MA180/186/190 Calculus Semester 2. Week 8: ``No one shall expel us from the paradise that Cantor has created for us''
Welcome to Week 8.
We have two key points this week. 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 Wednesday 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 Lecture 14 (Wednesday March 31).
The content of Wednesday's lecture tells us that (non-empty) bounded open intervals have the same cardinality as ℝ, but not whether this cardinailty is the same as that of the set of rational numbers or integers. The question of whether the set of real numbers is countable is our topic for Thursday. We will discuss the famous "Cantor diagonal argument", which establshes the uncountability of ℝ. This development had an enormous influence on the evolution of mathematics in the 20th century.
Here is Lecture 15 (Thursday April 1).
Slides for Lecture 15 without annotation, and annotated.
Relevant sections of the lecture notes this week are Sections 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 8
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!