Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Can anyone give me a (as much as possible) formal and general definition of natural domain of a function?

Let's say that a function is a triplet $(X,Y,f)$ where $f \subseteq X \times Y$ such that

\begin{eqnarray} \forall x \in X, \exists ! y \in Y | (x,y) \in f \end{eqnarray}

where $X$ is called the domain and $Y$ the codomain. This is rather general and formal: it does not specify anything about $X$, $Y$ and how we form the pairs in $f$. In the same manner it is possible to define what the image of a subset $A \subseteq X$ and what the preimage of $D \subseteq Y$ are as

\begin{eqnarray} &f(A) = \{y | (y \in Y) \wedge (\exists x \in A : y = f(x)\} \\ &f^{-1}(D) = \{x | (x \in X) \land (f(x) \in D)\} \end{eqnarray}

(using the notation $y=f(x)$ here just for convenience)

About the natural domain, I came up with something like:

The domain $X$ and the codomain $Y$ are often assumed from the context (or explicitly mentioned) to be some improper subset of some standard sets (like $N$, $R$, $R \times R = R^2$, $C$, etc.). The natural domain of a function $(X,Y,f)$ is the biggest superset $\hat X$ of $X$ for which $(\hat X, K, f)$ is still a function, where $K$ is one of those standard sets assumed from the context and $\hat X$ is a subset of one of those standard sets.

I know that this is far from formal and doesn't make great sense because $f$ remains the same set of ordered pairs here, so $(\hat X, Y, f)$ is not a function unless $\hat X = X$.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.