Archive for the ‘General concepts’ Category

Equivalence relations

October 30, 2011

Equivalence relations are in a way a fairly simple mathematical concept. After all, it’s not that hard to learn what reflexive, symmetric and transitive mean and to remember that if you’ve got all three properties then you’ve got an equivalence relation. However, equivalence relations do still cause one or two difficulties. One, which I’ll discuss first, is that many people find that the proof that equivalence classes always form a partition is rather complicated and hard to remember. The other is that equivalence relations are closely connected to the notion of quotients, which appear in many places in mathematics, and which many people find quite hard to grasp.


Alternative definitions

October 25, 2011

Something that happens very often in lecture courses is that you are presented with a definition, and soon after it you are told that a certain property is equivalent to that definition. This equivalence means that in principle one could have chosen the property as the “definition” and the definition as an equivalent property. To put that differently, suppose you are developing a piece of theory and have some word you want to define. To pick an imaginary example, suppose you have a notion of a set being “abundant”. Suppose that a set is defined to be abundant if it has property P, and that property P is equivalent to property Q. There may well not be much to choose between the following pair of alternatives. On the one hand you can say, “Definition: A set is abundant if it has property P,” and follow that with, “Proposition: A set is abundant if and only if it has property Q,” while on the other you can say, “Definition: A set is abundant if it has property Q,” and follow that with, “Proposition: A set is abundant if and only if it has property P.”


October 23, 2011

By now you will have seen several definitions in lectures. Many of them will be written in the form

Definition. A blah is …

That is, the definition is displayed and the word being defined is in italics (or underlined if somebody is writing by hand). Sometimes, one doesn’t bother with the display, and simply says, during a discussion, “We define a blah to be …”

What is likely to have been emphasized less is that there are several different kinds of definition. In this post I’d like to enumerate some of them and give examples. It’s very much worth being aware, each time you meet a definition, what kind it is.

Domains, codomains, ranges, images, preimages, inverse images

October 13, 2011

If I were writing a textbook, I would have discussed the basics of functions before talking about injections and surjections, but this is not a textbook — it is a series of blog posts that provide a kind of commentary on some of the lecture courses. However, now that I have got on to the subject of functions, it probably makes sense to discuss them a bit more, especially as I hear that they have made an appearance in Numbers and Sets.

Let me start with the most basic question of all: what is a function? This is one of the first examples (of many, unfortunately) of a concept that you were probably reasonably happy with until your lecturer explained it to you. This is absolutely not a criticism of your lecturer (who is excellent, by the way). It’s more like a criticism of an entire mathematical tradition that goes back to the days when the foundations were laid for our subject in terms of set theory.

Injections, surjections and all that

October 11, 2011

My spies tell me that the second lecture of the Group Theory course contained a discussion of functions, and in particular bijections — for which it was necessary to prove a few results about injections and surjections. Since a good understanding of functions is essential throughout mathematics, perhaps a post on the topic would be in order. My aim in these posts is not to cover the material all over again, but rather to stress the points that you need to understand in order to be able to write proofs. So let me give a few thoughts about functions, in no particular order.