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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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part 2 & 3 | Comment 2 of 8 |
part 2)
represent all positive rational numbers as p/q with gcd(p,q)=1
now list them in this manner:
start with all the rations with q=1
associate these with 2^p
next go with all q=2 and associate with 3^p

in general, associate p/q with the p'th power of the q'th prime number.

This creates a mapping from the positive rationals to a unique positive integer.

part 3)

use the same mapping as part 2 with the following alteration
for each value of q, associate positive p values with the p'th power of the (2q-1)'th prime and the negative p values with the |p|'th power of the (2q)'th prime.

  Posted by Daniel on 2016-03-15 11:44:42
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