Learning Math Proofs in College for New Students

[Sheneron]

Hi,
I finished calculus 1 in college this past year, and I was reviewing it in the summer to make sure I understand it and have a solid foundation for when continue taking math classes this upcoming year. My math has always been lacking a little from my high school past where I never paid attention. I remember doing small proofs in geometry in high school, but that was in my first year, I didn't learn it, and of course I don't remember them now.

But that brings me to my question. This past year during calculus we didn't have to prove anything really. We would just be told what something was and then worked examples. However, now when I am going back to look over things I will see a theorem accompanied by a proof. Most of the times these proofs seem a little over my head, and I would never be able to make a proof such as those from scratch. I am wondering whether I should know how to prove things a little bit now, or do they teach you proofs in a higher math class?

Also, if I am indeed behind, what is a good way to go about learning proofs, and some good book suggestions? And if I am not behind would it be a good idea to start learning now?

 

[arildno]

Hi, Sheneron!
The first thing you need to know about proofs are that they are not some mythical, etheric beings whose purity and rarefied essence only the professional mathematician can comprehend.

Instead, you should think of proofs somewhat analogously to saying&showing that a game of football was played according to the rules. (Or not, if a rule-breaking occurred!) .
Of course, lots of wildly different football matches can be shown to have been played according to the official rules, and similarly, lots of different maths can be played, according to ITS rules, and proofs are merely the demonstration that a particular math game did, indeed, play out according to its underlying rules.

Is this analogy somewhat enlightening?

In particular, do you see the central importance of, albeit arbitrariness of, the underlying RULES (in maths called axioms)?

 

[Sheneron]

I think so.

But, if someone were to say, hey prove to me L'hopital's rule is correct. I would say that I can't but mathematicians are very trustworthy. I am wondering if I should know how to prove things by now. Intuitively the things make a lot of sense, but I can't explain why something is the way it is other then, well they said so, and it makes sense.

Also, this is a little off topic but I was looking at a proof, yesterday, of the limit laws. Each proof of a limit law used another limit law to in its proof. For instance, let's say I wanted to prove a b and c. And I said a is true because of b, b is true because of c, and c is true because of a. Would that mean a is true?

 

[Howers]

Hi,
I finished calculus 1 in college this past year, and I was reviewing it in the summer to make sure I understand it and have a solid foundation for when continue taking math classes this upcoming year. My math has always been lacking a little from my high school past where I never paid attention. I remember doing small proofs in geometry in high school, but that was in my first year, I didn't learn it, and of course I don't remember them now.

But that brings me to my question. This past year during calculus we didn't have to prove anything really. We would just be told what something was and then worked examples. However, now when I am going back to look over things I will see a theorem accompanied by a proof. Most of the times these proofs seem a little over my head, and I would never be able to make a proof such as those from scratch. I am wondering whether I should know how to prove things a little bit now, or do they teach you proofs in a higher math class?

Also, if I am indeed behind, what is a good way to go about learning proofs, and some good book suggestions? And if I am not behind would it be a good idea to start learning now?

Depending on your major, proofs may or may not be a part of your curriculum. If say, you are an engineer or a applied scientist (ie. chemist, biologist) you probably won't see any proofs unless you do linear algebra. If you are a mathematician or a phycist, proofs will be abundant in upper year math. It seems your calc course was the humanities/science type, ie. something out of stewart where they don't really expect fluency in math.

Proofs are difficult when encountered for the first time because of how condensed they are and because of the un-intuitive techniques employed. They get easier with time. I remember too the limit laws... I thought there was no way in hell I would ever pull something like that off. But later I learned it was just following the definition of a limit and finding the existence of some delta (just like algebra!). Very easy and mechanical, albeit clever.

The best way to get started is to get a book called "How to Prove It" by Velleman and do atleast the first 3 chapters. And learn induction somewhere. This book will give you the logic behind proofs and some of the most common techniques. Supplement this with a book on elementary linear algebra from the library that proves properties of vectors and matrices. It will take you ~3 months before you understand proofs. The key thing now is to re-read them many times until you have a clear cut picture of why they work. Proofs won't be taught in upper years unless your school offers a course in it. All forms of algebra will assume you will "pick up" on proofs, a method I do not reccommend.

 

[arildno]

I think so.

But, if someone were to say, hey prove to me L'hopital's rule is correct. I would say that I can't but mathematicians are very trustworthy.

A good enough reply in the sense that it is impossible for a single individual to validate all statements that have being made through rigorous proofs, or at least, for non-mathematical issues, study and evaluation of the empirical record.
For example, if an elevator expert has given his certificate of approval for a particular elevator, we do not bother to backtrack and reiterate his inspection (let alone know how that ought to be done!)
A certain level of trust between humans is indispensable, and hence, rational.

For elementary ideas like L'Hopital, though, you should understand the proof, if you are to achieve some competence within the field.

Also, this is a little off topic but I was looking at a proof, yesterday, of the limit laws. Each proof of a limit law used another limit law to in its proof.

Correct.
They pile up on top on each other, based upon the bedrock of DEFINITIONS of the limit, and how these are related to the even lower level axioms.

For instance, let's say I wanted to prove a b and c. And I said a is true because of b, b is true because of c, and c is true because of a. Would that mean a is true?

Caution:
You may prove that a, b and c are logically equivalent statements, but it doesn't follow from this that they must be "true". Rather, what you may show is that their truth values rise and falls identically.

When proving that b implies a, that means: IF b is true, then a is necessarily true as well.
Furthermore, if c implies b means that IF c is true, then b is true as well.
The last link shows that a implies c, which means, specifically, that a must be logically EQUIVALENT to, say, b, since you have established:
IF a true, then c true, therefore b true (c true implies b true, remember!)
But this is nothing but a fancy way of saying IF a, then b!

In addition, you had at the beginning IF b, then a, whereby together, you have shown the equivalence relation between a and b, namely:
a true IF and ONLY IF b true.


But a,b,c might still all be untrue statements, but they have to have the same truth value (either all false or all true)