Cardinality and Infinite Sets
This week, 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 Wednesday 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 (an old version of) Lecture 13.
A famous demonstration/explanation of some of the ideas in Lecture 13 is Hilbert’s Hotel.
In Lecture 14, we will discuss the concept of boundedness for subsets of ℝ. A set of real numbers is bounded if it is enclosed within an interval of finite length in the number line. In this sense, bounded sets are “smaller” than unbounded sets, but not necessarily in terms of their cardinality. On Thursday we will see that all open intervals of the form (a,b), where a<b, have the same cardinality as each other, and as the full set of real numbers. The tan function provides us with a very useful mechanism to demonstrate this, as it gives us a bijection between ℝ and the open interval (-π/2,π/2).
Here is (an old version of) Lecture 14
Relevant sections of the lecture notes this week are Sections 2.2 , 2.3 and 2.4 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.