MA180/186/190 Calculus Semester 2. Week 7: Cardinality and Infinite Sets
Welcome to Week 7. 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 main theme this week.
In Wednesday's lecture, we will look at the concept of bijective correspondence and its connection to cardinality, mostly for finite sets. Bijections (or bijective correspondences) arise very frequently in mathematics, and there are different ways of thinking about them that are useful in different contexts. Sometimes it is useful to think of a bijection as a function that has an inverse, or is both injective (one-to-one) and surjective (onto). For our context, it might be appropriate to think of a bijection as a "pairing" of elements of one set with elements of another, so that every element in each set has exactly one "companion" in the other - no element is overused or left out.
Here is Lecture 12 (Wednesday March 24).
Slides for Lecture 12 (we had no annotations on this one).
On Thursday, we will look at bijective correspondences between infinite sets. Two infinite sets are considered to have the same cardinality if a bijective correspondence exists between them. On Thurdsay we will show that the set of integers and the set of rational numbers have the same cardinality, even though the rational numbers are densely packed into the number line and the integers are sparsely spread. This will (hopefully) become even more interesting in Week 8, when we show that not every infinite set has this property of admitting a bijective correspondence with the set of integers.
Here is Lecture 13 (Thursday March 25).
Relevant sections of the lecture notes this week are Sections 2.2 and 2.3 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.
Weekly Problem 7
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!