Tag Archives: Proofs

Things you need to be told at the beginning

July 12, 2016 Adrian Leave a comment

These quotes are from pages 89 and 90 of Velleman's "How To Prove It". If only I'd read all this when I was first introduced to a proof, I wouldn't have been so stressed!

"When mathematicians quote proofs, they usually just write the steps needed to justify their conclusions with no explanation of how they thought of them."

"Although this lack of explanation sometimes makes proofs hard to read, it serves the purpose of keeping two distinct objectives separate: explaining your thought processes and justifying your conclusions."

"The primary purpose of a proof is to justify the claim that the conclusion follows from the hypotheses, and no explanation of your thought processes can substitute for adequate justification of this claim. Keeping any discussion of thought processes to a minimum in a proof helps to keep this distinction clear."

"Don't worry if you don't immediately understand the strategy behind the proof you are reading".

I could hug this book right now.

Bertrand Russell, Books, Proofs, Text Books

End of the Quantifiers

July 7, 2016 Adrian Leave a comment

A month and a week, and I've just come to the end of the second chapter. Reasonably happy with the progress, but I could be going a bit quicker... Mind you, over just two chapters I've now created 46 pages of A4 of exercises. So there has been a LOT of material to go through. Frankly, just these first two chapters have worked wonders for my understanding of logic and what proofs are founded upon.

This second chapter mainly introduced quantifiers. The concept of "for all x" and "there exists at least one x...", but quickly branched off into more involved set theory.

The biggest issue I had towards the end of the second chapter was that on a couple of occasions, I don't think I thought carefully enough about the kind of answer the questions required. ie: in this context, whether the answer was required in logical notation, or whether it was required in set theory notation. Translating between the two is something I certainly found tricky. As such, I decided to write my own definitions of notation in the form of a list (thanks Lara Alcock!). Though the lack of lists of definitions could be considered a slight shortfall of the book, I think I benefited from constructing my own notes and definitions.

I found that towards the end of the questions (because of the more lengthy logical notation) I was concentrating more on the definitions than what the notation actually meant. Not convinced this is so good for the learning process, but at least I'm mindful of it now.

Last little question the second chapter covered was Russell's Paradox, as discovered by Bertrand Russell in 1901. The fact that I'm being introduced to stuff like this in the second chapter is pretty cool. Very enjoyable!

Next up, proof technique!

Books, General, Proofs

The Joy Of Sets

May 31, 2016 Adrian Leave a comment

54 pages, and 5 large exercise sections later, I've finally finished the first chapter of "How To Prove It". With the first chapter being about sentential logic, I've now covered truth tables, derivations of logical operations, set theory, and the conditional and bi-conditional connectives.

The next chapter covers further foundational logical concepts and only in Chapter 3 are the intricacies of actual proofs discussed. Having taken this long to cover the first chapter, and looking at the amount of paper I've used to do all the exercises so far, I'm not that surprised I was finding proofs so difficult. It turns out my intuition was right, I was missing a lot of foundational knowledge.

So far, it's all been going well. Nothing I've looked at in this first chapter has left me mystified and overall I feel like I'm learning. This is exactly where I wanted to be... Just need to up the pace, perhaps...

Proofs, Text Books

Books For Understanding Books - Part 2

April 4, 2016 Adrian Leave a comment

So this is getting ridiculous. I know, I can only apologise. I'll write some maths on here at some point, I promise.

It turned out that the super-valuable forum post I had on the OU forums has now been deleted. Apparently if a post isn't pinned it gets auto-nuked after two months. So now all that valuable information is gone.

But let's not dwell on it. Especially when there's a new book looming!!!!!

howToProveIt

Now, despite the fact that I've read Lara Alcock's books about how to learn Analysis, and started Brannan's Analysis book (see earlier posts) I realised I was missing more foundation-level knowledge. How To Prove It by Daniel J. Velleman looks like it'll be the book to give it to me. I remember it as being recommended on the deleted forum post, and the reviews generally are very very positive.

Already I've come to the end of the first (admittedly short) section and I can actually attempt all of the exercises! I totally understand everything he's saying and I really feel like I'm learning something with every page. At last!

More of a proper review of this on the way, but for the moment I'll be nose-deep in this for the next couple of months...

Analysis, General, Proofs

Readjusting Learning Methodologies

November 18, 2015 Adrian Leave a comment

I've just finished reading Lara Alcock's book on how to learn about Analysis. Or rather, I've finished reading the first part, and the part on the real number system.

Overall, the book has led me to reconsider my current learning technique. So much so, I've compiled a list of steps to follow depending on whether I read about a new definition, theorem or proof.

In turn, this has made me realise I may benefit from starting my main book on Analysis again from the beginning, but applying these new steps as I go. After all, I am still only at the beginning (kind of), and I don't have any kind of deadline looming over me (which is really nice). Overall it seems like the perfect opportunity to try out some new learning methodologies!

Out of the handful of additions, there are two really big changes for me.

The first being mind maps. As I go through my Analysis, I'll be creating a mind map of concepts, seeing how one builds on another. I'd tried to use mind maps before at university and they'd largely proved completely useless. Here, however, mind maps appear to offer a perfect way to visualise the building of concepts into larger concepts. Here's the beginning of my first mind map!

I'm using draw.io as the tool of choice. It seems flexible enough for what I need it for, it can save as XML, and it supports mathematical notation! (mathjax latex formatting I believe)

The second big change to my learning involves learning by self-explanation. This technique, mentioned in Lara Alcock's book, appears to be one of the key processes involved in truly understanding and appreciating Analysis. You can find out more about self-explanation training from Loughborough University's Mathematics Education Centre website. My difficulty here will lie in concentrating on actually doing self-explanation, rather than just paraphrasing (turns out, it's a very easy trap to fall in). So long as I do it regularly enough, self-explanation will be more likely to come to me naturally.

Expected result: More effective learning and better notes!

Maths, Proofs

Reading Mathematics

October 25, 2015 Adrian Leave a comment

Still working through my book on Analysis (which I will be until July 2016, so I should probably stop mentioning it...), but I recently came across another proof that I had difficulty understanding. I had to reach out to the mathematics forums in the end, but after getting a reply and working through some further steps that they mentioned, everything fell into place.

I felt really good that I'd finally understood the proof, but I also tried to work out what I could've done to push for that answer myself (so hopefully next time, I won't have to post a question to the forum).

I realised that I might not be reading through the mathematics effectively enough. After reading through Lara Alcock's book I realised how important it is to make effective notes whilst reading through "all the symbols". When reading through proofs I have this nasty habit of reading them like a novel, keeping this story of logic in my head... and then very quickly becoming confused because I didn't see how you could logically progress from one sentence to another.

This sounds really simple, and almost obvious, but I think all it really takes is to sit down with a pen and go through the mathematics of the proof in gritty detail, liberally re-arranging things as you go. It's worth mentioning that this is perhaps quite a different act than just "taking notes".

Being completely confounded by something only to solve the issue entirely on your own is enormously satisfying. The hope is that if I stay mindful and remain aware as to when to write the right kinds of notes, not only will I be able to solve more complicated problems on my own, but also in time these seemingly large logical steps will become second nature.

Anyway, doing independent reading of mathematics is proving, generally, to be really satisfying. Let's see if I can become a more effective reader...

Analysis, Maths, Proofs

Proof of Inequalities by Mathematical Induction

October 2, 2015 Adrian Leave a comment

Still reading though my book in Analysis, I've come across a section on proving inequalities. I'm glad to say that all of this made sense... until I reached a sub-section on proving an inequality by mathematical induction.

As I've written previously, I find that proofs are notoriously unintuitive. In the past however, I have been particularly puzzled by the logical steps involved in proving an inequality by mathematical induction.

To explain my difficulties, let's have a look at the example provided in the book:

Prove that $2^n \geq n^2$ , for $n \geq 4$ .

If we're proving this by mathematical induction, we generally follow these steps:

Let $P(n)$ be the statement $2^n \geq n^2$ .
Show that $P(4)$ is true.
Assume that $P(k)$ is also true for $k \geq 4$ .
Show that $P(k) \Rightarrow P(k+1)$ . Or rather, show that if $P(k)$ is true, then $P(k+1)$ is also true.

Step 4 is the key step here in the proof as it shows that if any number is true, and the next number is also true, then you can apply this rule forever, and your original statement must be true for all numbers!

Anyway, lets jump to step 2. Show that $P(4)$ is true. Well if $n=4$ , then $2^4 = 16$ and $4^2 = 16$ . So $P(4)$ is true! Easy.

Let's look at step 3. Let's ASSUME that $P(k)$ is true for some $k \geq 4$ .

Now, this is the part that caught me by surprise... At this step, the text in the book reads as follows:

"So, we are assuming that $2^k \geq k^2$ . Multiplying this inequality by 2 we get:

$2^{k+1} \geq 2k^2$

,

so it is therefore sufficient to prove that $2k^2 \geq (k+1)^2$ ."

Wait, what? How is it that all of a sudden, all we need to prove is that $2k^2 \geq (k+1)^2$ ? This isn't explained explicitly in the text so I had to close the book and do a bit more thinking.

First thing I had to realise here is that the "Step 4" I've listed above requires a bit more detail... What you're actually trying to do is show that you can progress naturally from $P(k)$ to $P(k+1)$ . ie: We should be able to show that we can progress naturally from:
$2^k \geq k^2$
to:
$2^{k+1} \geq (k+1)^2$ .

Now, if we multiply $2^k \geq k^2$ by 2, as mentioned in the text, we do arrive at:
$2^{k+1} \geq 2k^2$

This is good, as we've managed to get the $2^{k+1}$ we were looking for on the left-hand side of the inequality. But the right-hand side looks nothing like the right-hand side of $P(k+1)$ ie: $(k+1)^2$ .

Here's the key though... It doesn't matter they they're not the same. We only need to see how $2k^2$ and $(k+1)^2$ relate to each other. Look back at Step 3. Part of this assumption is that $k \geq 4$ . Just as a test, let's try $k=4$ :

$2k^2 = 2 \times 4^2 = 32$
and
$(k+1)^2 = (4+1)^2 = 25$

Well this is interesting. It's looking as if $2k^2 \geq (k+1)^2$ . This is exactly what was written in the text!

But to really ram it home, what we really have now is the following:

$2^{k+1} \geq 2k^2 \geq (k+1)^2$

So.... this show us that IF we can prove that last bit $(2k^2 \geq (k+1)^2 )$ is true for all $k \geq 4$ , and not just $k=4$ we have managed to prove that we can get from $P(k)$ to $P(k+1)$ !!! This is exactly why the text in the book said "so it is therefore sufficient to prove that $2k^2 \geq (k+1)^2$ ."

I'm sure in future I'll jump on this immediately and say "oh yes, of course that's all we need to do now", but working through the derivation of why it was sufficient was extremely useful. Long-winded... but useful.

Ah, the learning process...

…is learning mathematics.