When is it necessary to prove well definedness?

Question

Last semester, we tackled group theory. Our professor did not require us to show that a mapping $\phi:G\to G'$ is well-defined before proving that it is a group homomorphism. Now, our new professor requires us to show that $\phi$ is well-defined first before proving that $\phi:R\to R'$ is a ring homomorphism. So, in general, when is it necessary to show that a mapping is well-defined?

Answer 1

When there's a real chance that the map is not well-defined. For instance, the map $\psi\colon\mathbb{Z}_2\longrightarrow\mathbb Z$ defined by $\psi(\overline x)=x$ is not well-defined, since $\overline 0=\overline 2$ but $\psi(\overline0)\neq\psi(\overline2)$.

Answer 2

On a general note: A mathematical object, presented in some way, is well-defined if it really is what it’s implicitly or explicity claimed to be in its presentation. So saying “The map $φ \colon G → G'$ is a well-defined group homomorphism,” means “The map $φ \colon G → G'$ really is a map and, moreover, a group homomorphism.”

Proving well-definedness of a mathematical object is therefore appropriate whenever there’s reasonable doubt of it, as José Carlos Santos has already pointed out.

So bottom line: You have to apply common sense.

One way to do this is the following: Play devil’s advocate and try to find an ill-defined object in your writing, being very nitpicky about it. If you spot any chance of exposing something to be ill-defined and you tackle it, but it withstands and really turns out to be well-defined, but you had to convince yourself first (perhaps even by writing down an argument) – that’s when you should give the argument to prove the object is well-defined.

Answer 3

There are a number of standard forms for defining functions, such as:

• a pointwise formula $f(x) = \ldots$ where $x$ is a generic variable ranging over the domain of $f$
• setting $f$ equal to the result of an operation whose values are functions, such as function composition.
• defining a linear transformation by specifying its values on a basis

If you've defined a function in one of these ways, no further comment is necessary.

The issue is when you specify a relation or more general correspondence; e.g. by defining $f$ to be a correspondence (where $p$ is an integer and $q$ is a nonzero integer)

$$ f\left( \frac{p}{q} \right) = \frac{p^2 + pq + q^2}{p^2 - 2q^2} $$

This is not one of the standard forms; you don't have anything telling you that this defines $f$ as a function. To show that $f$ is a well-defined function means to show that this correspondence actually satisfies the definition of a function. In this case, you would need to check:

• Every rational number can be expressed as $p/q$
• Any two ways to write the same rational number give the same value on the right hand side