Proofs and definitions.

Question

I am a first-year university student and even with help from tutors I have a difficult time understanding proofs. In particular I notices that when using proofs (be it by induction or contradiction) the examples always use definitions. Do I need to memorize these for exams, if so can anyone link me a revision tool to learn these definitions. Thanks.

Answer 1

Maybe professor Sidney Morris series of video lectures about how to write proofs will help you.

Writing Proofs in Mathematics

I really liked them when i started out. Its a good introduction.

Answer 2

Often a proof contains a key idea, which is captured in a definition. Once you fully understand the proof, the definition should be rather natural.

For example, the proof of the Mean Value Theorem by "tilting" Rolle's Theorem (that if $f(a) = f(b)$ and $f$ is continuous on $[a,b]$ and differentiable on $(a, b)$, then $f'(x) = 0$ for some $x \in (a, b)$).

Given $f: [a, b] \to \mathbb{R}$ continuous on $[a,b]$ and differentiable on $(a, b)$, we need to create a function to which we may apply Rolle. We need to tilt $f$ over so that $f(a)$ is equal to $f(b)$, which it might not be at the moment.

So we define a function $g: [a, b] \to \mathbb{R}$ by setting $g(x)$ to be $f(a)$ at $x=a$, and $f(a)$ at $x=b$, and some kind of linear tilt along the way. The correct way to do this is by setting $$g(x) = f(x) - x \times \frac{f(b) - f(a)}{b-a}$$

The rest of the proof of MVT is simply applying Rolle to the function $g$.

This definition has not just appeared out of the blue. It was constructed by a definite series of steps of reasoning. Wherever a definition is made, you should try and understand why it has been made.

Answer 3

Do you need to memorize definitions? Absolutely. Mathematics is, naievely speaking, building theorems from nothing but logic and predefined objects. Without definitions you aren't studying maths. Every question you will be asked will require you to extend a definition, even if it's not directly obvious.

Do you need to memorize the proofs? No, that is a horrible way to learn mathematics and that will only take you so far. You won't be able to pass a senior mathematics course trying to memorize without understanding. Start learning maths correctly from the beginning. Go over each proof carefully, and try to understand the links between each step. It might take you longer but once you understand how a proof hangs together you'll always be able to remember it. You just need to memorize how each proof starts (as some theorems can sound quite similar) and then, because you understand it you will be able to reconstruct it in the exam.

Learning proofs this way will give you a huge advantage in all other areas of the exam, as in struggling to understand a proof you generally begin to gain a much deeper understanding of that section of work. This understanding will allow you to calculate tricky examples that you wouldn't have been able to do without that understanding.

Finally be patient with yourself. Mathematics is exceptionally abstract and our primate brains have evolved primarily to find fruit, so it is no wonder that most of us struggle at first (unless your name happens to be Terence Tao). So you will not quite grasp everything at first, or even at second. The key is to keep practicing. Go over every theorem and try to understand each step. You might not succeed at first but that's fine. Each piece of sincere effort you put in grows the abstract reasoning part of your brain, so that the next time you will find understanding comes just a bit easier. Unfortunately there is no substitute for hard work, but just because you haven't built Rome in a day doesn't mean that you can't build it eventually.

(This book seems fairly interesting and may help)

Answer 4

Yes, you will have to learn definitions.

First learn them by heart (e.g., "a prime is a natural number $>1$ divisible only by $1$ and by itself"). Then again, definitions are not l'art pour l'art, but rather we tend to make definitions for objects or properties that are specifically interesting, perhaps because they occur surprisingly often elsewhere ("A finite cyclic group is simlpe iff it is of prime order") or because they are equivalent to somewhat unexpected other properties ("A natural number $>1$ is prime iff, whenever it divides a product of integers then it divides one of the factors"). Once you've come that far, it will be much easier for you to remember definitions even if you don't remember the complete versions at once ("A prime, well, that's an number divisible only by $1$ and itself ... maybe I should say natural number as otherwise divisibility makes little sense ... and wait, does that make $1$ a prime? I'd better add an exception cause calling $1$ a prime spoils the unique prime factorization theorem")

Answer 5

Most of my exams at that level were open book. So, you don't have to memorize the definitions, you need to know where to find them in the book.

It is a pretty good rule in the early-going to write down the definitions of the terms you are using as you write your proof. It frequently gives an idea of what you will need to demonstrate. If the definition says "for any epsilon there exist a delta...." then you know that you will need to show how to bound delta in terms of epsilon.

Some definitions are not particularly useful as they are written. The definition of a compact space, "every open cover has a finite sub-cover." It took me a while to understand what that actually meant. What is an "open cover", and a "sub-cover", and what is the implication of finite? Closed and bounded is more simple-minded.