Short notation for ordered set

Question

In Game Theory, we often denote $\textbf{X}=(X_1,X_2....X_n)$ as a profile of strategies. Obviously $\textbf{X}$ is a tuple/ordered set. In my case $\forall i$ $X_i$ is a set. Two questions:

1. Does the notation $\textbf{X}=(X_i)_{i=1}^n$ makes sense?
2. Is there a short way to represent $\{l\in X_i|1\leq i \leq n \}$?

Thanks!

Answer 1

1. $\textbf{X}$ is a finite sequence. So this makes sense.

2. $\displaystyle\{l\in X_{i}|1\leq i\leq n\}=\bigcap_{i=1}^{n}X_{i}$.

Answer 2

Some ideas and personal opinions from graph theory:

• If you are using index sets like $\{1,2,\ldots,k\}$ very often, introduce a special notation for it, for example $[k]$ is quite common. Don't forget to specify what it really means (e.g., a footnote could be enough).
• Then you can use $$\mathbf{X} = \big(X_i\big)_{i \in [n]}, \quad \quad \bigcap_{i\in[n]}X_i.$$

• I would understand $$\{l \in X_i \mid 1\leq i \leq n\} \quad\text{ as }\quad\big\{l \ \big|\ \exists i \in [n]. i \in X_i\big\} = \bigcup_{i\in[n]}X_i$$ that is, using existential rather than universal quantifier. I urge you to clarify that aspect in your writing.

• When the index set is really, really clear from the context, you can sometimes omit it, e.g. $(X_i)_i$ or $\bigcup_i X_i$. This leads to ugliness, but I have seen cases where it might be preferable.

• When using a big number of parentheses, brackets and braces, the display-style formulas are much easier to read when the sizes differ, e.g. $$(\{l \in X_i \mid i \in [k]\})_{k \in [n]} \text{ could be written as } \Big(\big\{l \in X_i \ \big|\ i \in [k]\big\}\Big)_{k \in [n]}.$$ Also, don't forget about the spacing around the bar in the set notation, compare $$\{a\in\mathbb{N}|2a=p\}, \{a \in \mathbb{N}\mid 2a=p\}, \Bigg\{f(a)\Bigg|\sum_{i \in[k]}f_k(c)=d\Bigg\}, \Bigg\{f(a)\ \Bigg|\ \sum_{i\in[k]}f_k(c)=d\Bigg\}.$$

• If you are using $\LaTeX$, then it's nice to define things like

  \newcommand{\tuple}[1]{\mathopen{}\left\langle {#1} \right\rangle\mathclose{}}
  \newcommand{\Bigtuple}[1]{\mathopen{}\Big\langle {#1} \Big\rangle\mathclose{}}
  \newcommand{\set}[2]{\mathopen{}\left\{ {#1} \ \middle|\ {#2} \right\}\mathclose{}}
  \newcommand{\Bigset}[2]{\mathopen{}\Big\{ {#1} \ \Big|\ {#2} \Big\}\mathclose{}}
  

  and then use \tuple{a,b,c} or \Bigset{a \in \mathbb{N}}{f(a)=b}.

• Finally, plain text is easier to read than formulas (unless you can understand them at first glance). For more complex concepts use both, the redundancy is a blessing, esp. in conference papers where typos are quite common.

I hope this helps $\ddot\smile$