Question:
For any bijection $\alpha:A\rightarrow B$, define a bijection $\beta:B\rightarrow A$ such that $\alpha\beta$ is the identity function $I:A\rightarrow A$ and $\beta\alpha$ is the identity function $B\rightarrow B$.
Prove that either $\alpha\beta=I:A\rightarrow A$ or $\beta\alpha=I:B\rightarrow B$ determines $\beta$ uniquely.
So bijections have two-sided inverses.
Source: Groups: A Path To Geometry by R. P. Burn Chapter: 1 Question: 30
My proof:
$\alpha$ is a bijection $\Rightarrow$ For each $b\in B$ there exists a unique $a\in A$ such that $a\alpha=b$.
We define a bijective function $\beta$ such that for each $a\in A$ there exists a unique $b\in B$ such that $b\beta=a$.
For $I=\alpha\beta$, we have $(a\alpha)\beta=a$.
For $I=\beta\alpha$, we have $(b\beta)\alpha=b=a\alpha$.
We have $(a\alpha)\beta=a$ and $b\beta=a\Rightarrow\beta$ is determined uniquely. In other words, the left inverse is equal to the right inverse.
I was a bit confused with the wording of the question. Are all my statements correct?