Welcome to Week 6. This week we will look at some examples of finite and infinite sets of real numbers. When dealing with a finite set, you can (in principle at least) say how many elements it has. This is its cardinality. For finite sets, cardinality is a pretty straightforward concept, it’s just the number of elements in the set. So two finite sets have the same cardinality if they have the same number of elements, and we can often decide that by counting the number of elements in each. Alternatively, we could say that two finite sets have the same cardinality if it’s possible to match each element of one set with an element of the other, in such a way that every element gets matched with exactly one “partner” in the other set. We will emphasize this latter interpretation – it looks like a lot of fuss about an idea that was not complicated in the first place. But it has two advantages. One is that it might just be possible to establish that two finite sets have the same cardinality without knowing the number of elements in either of them, an idea that has lots of traction in the field of enumerative combinatorics for example. The second advantage, and the more important one for us at the moment, is that we can use this “matching” idea to investigate the question of whether two infinite sets (like ℝ and ℚ) have the same cardinality. That will be our theme in Week 7.
Relevant sections of the lecture notes this week are Sections 2.1 and 2.2 in Chapter 2. There are more examples in the lecture notes than we will discuss in our lectures, and more detailed explanations in some places.
Here is (an old version of) Lecture 11.
Slides for Lecture 11, without annotation, and annotated.
Here is (an old version of) Lecture 12.
Slides for Lecture 12 (we had no annotations on this one).
Weekly Problem 6
The weekly problems are just for fun. They have nothing much to do with our curriculum. Please send me an email if you have a solution that you would like to share with the class!