I want to prove $\mathbb{N} \sim \mathbb{Z}$ by indication of a bijection, thus the equipotency of the two sets. I know that I have to prove reflexivity, symmetry and transitivity. The reflexivity would be the identity map $f(x) = x$ on $\mathbb{N}$. For symmetry I would have to show two functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ and $f^{-1}: \mathbb{Z} \rightarrow \mathbb{N}$. But what would be such functions? I also don't know how to show transitivity, since I don't have a third set.
How would I prove symmetry and transitivity?
No, you don't have to prove reflexivity, symmetry and transitivity. You need to show a bijection exists, which (if you are exhibiting one) means proving surjectivity and injectivity. You want a function $f: \Bbb {N \to Z}$ that meets these requirements. If you take the odd numbers of $\Bbb N$ to the nonnegative ones of $\Bbb Z$ and the evens of $\Bbb N$ to the negative ones of $\Bbb Z$ you can get there.