1. Finite and Infinite Sets
Section 1.4
Wed, Sep 7, 2005

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 nonnegative 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

4. 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

5. 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

6. 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
Uncountable

7. 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

8. 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.

9. Sets and Loops Suppose we have a function bool isPrime(int n);
that will determine whether n is prime.
Finite set – Process the numbers 1 to 100 in order, printing the ones that are prime.
Countably infinite set – Process the positive integers in order, printing the ones that are prime.

10. Sets and Loops
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,
printing the ones that are prime.
What is wrong with that?

11. Sets and Loops
Uncountably infinite set – Process all real numbers x between 0 and 1, printing those for which x2 is rational.
This is impossible.

12. 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.
However, N has only a countable number of finite subsets.
How could we list them?