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?

Create the sequence S=0,-1,1,-2,2,-3,3,...  In general this is S(n) = (n-1)/2 for odd n and -n/2 for even n.  This establishes a bijection from positive integers to all integers.  Therefore the set of all integers is countable

Posted by Brian Smith on 2016-03-15 10:45:23