Proofs involving Well-Defined and One-to-One 
Chartrand, 3rd Ed, P224-225: Define a relation $R$ as a relation from A to B.
$R$ is well-defined means: $(a,b), (a,c) \in R \implies b = c$.
P220: A function $f: A \to B$ is one-to-one means:
For all $x, y \in A$, if $f(x) = f(y)$, then $x = y$.

I've observed that in the proofs of some functions, one can prove injectivity merely by reversing all the steps in the proof of the definition of well-defined.
$1.$ Is this always admissible and convenient? If not, when and why not?
$2.$ Is the converse true? Could one equally have started with proving the definition of well-defined and then reversed every step to prove injectivity?

Let's represent function $f:A\to B$ as a set of ordered pairs $f=\{(a_1,b_1), (a_2,b_2)\ldots\}$.
If this is a one-to-one function, then:


*

*If $(a,b)\in f$ and $(c,b)\in f$, then $a=c$.

*For all $a\in A$, there is exactly one $b\in B$ such that $(a,b)\in f$.
You have correctly noticed that if we reverse all the ordered pairs, then condition (1) corresponds to well-defined.  However we have this extra condition (2), which distinguishes the two definitions.
