Proof of equal cardinality $|\Bbb N \times\Bbb N \times\Bbb N| = |\Bbb N|$

Question

How do I prove that the following sets have equal cardinality?

1. $|\Bbb N \times\Bbb N \times\Bbb N| = |\Bbb N|$ ($|\Bbb N \times\Bbb N| = |\Bbb N|$ also for that matter)
2. $|\Bbb Z \times\Bbb Z| = |\Bbb Z|$
3. $|\Bbb R \times\Bbb R| = |\Bbb R|$

Thank you!

Answer 1

Note that having size $|N|$ means something is infinite but enumerable; meaning, roughly, that you can come up with a way of listing all the elements in a single row.

Note: I am abiding by $N = \{0, 1, 2, 3, \ldots\}$.

To show $|N \times N| = |N|$, you could start by listing all the elements $(m, n)$ with $m,n \in N$ such that $m + n \leq 0$. Next, list all the elements with $m+n \leq 1$, and so forth. In this way, you produce a list of the elements; a list of the form:

$a_0, a_1, a_2, a_3 \ldots$ which can be matched up with elements of $N$ using $a_n \rightarrow n$.

The case for $N \times N \times N$ is similar, as is the case for anything $Z$ related.

Your third problem is a bit more involved, but one place to start is by thinking about decimal expansions. I won't say anything more unless you post what you have tried thus far.

Answer 2

It is easy to produce a sequence that includes all the elements in $\Bbb N\times\Bbb N\times\Bbb N$.

E.g. $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(0,0,1)$, $(2,0,0)$, $(1,1,0)$, $(1,0,1)$, $(0,2,0)$, $(0,1,1)$, $(0,0,2)$, $(3,0,0)$, $(2,1,0)$, $(2,0,1)$,...

This can be adjusted to the case of $\Bbb Z\times\Bbb Z\times\Bbb Z$ and/or any finite number of copies of $\Bbb N$ or $\Bbb Z$, or even $\Bbb Q$.

To show that $|\Bbb R|=|\Bbb R\times\Bbb R|$ requires, of course, a different argument.

Answer 3

The function:$$f:\mathbb{Z}^+\times\mathbb{Z}^+\times\mathbb{Z}^+\rightarrow \mathbb{Z}^+$$ that sends $(i,j,k)$ to $(2(2j-1)2^{i-1}-1)2^{k-1}$ is a bijection.