Countable Set - Formal Definition and Properties

Formal Definition and Properties

By definition a set S is countable if there exists an injective function

from S to the natural numbers

It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view is not tenable, however, under the natural definition of size.

To elaborate this we need the concept of a bijection. 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

Since every element of { a, b, c } is paired with precisely one element of { 1, 2, 3 }, and vice versa, this defines a bijection.

We now generalize this situation and define two sets to be of the same size if (and only if) there is a bijection between them. For all finite sets this gives us the usual definition of "the same size". What does it tell us about the size of infinite sets?

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, using n ↔ 2n, so that

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 ordered pairs of natural numbers is countably infinite, as can be seen by following a path like the one in the picture:

The resulting mapping is like this:

0 ↔ (0,0), 1 ↔ (1,0), 2 ↔ (0,1), 3 ↔ (2,0), 4 ↔ (1,1), 5 ↔ (0,2), 6 ↔ (3,0) ....

It is evident that this mapping will cover all such ordered pairs.

Interestingly: if you treat each pair as being the numerator and denominator of a vulgar fraction, then for every positive fraction, we can come up with a distinct number corresponding to it. This representation includes also the natural numbers, since every natural number is also a fraction N/1. So we can conclude that there are exactly as many positive rational numbers as there are positive integers. This is true also for all rational numbers, as can be seen below (a more complex presentation is needed to deal with negative numbers).

Theorem: The Cartesian product of finitely many countable sets is countable.

This form of triangular mapping recursively generalizes to vectors of finitely many natural numbers by repeatedly mapping the first two elements to a natural number. For example, (0,2,3) maps to (5,3) which maps to 39.

Sometimes more than one mapping is useful. This is where you map the set which you want to show countably infinite, onto another set; and 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 because p/q maps to (p, q).

What about infinite subsets of countably infinite sets? Do these have fewer elements than N?

Theorem: Every subset of a countable set is countable. In particular, every infinite subset of a countably infinite set is countably infinite.

For example, the set of prime numbers is countable, by mapping the n-th prime number to n:

• 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.

What about sets being "larger than" N? An obvious place to look would be Q, the set of all rational numbers, which intuitively may seem much bigger than N. But looks can be deceiving, for we assert:

Theorem: Q (the set of all rational numbers) is countable.

Q can be defined as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto the subset of ordered triples of natural numbers (a, b, c) such that a ≥ 0, b > 0, a and b are coprime, and c ∈ {0, 1} such that c = 0 if a/b ≥ 0 and c = 1 otherwise.

• 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)
• ...

By a similar development, the set of algebraic numbers is countable, and so is the set of definable numbers.

Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable.

For example, given countable sets a, b, c ...

Using a variant of the triangular enumeration we saw above:

• a₀ maps to 0

• a₁ maps to 1
• b₀ maps to 2

• a₂ maps to 3
• b₁ maps to 4
• c₀ maps to 5

• a₃ maps to 6
• b₂ maps to 7
• c₁ maps to 8
• d₀ maps to 9

• a₄ maps to 10
• ...

Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem.

Also note that the axiom of countable choice is needed in order to index all of the sets a, b, c,...

Theorem: The set of all finite-length sequences of natural numbers is countable.

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.

Theorem: The set of all finite subsets of the natural numbers is countable.

If you have a finite subset, you can order the elements into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.

The following theorem gives equivalent formulations in terms of a bijective function or a surjective function. A proof of this result can be found in Lang's text.

Theorem: Let S be a set. The following statements are equivalent:

1. S is countable, i.e. there exists an injective function

   .

2. Either S is empty or there exists a surjective function

   .

3. Either S is finite or there exists a bijection

   .

Several standard properties follow easily from this theorem. We present them here tersely. For a gentler presentation see the sections above. Observe that in the theorem can be replaced with any countably infinite set. In particular we have the following Corollary.

Corollary: Let S and T be sets.

1. If the function

   is injective and T is countable then S is countable.

2. If the function

   is surjective and S is countable then T is countable.

Proof: For (1) observe that if T is countable there is an injective function Then if is injective the composition is injective, so S is countable.

For (2) observe that if S is countable there is a surjective function Then if is surjective the composition is surjective, so T is countable.

Proposition: Any subset of a countable set is countable.

Proof: The restriction of an injective function to a subset of its domain is still injective.

Proposition: The Cartesian product of two countable sets A and B is countable.

Proof: Note that is countable as a consequence of the definition because the function given by is injective. It then follows from the Basic Theorem and the Corollary that the Cartesian product of any two countable sets is countable. This follows because if A and B are countable there are surjections and . So

is a surjection from the countable set to the set and the Corollary implies is countable. This result generalizes to the Cartesian product of any finite collection of countable sets and the proof follows by induction on the number of sets in the collection.

Proposition: The integers are countable and the rational numbers are countable.

Proof: The integers are countable because the function given by if n is non-negative and if n is negative is an injective function. The rational numbers are countable because the function given by is a surjection from the countable set to the rationals .

Proposition: If is a countable set for each then is countable.

Proof: This is a consequence of the fact that for each n there is a surjective function and hence the function

given by is a surjection. Since is countable the Corollary implies is countable. We are using the axiom of countable choice in this proof in order to pick for each a surjection from the non-empty collection of surjections from to .

Cantor's Theorem asserts that if is a set and is its power set, i.e. the set of all subsets of, then there is no surjective function from to . A proof is given in the article Cantor's Theorem. As an immediate consequence of this and the Basic Theorem above we have:

Proposition: The set is not countable; i.e. it is uncountable.

For an elaboration of this result see Cantor's diagonal argument.

The set of real numbers is uncountable (see Cantor's first uncountability proof), and so is the set of all infinite sequences of natural numbers. A topological proof for the uncountability of the real numbers is described at finite intersection property.