# How to formulate a theorem about bijections between several sets

I have several sets $A_i$ and bijections between them. (As stated in my theorem) no composition of these bijections produces a permutation of $A_i$ not equal to identity. So every bijection is identified by the pair of sets between which it acts.

It would be to cumbersome to denote every bijection with a special letter (such as $\Phi$). I want to write it in some consise way.

For example I could denote the bijection from $A_i$ to $A_j$ as $\phi_{A_i,A_j}$ but this is not formally right as I would first to prove $A_i\ne A_j$. Then I would denote it $\phi_{i,j}$ but this way I would need explicitly number my sets, but I'd better to use English names or maybe letters to denote the sets not numbers.

The best solution I found insofar is to denote every set with some letter and denote my bijections as $\phi_{\alpha,\beta}$, where $\alpha$ and $\beta$ determine some sets. This solution is not ideal, because it would involve for each considered set two different notations to denote it: say $A_1$ and $\alpha$.

This is insofar the best solution I know. But maybe somebody may suggest me a better language to formulate my theorem?

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Why not call the sets $A_\alpha$ or $\alpha$? –  Johannes Kloos Mar 16 '12 at 20:59
Why not use something like $(A,B)$, where $A$ and $B$ are your sets? If you don't like parentheses, you can use brackets, or maybe you can use a bar $A|B$, $A*B$, or maybe just a comma, $A,B$ or a semicolon or just a colon without any braces whatsoever. –  Braindead Jun 5 '12 at 21:22

## 3 Answers

Each bijection $\phi$ determines its domain and range. Why not phrase your theorem in terms of the indexed set of bijections $\phi_j$ where $j$ ranges over some arbitrary index set $I$? You can define $D_j$ and $R_j$ (say) to be the domain and range of $\phi_j$ respectively. The sets $D_j$ and $R_j$ are your $A_i$ but you don't have to worry about whether they are distinct.

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An interesting idea. But this way we would have several distinct notations for a set $A_i$. –  porton Mar 17 '12 at 12:23
@porton: if my suggestion doesn't help, then you need to tell us more about what you are actually trying to say. –  Rob Arthan Mar 17 '12 at 14:27

If I understand correctly, your sets $A_i$ are already indexed, since otherwise you probably wouldn't be calling them "$A_i$". So, I think it would be the best to use the notation $\phi_{i,j}:A_i\to A_j$ as you have suggested. As the sets themselves are already indexed, there is no need of using any extra numbering and you can also use English names or colours or whatever you want. I shall explain in more detail below.

So, you have a collection $\mathcal{A}$ of sets. You want to give them names (you may have done this already). This is achieved by choosing some index set $\mathcal{I}$ and a function $\Phi:\mathcal{I}\to\mathcal{A}$. This has the effect of naming the sets: for $i\in\mathcal{I}$ it is now convenient to write $\Phi(i)=A_i$. (And it is this what makes it possible to consider the same set multiple times in possibly different ways. Since, if $\Phi(i)=\Phi(j)$ is the same set for two different indices, you might consider this as two "instances" of the same set. If you want consider a set multiple times in a different way, naming it by two different names is, I think, basically unavoidable.) If you don't like numbers, you can take $\mathcal{I}$ here to be a set of English names, thus giving your sets English names.

But, as you are saying, your bijections are uniquely determined by the ordered pairs of sets. So, there might not in fact be any real need of indexing your sets and you might just work with the (non-indexed) collection $\mathcal{A}$. And if you want to index it anyway, simply use $\mathcal{I}=\mathcal{A}$ and define $\Phi:\mathcal{I}\to\mathcal{A}$ to be the identity map.

I think there is no need of doing this in your case, however. Since you have stated that the bijections are uniquely determined by the pairs of sets, you might just as well index these bijections by the pairs that uniquely determine them. For example, if $\phi:A\to B$ is the unique bijection from $A$ to $B$, you might just as well name it $\phi_{(A,B)}$. As the pair $(A,B)$ uniquely determines the bijection, such notation should pose no problems and is well defined in the sense that if $A=C$ and $B=D$, then $(A,B)=(C,D)$ is the same pair and thus uniquely determines the bijection $\phi_{(A,B)} = \phi_{(C,D)}$. (Note, that in this case, you will never really need to use two names for a single set.)

I hope any of these suggestions helps.

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The best idea I got is the following:

Consider only bijections of the set $A_0$ with $A_i$ for every $i\ne 0$. There are no need to explicitly say about bijections of every $A_i$ with $A_j$.

The rest bijections can be easily figured out by a reader of my writing.

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