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Countable sets (Posted on 2016-03-15) Difficulty: 3 of 5
A set is countable if and only if each of its elements can be associated with a different positive integer. Every finite set is countable. For example, the set {2, 3, 5, 7, 11} is countable.

1↔2
2↔3
3↔5
4↔7
5↔11

Infinite sets can also be countable. For example, the set of all prime numbers is countable.

1↔2
2↔3
3↔5
4↔7
5↔11
6↔13
7↔17
8↔19
9↔23
10↔29
...

1. Is the set of all integers countable?
2. Is the set of all positive rational numbers countable?
3. Is the set of all rational numbers countable?
4. Is the set of all positive real numbers countable?
5. Is the set of all real numbers countable?

No Solution Yet Submitted by Math Man    
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re: Interesting | Comment 6 of 8 |
(In reply to Interesting by broll)

I do recall reading about Hilbert's Hotel, but the infinite fleet of buses is not I^I - it is I*I.  I can create a bijection from all the passnegers in the entire fleet of buses to the natural numbers, no need to invoke Cantor's Diagonal.


The buses fleet are countable; assign each bus a number starting with 1.  The passengers in any one bus are countable; for each bus assign each passenger a number starting with 1.  Then each passenger will have a unique combination (Bus#, Pass#).  

Let an individual passenger's Index# = Bus#+Pass#.  There is one passenger with Index=2: {(1,1)}, two with Index=3: {(1,2), (2,1)}, three with Index=4: {(1,3), (2,2), (3,1)} etc.

Then create a sequence by appending Index=2, Index=3, Index=4, etc: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), etc.  The passengers are assigned hotel rooms according to their position in the sequence.

I'll have to think some more to create something to show 2^I = I^I.

  Posted by Brian Smith on 2016-03-16 10:17:19
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