Why are $-1$ and $1$ generators for the Set of integers under addition? I'm reading my textbook and I'm confused why $-1$ and $1$ are generators for the group of integers under addition.
For example for 1:
we have $1, 1+1=2, 1+1+1=3, 1+1+1+1=4,$ etc.
So shouldn't $1$ be a generator for only the group of positive integers under addition?
For $-1$: we have
$-1, -1+-1=-2, -1+-1+-1=-3$.  So shouldn't $-1$ be a generator for only the group of negative integers under addition?
Or is it that both $1$ and $-1$ are generators for the set of integers under addition because in the definition of cyclic subgroup $\{a^n; n\in \Bbb Z \}$, $n$ can take on negative powers? Or am I just confused? Thanks.

$(-1)=\{(-1)a; a\in \Bbb Z \}=\{-a; a\in \Bbb Z \}=\Bbb Z$ and $(+1)=\{(+1)a; a\in \Bbb Z \}=\{a; a\in \Bbb Z \}=\Bbb Z$, so $$(-1)=(1)=\Bbb Z$$
