

Trying to find a contradiction - suppose the theorem is true, and seeing if it breaks something else known to be true - is also valid. It might be more convenient, too, than breaking it via counterexample depending on the context.

If the question is just "is this true/false," I'd honestly assume "true" if I couldn't make a counterexample. If there's time later in the test, I'd come back to it and investigate more thoroughly, but odds are if it's not easily broken, it's not easily proven, and you don't want to use more time/work than necessary since it'll hurt your other areas of the test. (Even moreso if you get stuck.) Of course, this is all on the assumption that you aren't expected to give the proof if it's true.

So suppose you can't break it easily. What then? It still might not be true, but how do you know? Some theorems have held for various examples, but only have their counterexamples for extremely large integers, or very extreme circumstances; can't expect to handle such cases very easily.

All you can really, logically do at this point is try to prove it. Which is a lot harder and depends on the context. I can't really help you there, I'm hit and miss on proof-writing. It's ultimately about knowing what definitions and theorems to use and when. It always will help to keep in mind what you are given, and what you need to show as well. A lot of experience/exposure and practice - try working exercises from a textbook, or explaining each step in a proof you know to yourself to really cement the techniques and underlying logic in your mind.

And always start with the hypothesis. Most theorems are in a "if-then" format, even if it's not immediately clear. Knowing your starting and end points are absolutely essential, that cannot be emphasized enough in proof-writing. Write it down clearly as early as you can in the proof, even: "If (X), (Y), (Z), then (what you want to show)."

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