I know what "Well Defined" means, and I know how to show that something specifically isn't well defined — that is, by presenting a case where two different but equivalent forms of $x$ have different images $f(x)$… But if I'm given a function and am asked to prove that it IS well defined, what steps do I have to do to show this?
Take for example the assertion that you can add and multiply congruence classes as such:
- $[a]+[c]=[a+c]$, and
- $[a]\cdot[c]=[a\cdot c]$
My textbook says that both are well defined statements because the theorem that $a\equiv b\mod{m}$ and $c\equiv d\mod{m}$ imply that $a+c\equiv b+d\mod{m}$ and $ac\equiv bd\mod{m}$ implies they are… I just can't quite follow why.
In general though, neglecting this specific example, what steps do I need to follow to show that something is well defined?
Best Answer
Morally, stating that an object is "well-defined" means that it shouldn't matter what name we call it. Here, we might have issues, since $a$ and $a + m$ are two very different-looking names for the same object, since we're only considering numbers up to addition of multiples of $m$.
So in general, to check well-definition, you need to write down an object and an arbitrary name for it, and make sure that the particular name doesn't change the result of a function.
So here in particular, if we fix an integer $a$, its other names are all of the form $a + km$ with $k$ an integer. Likewise, every name for $c$ has the form $c + nm$ for some $n$. Now check that the result using $a$ and $c$ agrees with the result using $a + km$ and $c + nm$, and you're done.