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8/13/2019 Proving Countable Sets
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A good way of proving that a set is countable
Prerequisites
The definition of a countable set, function-related notions such as injections and surjections.
Quick description
If you can find a function from to such that every has finitely many preimages, then is
countable.
See also
A quick way of recognising countable sets
General discussion
Here is a quick informal account of a standard proof that the set is countable: we can list
its elements in the order , , , , , , , and so on. (This is informal
because I have talked about "lists" and have not actually defined how the sequence continues.)
We can view this proof geometrically as follows: in order to count through the set , whichforms an infinite grid in the plane, we note that each downward-sloping diagonal (that is, a set of
pairs of positive integers with constant sum) is finite, and then we count through each of thesesets in turn.
Here, we are making use of the very simple principle that a countable union of finite sets is
countable. To put this more formally, if is a set that can be written as a union for somecollection of finite sets, then is countable.
This is very easy to see informally: one just counts through each in turn (leaving out elements
that have already been counted). It is not important for this article, but for completeness let us
briefly see how we might prove it more formally. We could write each set as ,
where is the size of . And then we could define an ordering on all pairs with
by taking if and only if either (i) or (ii) and . And then we would
define a function by the following procedure. We would repeatedly choose the element
with minimal index for which was not yet defined, and we would define it to be the
minimal integer that was not yet assigned as a value of . It is not hard to check that one ends up
assigning a value to all elements of and we never assign the same value twice. Thus, is aninjection and is countable.
In particular, a set is countable if there is a function such that every has finitely
many preimages.(This means that for every there are only finitely many with
.) This follows because we can define to be , and then we have
expressed as a countable union of finite sets.
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This observation is not particularly interesting as a theoretical statement, but as a tool for givingquick proofs of countability it is extremely useful. Here are some examples.
Example 1
To prove that the rational numbers form a countable set, define a function that takes eachrational number (which we assume to be written in its lowest terms, with ) to the
positive integer . The number of preimages of is certainly no more than , so we
are done.
As another aside, it was a bit irritating to have to worry about the lowest terms there. For some
reason many mathematicians are afraid of multifunctions (that is, things that are like functionsexcept that each element in the domain can map to several elements in the range) and sweep
them under the carpet. But for the rationals it is convenient to have them. We define a
multifunction from to by mapping to . This looks as though it isn't well-defined,
which as a function it isn't, but if we think of it as a multifunction, then for instance the rational
number maps to all possible values of for which . (If you really don't like this,then you could rephrase it in terms of bipartite graphs or something like that.) Each positiveinteger still has only finitely many preimages (defined to be any rational number that has at
least one image equal to ), and this proves the result since any multifunction with that propertycan clearly be restricted to a function with that property (just select, for each element of the
domain, one of its images).
Example 2
To prove that the set of all finite subsets of is countable, how might we find a suitable
function? An obvious function to consider is , but this doesn't work since there are
infinitely many sets of any given size. (OK, apart from size zero.) But there's another very simplefunction that works: . Clearly, the number of sets with maximal element is finite
(in fact, it is ), so we are done.
Example 3
To prove that the set of all polynomials with integer coefficients is countable is a similarexercise, but slightly more complicated. It is tempting to consider the sum of the absolute values
of the coefficients, but then we notice that the polynomials all have coefficientswith absolute values adding up to 1. So we need to restrict the degree somehow. But that is very
easy indeed: given a polynomial we define to be the degree of plus the sum of the
absolute values of the coefficients of .
Example 4
To prove that the set of all algebraic numbers is countable, it helps to use the multifunction idea.Then we map each algebraic number to every polynomial with integer coefficients that has as
a root, and compose that with the function defined in Example 3. It is easy to check (using the
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fact that every polynomial has finitely many roots) that for every integer there are at mostfinitely many algebraic numbers that map to , and we are done.
General discussion
A tool that is more often presented in treatments of countability is the fact that a countable unionof countable sets is countable. That translates into the more general principle that if you can finda function such that each has at most countablymany preimages, then is
countable. The point of this article is that it is almost always possible to use a function with onlyfinitely many preimages for each , and that this often leads to short, easy proofs.
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