Hi Mac,
The concepts of countable and uncountable can be confusing and counterintuitive. The concept most mathematicians use to distinguish the size of infinite sets comes from Georg Cantor. I want to start with an example:
Suppose you are organizing a meeting in some hall and you have put out chairs for the people sit on. Quite a few people have arrived and you wonder if you need to get more chairs. One way to determine this without counting the chairs and people is to ask everyone to sit down. If everyone sits down and there are chairs left then there are more chairs then people. If all the chairs are occupied and some people are left standing there are more people then chairs. If everyone is seated and there are no unoccupied chairs then there are the same number of chairs as people.
The idea here is that you match chairs with people. Mathematicians call this a one to one correspondence and it is used to compare the size of sets. The word we use for size is cardinality.
Definition:
Let A and B be sets. If there is a one to one correspondence from A into B we say the cardinality of A is less than or equal to the cardinality of B. If there is a one to one correspondence from A onto B we say that the cardinality of A is equal to the cardinality of B.
Before getting to your questions I want to state the Cantor–Bernstein–Schroeder theorem
If the cardinality of A is less than or equal to the cardinality of B and the cardinality of B is less than or equal to the cardinality of A then the cardinality of A is equal to the cardinality of B.
Definitions:
A set A is finite if it is empty or its cardinality is the cardinality of {1, 2, ···, n} for some natural number n.
A set A is countably infinite if its cardinality is equal to the cardinality of the natural numbers N.
A set is uncountable if it is infinite and not countably infinite.
N = {1, 2, ···, } and Even = {2, 4, ···, 6} have the same cardinality because there is one to one correspondence from N onto Even. The correspondence, as you stated is n ⇒2n.
Two other useful results involving cardinality are:
If A and B have the same cardinality and B and C have the same cardinality then A and C have the same cardinality.
If A is a subset of B then the cardinality of A is less than or equal to the cardinality of B.
These two results might explain what you are asking in 4.
After you have read this look again at http://www.cs.xu.edu/csci250/06s/Theorems/powerSetuncountable.pdf
If you still have a question then write back.
Harley
