How can this definition be improved?

Question

On my new theorem I have to introduce some new notions that may be extrange in academics reading and in order to get it accepted by the reader I would like to be as clear and correct as possible. Please let me know how can I improve this definition, as well as it needs a proof or an example or an extended explanation.

The m-Crown of a set S, denoted by $S^m$, is the family of sets of every subset of its index set of cardinality $m$ no containing its index, such that

$\forall(X_i \in S^m : i \in S) \rightarrow X_i = \{ x \subset S :(i\notin x \land|x|=m) \} $

Answer 1

Generally you want to start from simpler objects before constructing more complex. Something like this:

Let $m$ be a cardinal number and $S$ a set. For $i\in S$ define $$T(S, i, m)=\big\{A\subseteq S\ \big|\ i\not\in A\text{ and }|A|=m\big\}$$ The family $\big\{T(S, i, m)\big\}_{i\in S}$ will be called the m-Crown of a set $S$ and we will denote it by $S^m$.

(I'm not a native english speaker, but this sounds quite ok to me)

I deliberately used $T(S,i,m)$ instead of vague $X$ to emphasize the connection between this set, $S$, cardinal $m$ and index $i$.

I also encourage you to not use formal, mathematical, symbolic sentences like the one you've used at the end. Unless absolutely necessary. It is hard to read and understand it.

// Edit: As suggested by @DanielWainfleet $S^m$ is already used in some other context. In order to avoid ambiguity it is a good idea to replace the symbol with something else, like for example $\text{Cr}(S,m)$.