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.

A famous demonstration/explanation of some of the ideas in Lecture 13 is Hilbert’s Hotel.

In Lecture 14 on Thursday, 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).

Slides for Week 7.

Here are (old) video versions of Lectures 13 and 14.

The relevant sections of the lecture notes this week is Sections 2.3. 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.