How do you in general prove that a function is well-defined?
$$f:X\to Y:x\mapsto f(x)$$
I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the following two things:
- Every element in the domain maps to an element in the codomain:
$$x\in X \implies f(x)\in Y$$ - The same element in the domain maps to the same element in the codomain:
$$x=y\implies f(x)=f(y)$$
At the moment I'm trying to prove this function is well-defined: $$f:(\Bbb Z/12\mathbb Z)^∗→(\Bbb Z/4\Bbb Z)^∗:[x]_{12}↦[x]_4 ,$$ but I'm more interested in the general procedure.
Best Answer
When we write $f\colon X\to Y$ we say three things:
So to say that something is well-defined is to say that all three things are true. If we know some of these we only need to verify the rest, for example if we know that $f$ has the third property (so it is a function) we need to verify its domain is $X$ and the range is a subset of $Y$. If we know those things we need to verify the third condition.
But, and that's important, if we do not know that $f$ satisfies the third condition we cannot write $f(x)$ because that term assumes that there is a unique definition for that element of $Y$.