The exact textbook question is:
Let $\mathbb{N}$ denote the set $\{1,2,3,\ldots,\}$ of natural numbers, and let $s:N \to N$ be the shift map,
defined by $s(n) = n + 1$. Prove that $s$ has no right inverse, but that it has infinitely many
left inverses.
The only piece of this throwing me off is the infinitely many left inverses part.
The map $s$ is injective and it clearly has a valid left inverse function:
$l(n) = n – 1$.
$l \circ s = \text{id}_\mathbb{N}$
However, I don't see how $s$ has infinitely many left inverses. It seems to have just one left inverse.
Best Answer
Hint: Your definition of $l(n)$ (which is otherwise correct) fails for $n=1$, so you need to define that case separately.