well defined mapping-function

Question

I would like to know how to show an mapping or function is well defined

i think in generale we use that :

-$f$ is well defined mapping iff $( x\in E\implies f(x)\in F)$

in particular when mapping have quotion set such as :

$$f: \mathbb{Z}_2 \rightarrow \mathbb{Z}, \overline{x} \mapsto f(\overline{x}) $$ we use : $$\forall x,y,\ x=y\Rightarrow f(x)=f(y)$$

Am i right ? and "what is mathematical formulas for well-defined for example :

by the way i read this https://en.wikipedia.org/wiki/Well-defined

Answer 1

Given a claimed function $f:A\to B$ that is given by a formula or algorithm for $f(x)$, we show it is well-defined by:

1. Showing that if $x\in A$ then the claimed method for finding $f(x)$ always gives a value. This prohibits things like $\frac 1x$ where $x=0$. This shows possibility and the correct domain.
2. Show that $f$ is consistent, i.e. that $x\in A\land x=y\implies f(x)=f(y)$. This shows lack of ambiguity. This step is what your example concentrates on.
3. Show if $x\in A$ then for the claimed value, $f(x)\in B$. This shows a correct codomain. This is what you emphasize in your own definition.

If some strange terminology or technique is used in the definition that also may be proven to be well-defined before $f$ itself can be shown to be so. If $f$ is defined in a different way you may need to show additional things such as $f$ is a set of ordered pairs.