Infinity Intuition Math Example 1

Solution

1. 1
   Define a function $f: \mathbb{N} \to \{2, 4, 6, \ldots\}$ by $f(n) = 2n$.
2. 2
   This is injective: if $2m = 2n$ then $m = n$. It is surjective: every even positive integer $2k$ has preimage $k \in \mathbb{N}$.
3. 3
   Since $f$ is a bijection, the two sets have the same 'size' (cardinality) — they are both countably infinite.
4. 4
   This seems paradoxical because the evens are a proper subset of $\mathbb{N}$, yet they pair up perfectly with all of $\mathbb{N}$.