Category Archives: Books

A Romance Of Many Dimensions

I decided to buy Flatland: A Romance Of Many Dimensions by Edwin A. Abbott. Glad to say I got exactly what I wanted out of it: a light read!

It mostly takes the form of a light societal allegory, using aspects of multiple dimensions and a little bit of philosophy thrown in there for good measure too. Though I feel it didn't take itself too seriously and did make me smile on occasion. Something this book doesn't do is help you think in four dimensions and higher, though it does make you think a little more about the dimensions we can visualise in our heads!

Entertaining all the way though, it's a great, light read should you be requiring one!

Feedback - 01

I received the marks back for my first monster assignment! Did quite well as it turns out! But this blog isn't about spouting about my success, it's about the learning process! So here's some of the things I screwed up...

First off, my algebra is clearly rusty as fuck. In one instance put a minus sign in the wrong place AND mysteriously lost a factor of 2 in the progress of my working. In future I really need to re-read my working really carefully (three or four times over it seems), both the hand-written and the full typed-up LaTeX...

Something else I lost marks for was the apparently simple task of graph sketching, either where I hadn't considered asymptotes or had not considered the limits of the domain. Overall I clearly need to be a lot more mindful of whether I'm dealing with \leq or <. When I read those symbols I see them both so often, I frequently gloss over them without properly considering their usage. Again, pretty basic stuff.

With complex numbers I apparently need to be more explicit with my declaration of forms. My polar form was implicit in the answer, but there wasn't anywhere I actually stated it. Silly boy.

I fell down on a proof of symmetry for an equivalence relation. I just wasn't mindful whilst answering this. It is assumed that x-3y=4n. This can be rearranged in terms of y as y=\frac{x}{3}-\frac{4n}{3}. So substituting y, in the symmetrical y-3x results in: 4 \left(-\frac{2x}{3}-\frac{2n}{3}\right). Of course, at this point, proving that what's inside the brackets is an integer is pretty difficult. But that's where I left it. A bit more play would've shown that I could easily have arranged the first equation in terms of x instead which would've resulted in 4\left( -2y-3k\right), which is rather obviously an integer given the initial variables. More exploration required in future...

Lastly, in my last post I mentioned how there was a distinct lack of symbolic existential or universal quantifiers in all this new material. After Velleman, I was so used to seeing them, and working with them appropriately but because they're now not around, I got totally burnt by assuming I had to prove "there exists" instead of "for all" for one question. I suppose I'll be able to get around this with making sure my notes explicitly state whatever quantifier we're actually talking about. Damned English language... Symbols are much more concise! 🙂

PENS DOWN!

So that's it! The Summer has ended!

How far has my extra study got me? Well I've managed to get through around 120 pages of "How To Prove It" by Velleman, and have generated just over 70 double-sided pages of A4's worth of exercises from the book. Not bad for an extra-curricular topic!

This book has helped me loads. It's succeeded in taking away a lot of the mystery involved in reading and writing proofs.

Every topic in the book up until now has flowed well, and allowed me to think about solutions to the various problems fairly naturally. What I mean by that is, I never became absolutely stuck and unable to answer a question.

Having said that, the sub-topic I'm finishing on is proofs involving quantifiers. This is the one area in which I'll admit I've been struggling. At this point in the book, I've learned so much about the number of ways in which to decon/reconstruct a problem that any possible method by which to prove a theorem has actually become less obvious.

Here's an example of how convoluted the scratch work of a basic proof has become. Here's question 14 from p.122:

Suppose  \{A_i\: |\: i \in I\} is an indexed family of sets. Prove that \cup_{i \in I} \mathscr{P} (A_i) \subseteq \mathscr{P}(\cup_{i \in I} A_i).

It's a short question, but this immediately looks like a nightmare to a beginner like myself. We've got a mix of indexed sets, a union over them, and power sets.

First off I need to properly understand the damn thing. Seems sensible to draw up an example using the theorem...

Let's say I is \{1,2\}. So we've got \{A_{1},A_{2}\}.

Now let's say that A_1 = \{1,2\} and A_2 = \{2,3\}.

Looking at the LHS of the thing I need to prove, it's actually pretty easy to break down:

\mathscr{P} (A_1) = \{ \varnothing, \{1\}, \{2\}, \{1,2\}\}

\mathscr{P} (A_2) = \{ \varnothing, \{2\}, \{3\}, \{2,3\}\}

Which means that the union of all the elements of the power sets of A_i is:

\cup_{i \in I} \mathscr{P} (A_i) = \{ 1,2,3\}

A little half-way recap: the theorem says that in my example, \{1,2,3\} should be equal to, or be a subset of \mathscr{P}(\cup_{i \in I} A_i) (the RHS).

Let's see if that's true shall we?

Within the parenthesis of the RHS we've got \cup_{i \in I} A_i. So this is the union of all elements of all indexed sets. In this example:

\cup_{i \in I} A_i = \{1,2,3\}

Only thing missing now is the power set of this:

\mathscr{P}(\cup_{i \in I} A_i) = \{ \varnothing, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}

And there we go. Now I understand exactly what the theorem means. \cup_{i \in I} \mathscr{P} (A_i) \subseteq \mathscr{P}(\cup_{i \in I} A_i), in this specific example, turns out to be :

\{1,2,3\} \subseteq \{ \varnothing, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}

which is obviously true. Theorem understood. Achievement unlocked. Tick.

As recommended by Velleman, I'll also try to construct the phrasing of the proof along-side the scratch work as I go. Like so:

Suppose a thing is true that will help to prove the theorem.

[Proof of theorem goes here]

Thus, we've proved the theorem.

Okay, let's start.

The theorem means that if x \in \cup_{i \in I} \mathscr{P} (A_i) then x \in \mathscr{P}(\cup_{i \in I} A_i). So one thing implies the other. We can then class \cup_{i \in I} \mathscr{P} (A_i) as a "given", and aim to prove \mathscr{P}(\cup_{i \in I} A_i) as a "goal". So blocking out our answer:

Suppose that x \in \cup_{i \in I} \mathscr{P} (A_i) .

[Proof of x \in \mathscr{P}(\cup_{i \in I} A_i) goes here]

Thus, \cup_{i \in I} \mathscr{P} (A_i) \subseteq \mathscr{P}(\cup_{i \in I} A_i).

So let's start analysing \cup_{i \in I} \mathscr{P} (A_i) , remembering not to go too far with the logical notation. With baby steps, the definition of a union over a family of sets (here, the outer-most part of the logic) is:

\exists i \in I (x \in \mathscr{P}(A_i))

Then, going one step further, using the definition of a power set:

\exists i \in I (x \subseteq A_i)

Now we could go further at this point, applying the definition of a subset, but I'll stop the logical deconstruction here. In this instance, I've found that if I keep going so the entire lot is broken down into logical notation it somehow ends up getting a bit more confusing that it needs to be.

With this as our given, I notice the existential quantifier. Here, I can use "existential instantiation" to plug any value I want into i and then assume that what follows is true. So at this point the new "given" is simply:

x \subseteq A_i

Nice and simple.

Let's update the outline of our proper proof answer:

Suppose that x \in \cup_{i \in I} \mathscr{P} (A_i) .

Let i \in I be such a value that x \subseteq A_i.

[Proof of x \in \mathscr{P}(\cup_{i \in I} A_i) goes here]

Thus, \cup_{i \in I} \mathscr{P} (A_i) \subseteq \mathscr{P}(\cup_{i \in I} A_i).

So let's now move on to our "goal" that we have to prove: \mathscr{P}(\cup_{i \in I} A_i).

Again, starting from the outside, going in,

x \in \mathscr{P}(\cup_{i \in I} A_i)

by the definition of a power set becomes:

 x \subseteq \cup_{i \in I} A_i

I can't do a lot with this on it's own so I'll keep going with the logical deconstruction. By the definition of a subset, this becomes:

\forall a (a \in x \to a \in A_i)

So now, using "universal instantiation" I can say that here, a is arbitrary (for the sake of argument, it really can be anything), and that leaves us with an updated "givens" list of:

x \subseteq A_i
and
a \in x

and a new "goal" of

a \in A_i

Hey, but wait a sec... Look at our "givens"! If a is in x... and x is a subset of A_i, then a must be in A_i! -and that's our goal!

So update our proof:

Suppose that x \in \cup_{i \in I} \mathscr{P} (A_i) .

Let i \in I be such a value that x \subseteq A_i.

Let a be an arbitrary element of x.

[Proof of a \in A_i goes here]

Therefore a \in \cup_{i \in I} A_i. As a is arbitrary, we can conclude that x \in \mathscr{P}(\cup_{i \in I} A_i).

Thus, \cup_{i \in I} \mathscr{P} (A_i) \subseteq \mathscr{P}(\cup_{i \in I} A_i).

So let's wrap this up.

Theorem:
Suppose  \{A_i\: |\: i \in I\} is an indexed family of sets. Prove that \cup_{i \in I} \mathscr{P} (A_i) \subseteq \mathscr{P}(\cup_{i \in I} A_i).

Proof:
Suppose that x \in \cup_{i \in I} \mathscr{P} (A_i) . Let i \in I be such a value that x \subseteq A_i, and let a be an arbitrary element of x. But if a \in x and x \subseteq A_i, then a \in A_i. Therefore a \in \cup_{i \in I} A_i. As a is arbitrary, we can conclude that x \in \mathscr{P}(\cup_{i \in I} A_i). Thus, we conclude that \cup_{i \in I} \mathscr{P} (A_i) \subseteq \mathscr{P}(\cup_{i \in I} A_i). Q.E.D.

Overall the task has involved unravelling the symbols into logic, making sure they flow together, and then wrapping them back up again.

See what I mean by convoluted? All that work for that one short answer. I must admit, I still don't know if my reasoning is 100% correct with this. Despite some parts of this seeming simple, this really is the very limit of what I'm capable of understanding at the moment. I picked this example to write up, as so far I've found it to be one of the most complicated.

The next section of the book seems to marry this quantifier work with another previous section about conjunctions and biconditionals, that I found to be quite enjoyable at the time. Then towards the end of the chapter, Velleman seems to sneak in some further proof examples using the terms epsilon and delta. I imagine this is a sneaky and clever way to get the reader comfortable with further Analysis study...

Alas, my study of Velleman's book will have to stop here. I understand a lot more than I did, though not everything there is to know. I feel it may be enough to give me a slightly smoother ride through my next module, which was the whole point of me picking this book up. It's been so good, I hope I have a chance to return to it. I feel later chapters would put me in an even better position for further proof work!

For now... the countdown begins for the release of my next module's materials!

Things you need to be told at the beginning

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.

End of the Quantifiers

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!

The Joy Of Sets

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...

Books For Understanding Books - Part 2

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...

Books For Understanding Books

I couldn't help myself.

I've bought another book.

The reason I bought it is to help me with the book I'm currently reading... but it's not as bad as it sounds.

The last Analysis query I had, I posted to the OU forums. As usual, my question was answered almost instantly in a concise and understandable way. Awesome!

However, there was another post a few days later suggesting that I read "How To Think About Analysis", another book by Lara Alcock.

howToThinkAboutAnalysis

Initially I thought it would be a bad idea to drop my current book about Analysis, and pick up a new one, but it turns out that after the first 50 pages, this book can be used as a companion to learning different sub-topics of Analysis. Specifically, after the first 50 pages, the remainder of the book is split into sections about Real Numbers, Sequences, Series, Continuity, Differentiability, and Integrability. Each sub-topic appears to be about 40-ish pages long, and can be read just before or during the actual study of each. Bite-size!

Although I've not quite finished the first section yet,  I've already added several new strategies to my set of learning techniques. Despite the fact that the knock-on effect is that the learning process may become slower, the idea is that my knowledge and understanding of the material will become much much deeper.

I'll try and give another review once I've completed the first part, the section on real numbers, and related it all to where I currently am in my larger Analysis text book.

Year Four Begins!

My new learning materials have finally arrived!

mu123
This is the amount of reading and exercises that I would normally have to work through over a nine-month period. Entirely doable.

However, as I've mentioned in previous posts, this module is merely to make up the credits for my first stage. Bad news is: it's all quite basic. As a result, I'll also be working through a nice fat book on analysis, to ease me in to my second stage that starts next year. As such, my nine-month work load actually looks a little more like this:

mu123plus
So we'll see how far I get with all that!

Flicking through the first couple of pages of these new materials, I realise that they cover some very basic areas of mathematics indeed. Perhaps even a little more basic than I was expecting. I think I'll have to keep reminding myself that this was my plan from the beginning, and just concentrate on submitting my assignments on time. This work will definitely be taking a back seat...

More importantly, over the past couple of weeks I've also been working through the book on analysis, and although I'm only a few pages in, it's proving to be really valuable; for one, I'm becoming a lot more comfortable with proofs and solution sets of inequalities.