As may be obvious from the sudden increase in my posting rate (which I don’t expect to be able to keep up) The Princeton Companion to Mathematics is now off my hands, which gives me the chance to devote a bit of attention to other projects, of which the Tricks Wiki is one. So in this post I’m going to discuss a relatively elementary piece of university mathematics, and will do so in the form of a sample article for that site. I’ll be a little careful about predicting when the site itself will be up and running, but let me just say that I’ve put some work into it recently and I don’t want to waste that work.
In what follows, I shall adhere to what I hope will be the basic format of an article on the site. The most important elements of that format are that there is a brief description, or “slogan”, that encapsulates the basic idea, and a general discussion of the idea that is illustrated by several clearly delineated examples.
Actually, I think I’ll give the discussion in the form of two sample articles, since it really contains two separate ideas.
The first one originates in a recent change of mind that I’ve had when teaching countability. I always used to stress that the main tool for this was the lemma that a countable union of countable sets is countable. But I now think that that is not in fact the best thing to stress (though it can of course be useful), because there’s another, much simpler, lemma that tends to lead to proofs that I prefer.
Article 1. A basic lemma for proving that a set is countable.
Prerequisites: The definition of a countable set, function-related notions such as injections and surjections.
Quick description: If you can find a function from to such that every has finitely many preimages, then is countable.
General discussion: Here is a quick informal account of a standard proof that the set is countable: we can list its elements in the order , , , , , , , and so on. (This is informal because I have talked about “lists” and have not actually defined how the sequence continues.) We can view this proof geometrically as follows: in order to count through the set , which forms an infinite grid in the plane, we note that each downward-sloping diagonal (that is, a set of pairs of positive integers with constant sum) is finite, and then we count through each of these sets in turn.
Here, we are making use of the very simple principle that a countable union of finite sets is countable. To put this more formally, if is a set that can be written as a union for some collection of finite sets, then is countable.
This is very easy to see informally: one just counts through each in turn (leaving out elements that have already been counted). It is not important for this article, but for completeness let us briefly see how we might prove it more formally. We could write each set as , where is the size of . And then we could define an ordering on all pairs with by taking if and only if either (i) or (ii) and . And then we would define a function by the following procedure. We would repeatedly choose the element with minimal index for which was not yet defined, and we would define it to be the minimal integer that was not yet assigned as a value of . It is not hard to check that one ends up assigning a value to all elements of and we never assign the same value twice. Thus, is an injection and is countable.
In particular, a set is countable if there is a function such that every has finitely many preimages. (This means that for every there are only finitely many with .) This follows because we can define to be , and then we have expressed as a countable union of finite sets.
This observation is not particularly interesting as a theoretical statement, but as a tool for giving quick proofs of countability it is extremely useful. Here are some examples.
Example 1. To prove that the rational numbers form a countable set, define a function that takes each rational number (which we assume to be written in its lowest terms, with ) to the positive integer . The number of preimages of is certainly no more than , so we are done.
As another aside, it was a bit irritating to have to worry about the lowest terms there. For some reason many mathematicians are afraid of multifunctions (that is, things that are like functions except that each element in the domain can map to several elements in the range) and sweep them under the carpet. But for the rationals it is convenient to have them. We define a multifunction from to by mapping to . This looks as though it isn’t well-defined, which as a function it isn’t, but if we think of it as a multifunction, then for instance the rational number maps to all possible values of for which . (If you really don’t like this, then you could rephrase it in terms of bipartite graphs or something like that.) Each positive integer still has only finitely many preimages (defined to be any rational number that has at least one image equal to ), and this proves the result since any multifunction with that property can clearly be restricted to a function with that property (just select, for each element of the domain, one of its images).
Example 2. To prove that the set of all finite subsets of is countable, how might we find a suitable function? An obvious function to consider is , but this doesn’t work since there are infinitely many sets of any given size. (OK, apart from size zero.) But there’s another very simple function that works: . Clearly, the number of sets with maximal element is finite (in fact, it is ), so we are done.
Example 3. To prove that the set of all polynomials with integer coefficients is countable is a similar exercise, but slightly more complicated. It is tempting to consider the sum of the absolute values of the coefficients, but then we notice that the polynomials all have coefficients with absolute values adding up to 1. So we need to restrict the degree somehow. But that is very easy indeed: given a polynomial we define to be the degree of plus the sum of the absolute values of the coefficients of .
Example 4. To prove that the set of all algebraic numbers is countable, it helps to use the multifunction idea. Then we map each algebraic number to every polynomial with integer coefficients that has as a root, and compose that with the function defined in Example 3. It is easy to check (using the fact that every polynomial has finitely many roots) that for every integer there are at most finitely many algebraic numbers that map to , and we are done.
End of Article 1.
Note that the above article didn’t really describe a “trick” so much as a reasonably straightforward technique. That was deliberate: despite its name, the Tricks Wiki is not supposed to be exclusively for demonstrating mathematical sleight of hand, though there is a place for that. The main distinguishing feature of the articles is that they should tell you how to prove things rather than what is true. In that respect, the quick description was misleading, since I stated it as a lemma. But in this case I think it goes without saying that what I really meant was not the lemma itself but rather, “If you want to prove that a set is countable, try to find a function from to such that the inverse image of every is finite.”
The second article is meant to illustrate a principle that is less a proof technique (though it can in theory be used for that) and more a quickly-seeing-what-is-true technique.
Article 2. How to see easily that a given countable set is countable.
Prerequisites: The definition of a countable set, Article 1.
Quick description: A set is countable if there is a way of giving a finite description to every element of .
General discussion: It is not hard to prove that the set of all finite strings of symbols taken from a fixed finite alphabet is countable. Indeed, we can count in turn the strings of length 0, 1, 2, and so on. If the size of the alphabet is then for each there are strings of length , and in particular finitely many. Therefore, by the simple lemma discussed in Article 1, we are done (either by saying that we have expressed the set of all finite strings as a countable union of finite sets or by considering the function that takes each string to its length).
It follows from this that if is any set, and if we can come up with a procedure for giving a finite description (in, say, the English language augmented by a few convenient mathematical symbols) to each of the elements of , then is countable. After all, the description takes the form of a finite string of symbols taken from some fixed “alphabet” of symbols, and there are only countably many of those.
This is significant because it often gives an easy way of recognising that a set is countable. It also gives proofs of countability, but these proofs are usually slightly “silly” and better replaced by simpler ones. But if you have ever known instinctively that a set is countable before you have actually come up with a proof, then it may well be something like this principle that is in the back of your mind.
Example 1. The rationals are countable because every rational has a description such as -35792/1293879861. Here, the alphabet is .
Example 2. The set of all polynomials with integer coefficients is countable since (a very slight modification of) the normal way of writing down such a polynomial is a finite string of symbols taken from the alphabet . Here, the symbol is used instead of the normal notation for exponentiation, so for instance instead of writing we would write .
Example 3. The set of all algebraic numbers is countable, since each one is a root of a polynomial that can be described as above, and for any fixed polynomial we can specify which root we are talking about in terms of the ordering on . For example, we could describe as “the larger root of ”. If we allow complex numbers then we could choose an ordering of the roots by taking them in order of their modulus, and for roots of equal modulus we could order them by their arguments (defined to lie in the interval ). For example, could be defined as “the second root of the polynomial ”.
Alternatively, and more naturally, we could argue that since the set of polynomials with integer coefficients is countable and each such polynomial has finitely many roots, an easy deduction from the principle of Article 1 shows that the algebraic numbers are countable.
Example 4. The set of all non-increasing functions from to is countable. This time the fact that there is a finite description relies on the well-ordering principle for the natural numbers, which implies that every non-increasing function from to is constant from some point on. So every such function will have a description such as “, , and for every .”
Example 5. The set of all finite subsets of is countable, since each one can be finitely described by simply writing it in the usual way using the symbols, , , the integers from 0 to 9, and commas.
Conclusion: I cannot think of any explicitly defined countable set that isn’t completely obviously countable by this criterion.
End of Article 2.
These articles have some features that will not be typical of Tricki articles, so to give a more balanced picture I plan to put up a few more samples over the next few weeks. Comments are welcome, both about their form and about their content.