I'm doing some Maths revision, as I've got my GCSE exams coming up in a few weeks. I was hoping for some tips on mathematical proof. I understand proof when I'm shown it, but can rarely produce it myself.
What is the minimum that you can explain? And can you simply say that (for example) "due to the proven Circle Theorem, bla bla bla.."? Or must you prove everything?
Here's an example that I was hoping that you lot could give me an example on:
Prove:
2n - 1 = an odd integer
Now, I wouldn't really know how to *prove* this. I could explain it: 2 multiplied by any number is always an even number, 1 subtracted off an even number always resorts in an odd number... But, that's not enough 'proof' surely?
You see.. that makes great sense, but I would never have thought of that. lol. :-(
When you're given something that you need to mathematically prove, what is your first step? (ie: what do you look for, or think?)... Or does it *completely* differ on the question?
I really don't know and in many cases (when I ws in school) I too was stumped.
In some cases I will assume the assumption is true and see if that leads to a contradiction. Basically what I did here. I assumed it was true and therefore (2n-1) would be divisible by 2 and when trying to show that I got a contradiction.
Always think about what you know independent of what you are trying to prove. In this case that even nuumbers are divisible by 2.
Think about this one Prove the sum of 2 odd numbers is is an even number.
mlmcc
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I take it that you are not so much interested in the proof of your example as in what constitutes a proof in general.
You start with a set of postulates and then build on them. For any given proof you just have to go back to a previously proven proposition, not all the way back to the postulates. The result is an area of knowledge that may be useful. Examples: Euclidian geometry, set theory, algebras, spherical geometry, etc. New areas of study are opened up when new postulates are selected. From a logic stand-point it can become quite difficult because Gödel has shown that one cannot construct a consistent closed system.
However all this is probably too general for your purpose
From a practical exam point review a small book on formal Logic
Then get an elementary Euclidian geometry book. It will be replete with excellent examples of proofs. Study it with the goal, not of learning geometry, but of seeing formal proofs at work.
I can think of a few things that might be getting in your way:
1 logic
2 a negative belief about your mathematical abilities (this is very common)
3 not enough tools, i.e. not knowing enough facts from the area of math your proof is in
4 lack of experience with doing proofs
Have you tried this? Start with an assumption p, and see where it gets you, in other words, what inferences you can draw from it. When you reach some conclusion that looks nice, call it q. Voilà, you've made your own theorem p implies q. And you know how to prove it.
You could do this with non-mathematical ideas too, for practice.
The best way to read a math book is with pencil and scratch paper handy. Whenever the author presents a proof, see if you can convince yourself of the conclusion before reading the author's proof.
Often the first thing to do when making a proof is to write down any definitions you think might be relevant. You might also find it helpful to make a sketch or consider an example, as ways to develop some intuition about what's going on.
An even number is divisble by 2 with no remainder. Apply the test
Try (2n-1)/2
(2n-1) / 2 Distributing the division
(2n)/2 - 1/2
n - 1/2
Know 1 is not divisible evenly by 2 therefore (2n-1)/2 = n R1
so 2n-1 is not even
mlmcc