Injection from natural numbers to union of two aleph null sets. Let $X$ and $Y$ be disjoint sets, each having cardinality of the set of all natural numbers. How do I see that there is an injection $f: \mathbb{N} \to X \cup Y$?
If $f\colon \mathbb{N}\to X$ and $g\colon \mathbb{N}\to Y$ are your bijections, then $f\colon \mathbb{N}\to X\cup Y$ is also an injection. Or you can use $g$.
$$f: (\mathbb{N},\mathbb{N} )\to \mathbb{N}$$

$$(a,b)\to 2^{a}3^{b}$$

Obviously, it is injective