What are proofs in mathematics like?

[SrVishi]

Hello, I am a senior in high school wondering if I should major in mathematics. I am developing a strong interest in the subject and am currently enjoying and doing well in my AB AP Calculus course. The problem, however, is that I have read in many places (such as on these fantastic forums) that university mathematics is much different than what is taught in k-12. I hear that it is much more proof based and that these proofs are pretty much the major focus of undergraduate mathematics. My question is, may anyone provide an example of a typical proof question or problem that they would encounter in their studies? Any helpful and related responses would be greatly appreciated!

 

[espen180]

Proofs come in many forms. There is no "typical" proof as far as I am aware.

What kind of problems you encounter depend on what you study, and at what level. In your first couple of courses, you will probably do a lot of computation and little proof writing. Your later classes may be virtually all proof and no computation.

To answer your question, in an analysis class a few plausible proof problems might be:
- Show that if a function is differentiable at a point, it is also continuous there.
- Prove the chain rule.
- Derive the formula for integration by parts.
- Derive the substitution rule for integration.
etc...

 

[VertexOperator]

I think a "typical" proof is for example proving the irrationality of numbers like pi, e, sqrt2, series proofs, inequalities etc.

 

[Number Nine]

You might try the Wiki article on the Fundamental Theorem of Arithmetic, which is one of the canonical introductory proofs (so it's quite accessible). Don't worry if it seems difficult; University mathematics is designed to introduce you to these kinds of arguments, so you obviously don't have to be proficient with them now.

 

[WannabeNewton]

Prove that the following function $f:\mathbb{R}\rightarrow \mathbb{R},x \mapsto \begin{Bmatrix} \frac{1}{q} \text{ if x is rational, } x =\frac{p}{q} \text{ in lowest terms} & \\ 0 \text{ if x is irrational} & \end{Bmatrix}$ is continuous in the irrationals and discontinuous in the rationals. See Spivak's chapter on limits (ch. 5) for a discussion of the map, in his single variable calculus book; if you can prove the continuity properties of this map then you pretty much have epsilon delta proofs down as far as Spivak's Calculus is concerned :D.

 

[micromass]

A good thing to do s to buy Spivak's Calculus book. The first few chapters is filled with really nice problems that are actually challenging and proofy.

Some nice problems for you might consist out of proving elementary facts. For example, show that if $x^2 = y^2$, then $x=y$ or $x=-y$.
Or show that $\frac{x}{y}\cdot \frac{a}{b} = \frac{xa}{yb}$.
If your HS education was like mine, then you probably had to memorize these things, but you never really proved them. A good exercise for you would be to prove things like this.

Of course, you shouldn't start thinking that mathematics is all about giving proofs for facts you already know. But proving things in a context that you are familiar with has many advantages. Only if you know how to prove elementary facts, only then can you move on to more difficult and abstract stuff.

Some more difficult things would be like: find a formula for

$$1+2+4+8+...+2^n$$

and prove the formula is correct.

 

[Fredrik]

Here's an example:

Definition: For all real numbers a,b, the set of all real numbers x such that a<x<b, is called an open interval. This set is denoted by (a,b).

Definition: A sequence $x_1,x_2,\dots$ of real numbers is said to be convergent if there's a real number x such that every open interval that contains x contains all but a finite number of terms of the sequence. Such an x is said to be a limit of the sequence.

Theorem: Every convergent sequence of real numbers has exactly one limit.

Proof: Let $x_1,x_2,\dots$ be an arbitrary convergent sequence. The definition of "convergent" implies that this sequence has at least one limit, so it's sufficient to prove that it has at most one limit. We will do this by deriving a contradiction from the assumption that it has two different limits. So suppose that x and y are limits of this sequence, and that $x\neq y$. Define r=|x-y|/2. Since x is a limit of the sequence, the open interval (x-r,x+r) contains all but a finite number of terms. This implies that the open interval (y-r,x+r) contains at most a finite number of terms. (Note that r was chosen to ensure that this interval doesn't overlap the other one). This contradicts the assumption that y is a limit of x.

 

[zooxanthellae]

If you have not seen proofs before I strongly recommend Daniel Velleman's "How to Prove It". Spivak is a great book for bridging calculus and analysis but it would not be my first recommendation for learning about proofs (that would be Velleman).

 

[micromass]

If you have not seen proofs before I strongly recommend Daniel Velleman's "How to Prove It". Spivak is a great book for bridging calculus and analysis but it would not be my first recommendation for learning about proofs (that would be Velleman).

I disagree. I never really saw the point of proof books. When I see people who worked through proof books, then their proofs are always pretty weird. You always have to spend time unlearning from what you learned from proof books.

The best way to learn proofs is to get a proofy book like Spivak or Lang and work through it. Solve the exercises the best way you can and then (most important) present them to somebody knowledgeable and let him totally rip apart the proofs. After you've done about 10 proofs like this, I guarantee that you will know how to do basic proofs.

I have always found it much better to learn proofs from an actual math book, then about an artificial math book with artificial problems.

 

[zooxanthellae]

@micromass: Looks like we disagree here, then. At the very least most introductory proof-based courses start with a bit on logic and set theory, which Velleman has (and Spivak doesn't, if I remember correctly - I no longer own a copy). And I certainly don't feel like I had to unlearn anything when going from Velleman to Spivak. Instead I was glad I'd worked on it and in the "rigorous calculus" first-year course I used it for I think it gave me better footing than those who had never seen proofs before. I don't think basic set theory (which as I remember is most of what Velleman's book focuses on) constitutes "artificial problems" either, but, you know, everyone's different.

 

[micromass]

@micromass: Looks like we disagree here, then. At the very least most introductory proof-based courses start with a bit on logic and set theory, which Velleman has (and Spivak doesn't, if I remember correctly - I no longer own a copy). And I certainly don't feel like I had to unlearn anything when going from Velleman to Spivak. Instead I was glad I'd worked on it and in the "rigorous calculus" first-year course I used it for I think it gave me better footing than those who had never seen proofs before. I don't think basic set theory (which as I remember is most of what Velleman's book focuses on) constitutes "artificial problems" either, but, you know, everyone's different.

I'm not talking about basic set theory. Things like basic set theory are absolutely necessary. But Velleman's book contains a lot more than that. And that "a lot more" is usually unnecessary rubbish that doesn't help you prove things.

There are only a limited number of proof techniques that one has to know before he can go out and prove things himself. After that, it's just practice. And I prefer that people practice proofs in actual math books like Spivak.

I'm really not boasting when I say that I can teach somebody proofs in a very short time. All I will do is present statements that the other needs to prove and then I will rip the proof completely apart and tell them what is wrong with it. After a very minimal time, they really know how proofs work. There is no need to work through long books like Velleman. I'm sure they help, but there are more efficient ways.

 

[Fredrik]

I'm really not boasting when I say that I can teach somebody proofs in a very short time. All I will do is present statements that the other needs to prove and then I will rip the proof completely apart and tell them what is wrong with it. After a very minimal time, they really know how proofs work. There is no need to work through long books like Velleman. I'm sure they help, but there are more efficient ways.

I've been thinking this too. I have never even looked inside a book about proofs, so I can't comment specifically on any of those. There is a small number of huge blunders that people make at first, in particular ignoring the definitions of the terms used in the theorem, and not making it clear if the variables they're using are part of "for all" or "there exists" statements. It shouldn't take long to get them past this stage.

There is however a problem with this plan. It requires a teacher.