The following is an exercise from Jacobson's Basic Algebra I:
Let $S=\{1,2,\cdots\}$. Give an example of two maps $\alpha, \beta$ of $S$ into $S$ such that $\alpha \beta=1_{S}$ but $\beta \alpha\ne 1_{S}$. Can this happen if $\alpha$ is bijective?
I have a fairly good answer for this exercise, I think.
An Example
Let $\alpha(s)=|s-2|$ and $\beta(s)=s+2$ for $s \in S$. Thus, $(\alpha \circ \beta)(s)=|(s+2)-2|=|s|=s$, which implies $\alpha \beta=1_{S}$.
However, $(\beta \circ \alpha)(s)=|s-2|+2$. We see that $(\beta \circ \alpha)(1)=3$, hence $\beta \alpha\ne 1_{S}$.
Consequences of $\alpha$ Being Bijective
If $\alpha$ is bijective, there exists an inverse map $\gamma$ such that $\gamma \alpha=\alpha \gamma=1_{S}$. Since $\alpha \beta=1_{S}$, we have that $\alpha \beta=\alpha \gamma$. Composing both sides of the equation with $\gamma$, we have that $\gamma(\alpha \beta)=\gamma(\alpha \gamma)$. Next, using the law of associativity, we have that $(\gamma \alpha)\beta=(\gamma \alpha)\gamma$. Since $\gamma \alpha=1_{S}$, we thus have $1_{S}\beta=1_{S}\gamma$. Hence, $\beta=\gamma$.
Since we previously stated that $\gamma \alpha=1_{S}$ and we've shown that $\gamma=\beta$, it must be so that $\beta\alpha=1_{S}$. $\square$
My question
Are there more natural examples of $\alpha$ and $\beta$? Furthermore, is my proof accurate? I ask the second question only as I see it directly pertains to the first.
P.S. By natural, I mean elegant or insightful.