Finite and Infinite Sets

1. Finite and Infinite Sets Section 1.4 Fri, Sep 3, 2004

2. Equinumerosity • Let A and B be sets. We say that A and B are equinumerous if there is a bijection f : AB. • A set is finite if it is equinumerous with a set {1, 2, …, n} for some integer n. • Clearly, n is the number of elements in the set. • Two finite sets are equinumerous if and only if they have the same number of elements.

3. Infinite Sets • A set is infinite if it is not finite. • An infinite set is countably infinite if it is equinumerous with N. • Otherwise, it is uncountable. Countable Infinite Finite Countably Infinite Uncountable

4. Sets and Loops • A finite loop will process every element in a finite set and then quit. • A countably infinite loop will never quit, but each element will be processed at some finite time, if the loop is handled just right. • An uncountably infinite loop will never quit and most elements will never be processed (ever!). • This has ramifications in the study of what is computable and what is not computable.

5. Sets and Loops • Finite set – Process the numbers 1 to 100 in order. • Countably infinite set – Process the positive integers in order. • Countably infinite set – Process the positive integers by first processing 0, then all integers that begin with 1, then all integers that begin with 2, and so on, up to 9. (What is wrong with that?) • Uncountably infinite set – It is impossible to process sequentially the real numbers between 0 and 1.

6. Combining Countable Sets • Let A and B be countably infinite. • A B is countably infinite. • A  B is countably infinite. • 2A is uncountable. • It is impossible to list all subsets of N. • It is impossible to list all paths through an infinite binary tree.