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## Countable set
In mathematics, a set is called
The elements of a finite set can be listed, say {a To elaborate this we need the concept of a bijection. Do the sets {1,2,3} and {a,b,c} have the same size? "Obviously, yes." "How do you know?" "Well it's obvious. Look, they've both got 3 elements". "What's a 3?" This may seem a strange situation but, although a "bijection" seems a more advanced concept than a "number", the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets. This is where the concept of a bijection comes in: define the correspondence
- a ↔ 1, b ↔ 2, c ↔ 3
precisely one element of {1,2,3} (and vice versa) this defines a bijection.
We now generalise this situation and Consider the sets A = {1,2,3,...}, the set of positive integers and B = {2,4,6,...}, the set of even positive integers. We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recall that to prove this we need to exhibit a bijection between them. But this is easy: 1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ... As in the earlier example every element of A has been paired off with precisely one element of B, and vice versa. Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets, a situation which is impossible for finite sets. Likewise, the set of all pairs of natural numbers is countably infinite through the following "triangular" mapping: - (0,0) maps to 0
- (0,1) maps to 1
- (1,0) maps to 2
- (0,2) maps to 3
- (1,1) maps to 4
- (2,0) maps to 5
- ...
- (m,n) maps to 0.5*k*(k+1)+m, where k=m+n
THEOREM: The Cartesian product of finitely many countable sets is countable.This triangular mapping recursively generalizes to vectors of finitely many natural numbers by repeatedly mapping the first two elements to a natural number. For example, (2,0,3) maps to (5,3) which maps to 41. Sometimes it is useful to use more than one mapping. This is where you map the set you want to show as countably infinite to another set. You then map this other set to the natural numbers. For example, the positive rational numbers can easily be mapped to (a subset of) the pairs of natural numbers through p/q maps to (p,q).
What about subsets of infinite countable sets? Are those smaller than
(Proof omitted)
For example, the set of prime numbers is countable, by mapping the - 2 maps to 1
- 3 maps to 2
- 5 maps to 3
- 7 maps to 4
- 11 maps to 5
- 13 maps to 6
- 17 maps to 7
- 19 maps to 8
- 23 maps to 9
- etc.
N? An obvious place to look would be Q, the set of all rational numbers, which is "clearly" much bigger than N. But looks can be deceiving, for we assert
- 0 maps to (0,1,0)
- 1 maps to (1,1,0)
- -1 maps to (1,1,1)
- 1/2 maps to (1,2,0)
- -1/2 maps to (1,2,1)
- 2 maps to (2,1,0)
- -2 maps to (2,1,1)
- 1/3 maps to (1,3,0)
- -1/3 maps to (1,3,1)
- 3 maps to (3,1,0)
- -3 maps to (3,1,1)
- 1/4 maps to (1,4,0)
- -1/4 maps to (1,4,1)
- 2/3 maps to (2,3,0)
- -2/3 maps to (2,3,1)
- 3/2 maps to (3,2,0)
- -3/2 maps to (3,2,1)
- 4 maps to (4,1,0)
- -4 maps to (4,1,1)
- ...
For example, given countable sets Using a variant of the triangular enumeration we saw above:
`a`_{0}maps to 0`a`_{1}maps to 1`b`_{0}maps to 2`a`_{2}maps to 3`b`_{1}maps to 4`c`_{0}maps to 5`a`_{3}maps to 6`b`_{2}maps to 7`c`_{1}maps to 8`d`_{0}maps to 9`a`_{4}maps to 10- ...
a, b, c,... are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem.
This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem.
If you have a finite subset, you can order the elements to get a finite sequence. There are only countably many finite sequences, so there are also only countably many finite subsets. Further theorems about uncountable sets:
- The set of real numbers is uncountable, and so is the set of all sequences of natural numbers and the set of all subsets of
**N**(see Cantor's diagonal argument).
See also:
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