Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathbb{N}$ be the set of natural numbers, let countable sets be as in Countable Set (i) and finite sets as in Finite Sets (i), let maps be as in Functions, Values of a Function, and Functions from One Class to Another (iii) and surjective maps and bijections as in Injective, Surjective and Bijective Functions between Classes, let $\times$ denote the Cartesian product, which for two sets is a set by Membership in a Cartesian Product, and the Cartesian Product of Two Sets Is a Set (ii), let images of sets under maps be as in Sets and Maps: Ordinary Notation (vii), where they are sets, and let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and of rational numbers, as in The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals (ii) and The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals (iii). Let $X$ and $Y$ be sets.

(i) $\mathbb{N}$ is countable.

(ii) There is a bijection $\tau:\mathbb{N}\to\mathbb{N}\times\mathbb{N}$; in particular, $\mathbb{N}\times\mathbb{N}$ is countable.

(iii) If $X$ is finite, then $X$ is countable.

(iv) If $X$ is countable and $Y\subseteq X$, then $Y$ is countable.

(v) If $X$ is countable and there is a surjective map $g:X\to Y$, then $Y$ is countable.

(vi) Let $A$ and $B$ be sets and $g:A\to B$ a map. If $X\subseteq A$ and $X$ is countable, then the image $g(X)$ is countable.

(vii) If $X$ and $Y$ are countable, then $X\times Y$ is countable.

(viii) $\mathbb{Z}$ is countable.

(ix) $\mathbb{Q}$ is countable.

(x) $\mathbb{Q}\times\mathbb{Q}$ is countable.