Tuesday, March 27, 2007

'Me & My Maths'

I don't know when you actually started reading my blog, but if it was from the very beginning then you may or may not remember that I had naively called my blog 'Me & My Maths'. As you can tell this was a very silly thing to do, that's why I promptly changed it to 'Me Or My Maths'!

Some of you may be wondering what's the difference, so here's the explanation!

Given two sets 'Me' and 'My Maths' we can form the set of elements which lie in both 'Me' and 'My Maths'. This is called the intersection of 'Me' and 'My Maths' and is denoted: . Thus,

(you read this as, 'x such that, x belongs to Me and x belongs to My maths.)

Where x is probably the posts that I make. Since there have been things which I have posted that only belong to the set 'Me', then the initial name would have a been a lie. (I've got to stop doing that!). You only need one counter-example to prove something is false, and I can think of a few posts which would do the job. Hence, that is the reason why my blog got its new and cooler name!

You see given two sets 'Me' and 'My Maths' we can form the set of elements which lie in 'Me' or lie in 'My Maths'. This is called the union of 'Me' and 'My Maths' and is denoted: . Thus,

I'm sure you'd agree that this describes the content of my posts to a better degree! If you still need convincing then we can also use truth tables!
'Me' or 'My Maths' is called the disjunction of the two statements Me and My Maths. In everyday speech we often use 'or' in the exclusive sense. However, in mathematics we always use 'or' in the inclusive sense (to avoid ambiguity) determined by the above truth table (on the left). On the other hand, 'Me' and 'My Maths' is called the conjunction of the two statements Me and My Maths (table on the right). So if for instance, looking on the table on the right, for my post to be a 'true statement' I would always have had to write something about 'Me' and 'My Maths'. If I only wrote about 'Me' and not about 'My Maths' then the post would be false! That's why the or option is 'nicer', since if I wrote about 'Me' but not about 'My Maths' then the post would still be a true 'statement'. Obviously if I don't write about 'Me' or 'My Maths' then the post is false in both cases. (so my posts have been 'true statements!).

[I'm being a little 'cheeky' here, since these truth tables tend to deal with statements in general. Statements generally either denote predicates or propositions. A proposition is a 'sentence' which is either true of false, so 1+1=3 is a proposition. Whereas a predicate is of the type: m < n. They involve free variables, and once these are given values they become propositions. ]

So yes, some thought had gone into naming my blog! Since I've gone this far, I'm tempted to go slightly further! Intersections and unions are operations on sets, and you've probably come across them in stats (hmpf), and truth tables are about the basics of the language of maths (they can get ugly when you get more than two statements!).

Alas, I can't control myself! Given sets 'Me' and 'My Maths' we say that 'My Maths' is a subset of 'Me', written or , when every element of My Maths, is an element of Me, i.e. .(I don't suppose the converse is true!). If Me and My Maths are in addition unequal, so that Me contains some element not contained in My Maths then we say that My Maths is a a proper subset of Me and write:.

So that's the mathematics behind my blogs name (there will probably be more stuff when I move onto other topics, or when I really should be revising but rather think of stuff like this!). Now I don't really understand proper subsets, and well you've probably figured that I used my notes in order to create this post (but do tell if I made any errors). Like I said this is the 'language of set theory' and I don't particularly like it! I get confused when to use 'the proper subset' notation or just the 'subset' one. However, let's not get distracted about what I can or can't do.

I hope that I've now convinced you about why 'or' is ultimately the better word to use in my blog title. When someone offers you an 'apple' or 'banana' (could be anything in general), you could be cheeky and take both, and then innocently say, 'I thought you meant the inclusive or!'. This could either result in physical abuse or weird looks and someone who clearly is not amused! (although I would recommend trying it, just to amuse yourself if not for anything else.):D

Don't worry if this mumbo jumbo doesn't make sense, I was just erm.. amusing myself, but later I'll insert links of the 'mathematical' stuff that I've mentioned in case it's confusing. :) Oh and you're not allowed to give me them 'weird' looks, if you're a mathematician you should deal with the inclusive or at all times. That is all. No ifs or buts. Sacrifices have been made and more have to be made! Look at the big picture. (I'll stop now, in case I risk scaring you for life!).

(You've probably figured that the Internet is working and are hopefully relieved of this fact! It was the darn router).


Anonymous said...

Can't you just say that the blog as a whole is both about "me" and it's about "my maths", although a particular post might not be about both?

beans said...

I don't know, but then if a particular post isn't about both then can you say that it's part of this blog? But if I say it's about "me" and "my maths" then that's not technically correct is it? (I'm not sure to be honest.)

I like the subset idea though :o

(I think I was just amusing myself with all that fancy notation! And it gave me another excuse to use LaTeX.)

Jake said...

There is a famous book by the physicist Richard Feynman called 'Surely you're joking Mr. Feynman' which contained an amusing anecdote about the ambiguity between or and Xor (eXclusive or) in spoken English.

Feynman was at a formal afternoon tea at Princeton university and was asked by a woman if he wanted cream or lemon in his tea; being unaware of the social conventions, Feynman answered 'both please!' to which the woman retorted "Surely you're joking Mr. Feynman?" (Obviously drinking tea with cream and lemon is some class of social gaffe). Obviously the gag was that the woman said 'or' but really meant 'Xor' in the mathematical sense.

In fact the whole book is quite enjoyable; it isn't really so much about a famous physicist but rather, as the subtitle suggests, a collection of amusing anecdotes about a curious character. I definately recommend checking it out for some light reading.

beans said...

I'm glad that for once the name Richard Feynman rang a bell! (although a very faint one)

Ha, sounds like a book which I might enjoy. I'll look it up when I get the chance.

Have you ever tried the 'or' thing out? I tend to say it to annoy certain people, or to sometimes 'break' a conversation (with people who don't like maths)! I've got to say I'm easily amsued. :o (yes I'm sitting here laughing).

Jake said...

Have you ever tried the 'or' thing out? I tend to say it to annoy certain people, or to sometimes 'break' a conversation (with people who don't like maths)!

Yes, definately, especially as a child I was excruciatingly pedantic (not on purpose, I think just took some time to understand the concept of figurative speech).

When I use 'or' statements in my own speech I tend to say:
'Either X or Y' if I wish to imply an exclusive or and
'X, Y or both' if I wish to imply an inclusive or. I have found that phrasing or/Xor statements in this manner usually allows third parties to fully understand my intentions.

egm said...

Haven't read Surely You're Joking, but I have Pleasure of Finding Things Out. A very interesting Feynman book for sure.

beans said...

Where's the fun in wanting people to understand your intentions! It really depends on what you're saying (and who you're talking to!).

Reading these books is surely going to make my summer holidays more interesting (hopefully). (thanks for the heads up egm).

Jake said...

Haven't read Surely You're Joking, but I have Pleasure of Finding Things Out. A very interesting Feynman book for sure.

I would check out 'Don't you have time to think?' as well. It is a collection of Feynman's correspondence from over the years and makes for quite an interesting read.

Beans said...

I know this is a very very late reply, but anon, I actually get what you said now and misunderstood you a year ago!

What you say is absolutely true (and I like it).