0
$\begingroup$

I need confirmation for my suspicion that

Space = A $\rightarrow$ B

defines the domain of all functions mapping from A to B while

f : A $\rightarrow$ B

defines one function out of that space.

Also, is the usage of the term "define" correct in the latter case? After all I'm not giving a concrete definition of $f$.

$\endgroup$
1
$\begingroup$

"Space = $A \to B$" is not a commonly accepted notation for spaces.

Rather, one writes $B^A$: we "make $A$ choices from $B$".

When vertical space is scarce (particularly with iterations) I have occasionally resorted to $[A \to B]$.

However, this is likely idiosyncratic and should be defined very clearly whenever it is used.


As to "$f: A \to B$", it names a (up until now arbitrary) function from $A$ to $B$. Because we haven't specified what $f$ is, we can only reason with what follows directly from $f \in B^A$ (e.g. that its domain is $A$, to name something trivial).

| cite | improve this answer | |
$\endgroup$
1
$\begingroup$

Given two nonempty sets $A$ and $B$ one consider maps $f:\>A\to B$. The last formula does not "define" $f$; it just tells the reader that $f$ is a mathematical object having the properties associated with a map from $A$ to $B$. E.g., you may consider $f$ as a subset of $A\times B$ having certain properties.

Now the set of all maps $f:\>A\to B$ is just a set, not a "space". It is quite common to denote this set by $B^A$. If $A$ and $B$ are finite then the set of maps $f:\>A\to B$ has in fact $|B|^{|A|}$ elements. But this notation has typographical disadvantages, and, above all, is very counterintuitive, since the maps in question go in the direction $A\to B$.

If you have to handle a lot of such sets of maps in your paper you could write ${\rm Map}(A,B)$ instead of $B^A$, in the same way as one denotes the set of all homomorphisms between two groups $G$ and $H$ by ${\rm Hom}(G,H)$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.