The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable

Question

I'm examining a proof I have read that claims to show that the Cartesian product $\mathbb{N} \times \mathbb{N}$ is countable, and as part of this proof, I am looking to show that the given map is surjective (indeed bijective), but I'm afraid that I can't see why this is the case. I wonder whether you might be able to point me in the right direction?

Indeed, the proof begins like this:

"For each $n \in \mathbb{N}$, let $k_n, l_n$ be such that $n = 2^{k_n - 1} \left(2l_n - 1 \right)$; that is, $k_n - 1$ is the power of $2$ in the prime factorisation of $n$, and $2 l_n - 1$ is the (necessarily odd) number $\frac{n}{2^{k_n - 1}}$."

It then states that $n \mapsto \left(k_n , l_n \right)$ is a surjection from $\mathbb{N}$ to $\mathbb{N} \times \mathbb{N}$, and so ends the proof.

I can intuitively see why this should be a bijection, I think, but I'm not sure how to make these feelings rigorous?

I suppose I'd say that the map is surjective since given any $\left(k_n , l_n \right) \in \mathbb{N} \times \mathbb{N}$ we can simply take $n$ indeed to be equal to $2^{k_n - 1} \left(2l_n - 1 \right)$ and note that $k_n - 1 \geq 0$ and thus $2^{k_n - 1}$ is both greater or equal to one so is a natural number (making the obvious inductive argument, noting that multiplication on $\mathbb{N}$ is closed), and similarly that $2 l_n - 1 \geq 2\cdot 1 - 1 = 1$ and is also a natural number, and thus the product of these two, $n$ must also be a natural number. Is it just as simple as this?

I suppose my gut feeling in the proving that the map is injective would be to assume that $2^{k_n - 1} \left(2 l_n - 1 \right) = 2^{k_m - 1} \left(2 l_m - 1 \right)$ and then use the Fundamental Theorem of Arithmetic to conclude that $n = m$. Is this going along the right lines? The 'implicit' definition of the mapping has me a little confused about the approach.

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On a related, but separate note, I am indeed aware that if $K$ and $L$ are any countable sets, then so is $K \times L$, so trivially, taking the identity mapping we see trivially that this map is bijective and therefore that $\mathbb{N}$ is certainly countable (!), and thus so is $\mathbb{N} \times \mathbb{N}$. Hence, it's not really the statement that I'm interested in, but rather the exciting excursion into number theory that the above alternative proof provides.

Answer 1

Your intuition is correct. We use the fundamental theorem of arithmetic, namely the prime factorization is unique (up to order, of course).

First we prove injectivity:

Suppose $(k_n,l_n),(k_m,l_m)\in\mathbb N\times\mathbb N$ and $2^{k_n - 1} (2 l_n - 1 ) = 2^{k_m - 1} (2 l_m - 1)$.

$2$ is a prime number and $2t-1$ is odd for all $t$, and so we have that the power of $2$ is the same on both sides of the equation, and it is exactly $k_n=k_m$.

Divide by $2^{k_n}$ and therefore $2l_n-1 = 2l_m-1$, add $1$ and divide by $2$, so $(k_n,l_n)=(k_m,l_m)$ and therefore this mapping is injective.

Surjectivity it is even simpler, take $(k,l)\in\mathbb N\times\mathbb N$ and let $n=2^{k-1}(2l-1)$. Now $n\mapsto(k,l)$, because $2l-1$ is odd, so the powers of $2$ in the prime decomposition of $n$ are exactly $k-1$, and from there $l$ is determined to be our $l$. (If you look closely, this is exactly the same argument for injectivity only applied "backwards", which is a trait many of the proofs of this kind has)

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As for simpler proofs, there are infinitely many... from the Cantor's pairing function ($(n,m)\mapsto\frac{(n+m)(n+m+1)}{2}+n$), to Cantor-Bernstein arguments by $(n,m)\mapsto 2^n3^m$ and $k\mapsto (k,k)$ for the injective functions. I like this function, though. I will try to remember it and use it next time I teach someone such proof.

Answer 2

It is possible that the notation is getting in the way of seeing what's going on.

Every positive integer $n$ is a power of $2$ times an odd number. (Note that $2^0$ is a power of $2$.)

For example, $840=2^3 \times 105$ and $747=2^0 \times 747$.

In symbols, $$n=2^e \times a$$ where $e$ is a non-negative integer, and $a$ is an odd positive integer.

Moreover, the above representation is unique: If we know $e$ and $a$, we know $n$, and if we know $n$, we know $e$ and $a$.

Since $a$ is odd, it is of the form $2b+1$, where $b$ is a non-negative integer. As $a$ ranges over the odd positive integers, $b$ ranges over the non-negative integers.

Let $f$ be the function that takes $n$ to the ordered pair $(e,b)$. For example, since $840=2^3 \times 105$, we have $f(840)=(3,52)$. Similarly, $f(747)=(0,373)$.

Let $\mathbb{N}_0$ be the set of non-negative integers. Then $f$ is a bijective map from $\mathbb{N}$ to $\mathbb{N}_0 \times \mathbb{N}_0$. This is an immediate consequence of the fact that if we know $e$ and $b$, we know $n$, and conversely that if we know $n$, we know $e$ and $b$.

Not exactly what are looking for, but awfully close! And easy to fix, since $\mathbb{N}$ is just $\mathbb{N}_0$ pushed forward by $1$.

All we need to do is to map $n$ to $(e+1, b+1)$.

Answer 3

For a quick proof, why not take a pair of primes, like, say , 2 and 3, then inject a pair (a,b) inf: $\mathbb N \times \mathbb N$ to $2^a3^b$, and, for the opposite direction, inject n in $\mathbb N$ into $\mathbb N\times \mathbb N$ by $n\rightarrow(n,0)$, and then use Rick Schroder-Bernstein. It seems clear that if f(a,b)=f(a',b'), so that $2^a3^b=2^{a'}3^{b'}$, then $2^{a-a'}3^{b-b'}=1$ , forcing a=a' and b=b'(alternatively, if $2^x3^y=1$ then both $2^x$ and $3^y$ must divide 1, so that x=y=0, and injectivity follows); OTOH, if (n,0)=(n',0) , then clearly n=n'

EDIT: Cantor-Schroeder-Bernstein maps can be extended (uniquely) into bijections.

Answer 4

In the OP's question he writes

is a surjection from $\mathbb{N}$ to $\mathbb{N} \times \mathbb{N}$, and so ends the proof

That is a bit amusing since surjectivity is all that is needed - the image of any enumeration is either a finite or countably infinite set.

The OP also states

Hence, it's not really the statement that I'm interested in, but rather the exciting excursion into number theory that the above alternative proof provides.

So for another 'exciting' (or perhaps only humorous bordering on silly) excursion, we ask,

What is the minimal amount of number theory needed to build a surjective enumeration of
$\mathbb{N}$ onto $\mathbb{N} \times \mathbb{N}$?

Let $T = \{\,(2^j3^k,m) \, | \, j,k \in \Bbb N_0 \land m \in \Bbb N\}$.

We have an injective mapping $f$ from $\mathbb{N}$ into $T$ defined by

$\quad m \mapsto (2^0 3^0, m)$

We can define an equivalence relation $\sim$ on $T$,

$\quad (2^j 3^k, m) \sim (2^\bar j 3^\bar k, \bar m) \; \text{ if } 2^j 3^k m = 2^\bar j 3^\bar k \bar m $.

We have the quotient map $\rho$ defined by $\sim$,

$\quad \rho: T \mapsto \frac{T}{\sim}$

Any element in the image of $\rho$ can be represented by an ordered pair $(2^j 3^k, m)$ with $m$ minimal. To show that these representatives must be unique boils down to proving that

$\tag 1 \text{If } 2^s = 3^t \text{ Then } s = 0 \land t = 0$

So we can define a mapping $g$ from the quotient to $\mathbb{N} \times \mathbb{N}$, by taking the unique representative $(2^j 3^k, m)$ of a block and mapping it to $(j,k)$.

It is always true that $(2^0 3^0, 2^j 3^k) \sim (2^j 3^k, 1)$.

Putting this altogether we see that the mapping $g \circ \rho \circ f$ is a surjection.

So $\text{(1)}$ is the only number theory needed here.