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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
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8↔19
9↔23
10↔29
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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?

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Some Thoughts Sets of all Rationals (Parts 2 and 3) | Comment 3 of 8 |
Part 2: Positive Rational Numbers
Express each positive rational number in the form x/y with gcd(x,y)=1.  Let n=x+y.  For each positive integer n create a list of rational numbers ordered by x in increasing magnitude. For the smallest n=1, there is one rational in list 1: {0/1}. The list n=2 is {1/1}.  Continuing on n=3:{1/2, 2/1}; n=4:{1/3, 3/1}; n=5:{1/4, 2/3, 3/2, 4/1}; etc.

Create a sequence by appending list 1, list 2, list 3, etc: 0/1, 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, etc.  This establishes a bijection between the positive rationals and positive integers by mapping integer n to the nth member in the sequence.  Therefore the set of positive rational numbers are countable.

Part 3: All Rational Numbers
Modify the above process to include negatives in each list. n=1 is still {0,1} but all other lists are doubled in size: n=2:{-1/1, 1/1}, n=3:{-2/1, -1/2, 1/2, 2/1}, etc.  Then the sequence becomes 0/1, -1/1, 1/1, -2/1, -1/2, 1/2, 2/1, -3/1, -1/3, 1/3, 3/1, etc.  Then this establishes the bijection which proves the set of all rational numbers are countable.

  Posted by Brian Smith on 2016-03-15 12:10:44
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