What does the following statement in the definition of right inverse mean? ("For $b\in B$, $b\neq a\alpha$ for any $a$, define $b \beta=a_{1}\in A$") Question:
Let $A$ and $B$ be arbitrary sets, with $\alpha:A\rightarrow B$ an injection. Show how to define $\beta:B\rightarrow A$ such that $\alpha \beta$ is the identity function on $A$.
Solution:
For $a\in A$ define $(a\alpha)\beta=a$. For $b\in B$, $b\neq a\alpha$ for any $a$, define $b \beta=a_{1}\in A$.
Source: Groups: A Path To Geometry by R. P. Burn. Chapter: 1 Question: 24

The injection has the property $x\alpha=y\alpha \Rightarrow x=y$.
My problem lies in understanding this statement "For $b\in B$, $b\neq a\alpha$ for any $a$, define $b \beta=a_{1}\in A$".
Does "$b\neq a\alpha$ for any $a$" mean that no image of any $a\in A$ can be equal to itself? Why must this be true?

I’ll rephrase the solution in what I hope is a more understandable way. 

We need to define $b\beta$ for each $b\in B$. There are two kinds of elements of $B$: those that are in the range of $\alpha$, and those that are not. If $b=a\alpha$ for some $a\in A$, we define $b\beta=a$. Now fix a particular $a_1\in A$. If there is no $a\in A$ such that $b=a\alpha$, we define $b\beta=a_1$.

The last two sentences correspond to the statement about which you’re asking.
