I would like to give a class of 60 minutes to my undergraduate students about cardinality. I would like to begin with the definition of cardinality and end with one or two good application of this theory. I need some advices and some books suggestions.
Any help is welcome
Thanks in advance
On
I'm not sure I understand who these undergrads are. If this is in a course on set theory, then the text you are using already has material on cardinality. So I will assume it is some sort of survey course on math for non-science majors, who have not seen anything about cardinality before.
If that is the case, you can reasonably hope to show that the evens are one-to-one with the integers, that so are the rationals, and get to the diagonal proof that the reals are of greater cardinality than the integers. You can then generalize that proof to show that the power set $P(S)$ is of greater cardinality than $S$, so that there are "infinitely many levels of infinity".
Finally, I would ask the question "are there any sets of cardinality greater than the integers but less than the reals. And I would tell them that not only don't we know, but that it has been proven either answer is consistent with "math as we know it" (ZF).
On
When I present the notion of cardinality, I usually begins by reminding the students that without the need for counting, you can be sure that both hands have the same number of fingers (well, for most people), just by placing them one onto the other and seeing that every finger on the right hand is matched uniquely to a finger on the left hand.
Then we extend this to a flock of sheep. To know which one of us has more sheep, we need only to line up our sheep, one against one, and see if the two flocks match perfectly, or if one of them is finished first.
From this we come to the notion of equipotency using bijections. Because mathematically lining up these objects uniquely means exactly there is a unique ordered pair with those elements, which makes a bijection between the two sets.
From there we move to injections, the Cantor-Bernstein theorem, Cantor's theorem, and if the axiom of choice has been discussed, then the equivalence between ordering using injections and using surjections.
As for applications, showing that $\Bbb R$ is uncountable, and that the algebraic numbers are countable gives you the foremost use of cardinality. Most real numbers are no algebraic. Next, you can show that given a continuous function from $\Bbb R$ to itself, it is fully decided by the values it obtains on $\Bbb Q$. Since $\Bbb Q$ is countable, it means there are only $2^{\aleph_0}$ continuous functions. In particular "most" functions are not continuous.
If you present these ideas, skip the proof of the Cantor-Bernstein theorem. Note that both applications require nontrivial theorems (the uncountability of $\Bbb R$; and the "correctness" of cardinal arithmetic). I would cherry pick the proofs I'd like to show in just 60 minutes, perhaps omitting the second application completely, and just mentioning it without proof.
First, I think you shall start with finite sets. Introduce it as you would to children. Say that we can measure the quantity of apples and lemons in separate baskets by counting them.
Now, say we have a large number of apples and lemons. It will be seriously hard to count numbers especially since we may fall into error. Instead, we get a brilliant idea. We will put them in pairs of lemons and apples, and here is how we'll know if they're the same or not. If every lemon has an apple, every apple has a lemon, and no apple/lemon is shared, then we can trivially conclude that they are of the same quantity, the two baskets I mean.
Now, state this in terms of lemons and apples. Formalize equipollence and show that it applies to what we did earlier. Now, say that if we have an infinite set, and another infinite ones, how would we know if they have the same "quantity"? Explain that this measure called cardinality was created, and it works on the same principle as the "lemon/apple pairs". Well, from here on, you're the math teacher so it's all up to you.
For good and interesting applications, the best ones would be ones of mathematical nature. Let your students wonder whether $\mathbb{N}$ is of the same cardinality as $\mathbb{Q}$. I suppose most would answer wrongly, then amaze them by writing a bijection. Another application of the concept would be the cardinality of the continuum, that is it is not equipollent with $\mathbb{Q}$, the fact that the set of algebraic numbers is countable, and bijections from one interval into another.
For the last part, you might want to bring one of those big wooden set squares, and show that by orthogonal projection of the hypotenuse, we actually bijected the hypotenuse into a smaller line, one that can be arbitrarily small as the angle approaches $\frac{pi}{2}$. I think this should clear any trouble in understanding why the "big" $\mathbb{Q}$ is equipollent to the "small" $\mathbb{N}$, showing them that intuition can fail when dealing with infinite sets.
Good luck.