Category Archives: General

Large Intro

October 23, 2016 Adrian Leave a comment

Finally submitted my first assignment. It was monstrous. Just over 23 pages of mathematics and sketches of graphs. All of it typed up in LaTeX. Skipping ahead to look at the rest of the assignments, it looks as if this first assignment may very well be the biggest of the whole lot. This is a very good thing as I really don't think I could churn out that much work of a high quality every month.

Glad to say that most of this introduction section I was familiar with. Only really new topic was equivalence relations, which caused some problems initially.

Overall though, what I've found difficult is the apparent lack of logical notation. After reading "How To Prove It" I've become half-decent at making sense of and rearranging logical notation to solve a problem. The difficulty comes in looking at the plain-English description of something in the texts and then having to translate it into logical notation to allow my fussy brain to think about them logically.

Perfect example of this is the definition of a function being "onto". In the text, the definition reads:

"A function $f: A \longrightarrow B$ is onto if $f(A)=B$ ".

Which is fine, but the Wikipedia definition reads:

" $\forall y \in Y, \exists x \in X$ such that $y = f(x)$ "

Which for me, gives me a much better idea about how to go about proving if a function is onto. Why leave out the quantifiers? The Wikipedia definition tells me so much more. I suppose translating English into logical notation is just something I'll have to get good at!

Though even after this long intro section, I really feel I need more practise with proofs... I guess this may have to wait until revision time... Next up is the first section on group theory, with an assignment due on November 24th. Onward.

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...

General

Year Four Done!

April 9, 2016 Adrian Leave a comment

I just completed my first proper year of university! Only took me fours years!

Now I've got until September to work on my proofs and analysis preparation before my next module starts. -and looking at the schedule for my new module, I'll be needing as much preparation as possible. It turns out that (with December aside), there's an assignment due every month from October to May. Given that most of my assignments in the past have been 18 pages long on average and are typed in LaTeX, this concerns me a little bit...

All the more reason to be in the best position I can be for the start of my second (fifth) year! Argh!

Analysis, General

Analysis Battle

March 23, 2016 Adrian Leave a comment

It turns out that wrapping up a permanent job and starting a new one takes an enormous amount of time. Throw a holiday or two into the middle of it and you suddenly completely lack any spare time whatsoever.

Glad to say I'm back now though. Ready to tackle the final assignment of my intro module, and close things off for a Summer (of Analysis study).

Speaking of Analysis, I've spent a few more lunch hours running through the exercises of the end of Chapter 1 on the real number system and even after ALL the reading I've done I'm only able to complete about 50% of the questions. I feel my knowledge is lacking enormously. What makes things more frustrating is that David Brannan's book doesn't contain any solutions for the exercises at the end of each chapter. So I'm completely unable to unstick myself (or check to see if I've answered the question in the correct way).

As a consequence I feel like I'm slightly wasting my time with this book. Just banging my head against another dead-ender of a question. This is extremely unfortunate, as I was looking forward to finally working through as much as possible before September. Hoping it would give me an edge for my next module. But today, I tried to think about the situation logically: It feels as if I'm lacking knowledge. Knowledge and experience. There haven't been nearly enough examples in this book for me to get a feeling for the kinds of things I should be proving. So maybe that's what I need... more examples... more simple questions to attempt.

There was only one source I could think of that may deliver this... On the maths forum, I was given a link to proof exercises as supplied to kids studying pure mathematics at a-level. This might just give me a leg up...

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!

…is learning mathematics.