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Mathematical Proofs

rarbass

rarbass

Jul 3, 2008
1,156
47
#1
Has anyone around here taken a course called Mathematical Foundations? I'm not sure if that's what it's called universally, but essentially you do mathematical proofs.
Using terms like union, intersection, quantifiers, negation, etc.?

I'm not sure how universal the course is. Anyway, if anyone has or anyone knows how to do mathematical proofs well, could you please offer some advice? Something about them is not clicking... :bawl::help:

Again, not sure how universal this course is, but if ANYONE can give me some advice, it'd be much appreciated!
 
#4
rarisgod said:
Has anyone around here taken a course called Mathematical Foundations? I'm not sure if that's what it's called universally, but essentially you do mathematical proofs.
Using terms like union, intersection, quantifiers, negation, etc.?

I'm not sure how universal the course is. Anyway, if anyone has or anyone knows how to do mathematical proofs well, could you please offer some advice? Something about them is not clicking... :bawl::help:

Again, not sure how universal this course is, but if ANYONE can give me some advice, it'd be much appreciated!
Click to expand...

I have done that course but it's no use trying to explain you mathematical proofs in one post + I don't have time and patience to teach you mathematics for hours and hours to come. That is why math teachers exist.

Use Wikipedia?
 
#5
karrot-x said:
Union, intersections, quantifiers and negation are set theory. They are tools used in proofs.

What don't you get?
Click to expand...

I know what the tools are, I was only identifying them in case nobody knew what I was talking about. Set theory is included in our course.

I understand the methods of proving we've learned so far (except for induction, but that's the latest). The only problem is seeing how to come up with a proof for something. I never know exactly what to write down, and even if I come to what seems to be a right answer, it ends up being wrong.

How would you go about looking at a proof?
 
#6
RestInPieces said:
I have done that course but it's no use trying to explain you mathematical proofs in one post + I don't have time and patience to teach you mathematics for hours and hours to come. That is why math teachers exist.

Use Wikipedia?
Click to expand...

I know and this is the sort of answer I would expect anyway. Learning this material, I find, isn't always done best by explaining, but simply understanding.
 
#7
Sorry I can't really offer any help, I think that mathematics is something that must be seen in context repeatedly for one to gain understanding(like any language), and when new material is introduced new types of proofs may need to be learned(I am currently doing representation theory and have no idea what to do proof-wise yet)
 
#8
The course is universal. Writing proofs has been the same for hundreds of years.

Take the first proofs you cover, rewrite them in your own hand. Then do it again. Try to understand what it says. At some point it will click, but it may take a while. The key is understanding why the early proofs are proofs. Do not skip one. Understand each one. Ask why. The more you understand, the easier they get. After while you can look at something you want to prove and see how to proceed.

Sounds like you have the basic course. They get much more complicated. I took one my college called Analysis. We basically proved Calc 1 & 2. I did more work for that course than I had ever done before. Prof was a complete hardass who failed over 50% of the class, and he was the only one who taught it, plus he wrote the book we used. The course was required for a math degree. I hated him, but I learned more in that class than from any other course I ever took.
 
#9
Well, I have a BA & MS in math, so I guess I should weigh in. When I'm looking for the way to attack a proof, there are a couple of general approaches that I think about:

1. Assume the result is false and derive a contradiction. A famous example is the proof that the square root of 2 is irrational. You start by assuming there is a fraction a/b such that (a/b)^2 = 2, and manipulate that equation until you get something that is clearly false.

2. Think about the definition of the thing you're trying to prove, and about any theorems you already have that are sufficient to prove the result. So if you want to prove that all sets with property A have property C, and you already know that all sets with property B have property C, then it's sufficient to show that A implies B.

3. Sometimes you can break it into smaller pieces that are easier. If you need to show that a = b, can you show (separately) that a >= b, and b >= a? Then you're done.

Maybe an even more general approach is to look at the result you want to prove and ask yourself, "what would it mean for this result to be false? " For example if you want to show that x is the only solution to an equation, what would it mean for that to be false? It would mean there would have to be another number y, not equal to x, that solves the equation. OK, so what would that mean? Well, it means that x-y is not zero, and that x/y is not 1.00, so does that help? Does that lead to something that looks familiar or useful?

HTH and good luck.
 
#10
Learning mathematics is something like learning music. You do it, and do it again, and do it again, and then something clicks and you've moved up a level. It's not a smooth learning process - you have to keep banging your head against the wall for a while before it falls down and you get a new wall to bang your new understanding against.

Don't be discouraged if it isn't clicking. It will.
 
#11
I once read a proof that horses have an infinite number of legs.

First, it's a given that a horse has an even number of legs.
In the back, it has two hind legs. In the front, it has fore legs.
Four plus two equals six -- which is certainly an odd number of legs for
a horse to have. And the only number that is both odd and even is
infinity. Thus, a horse has an infinite number of legs. :ninja:
 
#12
taking a first proof-based math course really reshapes your brain. you've got to put a lot of time and effort in it and work with the prof/ta closely. go to office hours every day, work with friends in the class.... treat it like it's a big deal.
 
#13
A proof is not too much different than an algebra problem, except that you may be dealing with entities other than numbers, such as sets. You take a bunch of known facts, use rules (of logic) to create new facts, and again, until you reach the statement that you are trying to prove.

The challenge is figuring out how to decide which facts might apply to the statement that you are trying to prove. Here, language helps. Use your textbook as a reference. Look up the definitions for all of the terms in the proof statement. Look for axioms that use those terms. For many simple proofs, just reading some of those definitions and axioms to yourself will give you an idea for how you might use them.

This process will gradually cause you to commit all of those definitions and axioms to memory.

If you are assigned only a subset of chapter problems for homework, consider the other problems to be for your own use, to study and learn the material.
 

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