Bijections between Intervals: the Gap Map and the Reversal

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $[n]$ be as in Intervals of Natural Numbers §segment.

¶ Let $n\in\mathbb{N}_{0}$ and $j\in[n+1]$. Setting $g_{j}(k)=k$ for $k<j$ and $g_{j}(k)=k+1$ for $k\ge j$ defines a bijection $g_{j}$ from $[n]$ onto $[n+1]\setminus\{j\}$.

¶ Let $N\in\mathbb{N}$. The map $a\mapsto N+1-a$, with the difference $N+1-a$ defined since $a\le N+1$, is a bijection from $[N]$ onto $[N]$ and is its own inverse.