Infinity Intuition Math Example 4

Solution

1. 1
   The integers are $\ldots, -2, -1, 0, 1, 2, \ldots$ and the natural numbers are $1, 2, 3, \ldots$
2. 2
   Define $f: \mathbb{N} \to \mathbb{Z}$ by: $f(1) = 0$, $f(2) = 1$, $f(3) = -1$, $f(4) = 2$, $f(5) = -2$, \ldots (pair positive $n$ with $\frac{n}{2}$ and odd $n$ with $-\frac{n-1}{2}$).
3. 3
   This bijection shows $|\mathbb{N}| = |\mathbb{Z}|$: both are countably infinite with the same cardinality.