Unit 7: Sets, Part II
In this unit, you will study cardinality. One startling realization is that not all infinite sets are the same size. In fact, there are many different size infinite sets. This can be made perfectly understandable to you at this stage of the course. In Unit 7, you learned about bijections from set 
to set 
. If two sets and have a bijection between them, they are said to be equinumerous. It turns out that the relation equinumerous is an equivalence relation on the collection of all subsets of the real numbers (in fact on any set of sets). The equivalence classes (the cells) of this relation are called cardinalities.
Completing this unit should take you approximately 8 hours.
Upon successful completion of this unit, you will be able to:
- compute the image and inverse image of sets under a given function;
- determine if a function has an inverse function, and if so (and when possible), determine a formula for it;
- establish bijections between certain subsets of real numbers, and connect this to computing the cardinality of these sets;
- define the Cantor set, and prove that it is uncountable;
- recognize countable subsets of the real numbers; and
- establish properties of countable sets.
7.1: Cantor Diagonalization Theorem: The Existence of Uncountable Sets of Real Numbers
Watch this video. Make sure you understand how two sets A and B can be equinumerous.
Watch these lectures to continue your studies on sets.
7.1.1: Proof of the Theorem
Read this paper. Spend some time studying the proof of Cantor's Theorem on pages 3 and 4. Even though the proof is quite brief, this idea is new to you, and therefore is likely to be harder to understand.
Watch this video to supplement the written proof of Cantor's Theorem.
Watch this video about counting finite sets and Cantor's Diagonalization Theorem.
7.1.2: Even the Cantor Set Is Uncountable, the Base-3 Connection with the Cantor Ternary Set
Watch this video to see a proof that the Cantor middle third set is uncountable.
7.1.3: Other Examples of Uncountable Subsets of R
Read this document to learn about countable and uncountable sets. Focus on the several examples of uncountable subsets of R.
7.2: The Rational Numbers Are Countable
Watch this video to see a proof that rational numbers are countable.
7.2.1: The Proof
Study the proof on this webpage, which shows that rational numbers are countable.
7.2.2: The Algebraic Numbers Are Countable
Study this proof, which demonstrates that algebraic numbers are countable.
7.3: Other Bijections
Please read the paper, paying special attention to examples 2.2.5, 2.2.6, and two functions
and 
in the paragraph above example 2.7.1.
Unit 7 Assessment
- Receive a grade
Take this assessment to see how well you understood this unit.
- This assessment does not count towards your grade. It is just for practice!
- You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.
- Receive a grade
Take this assessment to see how well you understood this unit.
- This assessment does not count towards your grade. It is just for practice!
- You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.