Editing Well-defined expression (section)

==Independence of representative==
Questions regarding the well-definedness of a function often arise when the defining equation of a function refers not only to the arguments themselves, but also to elements of the arguments, serving as [[representative (mathematics)|representative]]s. This is sometimes unavoidable when the arguments are [[coset]]s and when the equation refers to coset representatives. The result of a function application must then not depend on the choice of representative.

===Functions with one argument===
For example, consider the following function:
:<math>
\begin{matrix}
f : & \Z/8\Z         & \to     & \Z/4\Z\\
    & \overline{n}_8 & \mapsto & \overline{n}_4,
\end{matrix}</math>
where <math>n\in\Z, m\in \{4,8\}</math> and <math>\Z/m\Z</math> are the [[modular arithmetic|integers modulo ''m'']] and <math>\overline{n}_m</math> denotes the [[modular arithmetic#Congruence classes|congruence class]] of ''n'' mod ''m''.

N.B.: <math>\overline{n}_4</math> is a reference to the element <math>n \in \overline{n}_8</math>, and <math>\overline{n}_8</math> is the argument of ''<math>f</math>''.

The function ''<math>f</math>'' is well defined, because:

:<math>n \equiv n' \bmod 8 \; \Leftrightarrow \; 8 \text{ divides } (n-n') \Rightarrow \; 4 \text{ divides } (n-n') \; \Leftrightarrow \; n \equiv n' \bmod 4.</math>

As a counter example, the converse definition:

:<math>
\begin{matrix}
g : & \Z/4\Z         & \to     & \Z/8\Z\\
    & \overline{n}_4 & \mapsto & \overline{n}_8,
\end{matrix}</math>
does not lead to a well-defined function, since e.g. <math>\overline{1}_4</math> equals <math>\overline{5}_4</math> in <math>\Z/4\Z</math>, but the first would be mapped by <math>g</math> to <math>\overline{1}_8</math>, while the second would be mapped to <math>\overline{5}_8</math>, and <math>\overline{1}_8</math> and <math>\overline{5}_8</math> are unequal in <math>\Z/8\Z</math>.

===Operations===
In particular, the term ''well-defined'' is used with respect to (binary) [[operation (mathematics)|operation]]s on cosets. In this case, one can view the operation as a function of two variables, and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some ''n'' can be defined naturally in terms of integer addition. 
:<math>[a]\oplus[b] = [a+b]</math>

The fact that this is well-defined follows from the fact that we can write any representative of <math>[a]</math> as <math>a+kn</math>, where <math>k</math> is an integer. Therefore, 

:<math>[a]\oplus[b] = [a+kn]\oplus[b] = [(a+kn)+b] = [(a+b)+kn] = [a+b];</math>

similar holds for any representative of <math>[b]</math>, thereby making <math>[a+b]</math> the same, irrespective of the choice of representative.