Let
- $A$ be a countable set of subsets of $\mathbb{R}$
- $f:A \rightarrow \mathbb{Z}$ be a bijection
- $O= \{x_k:\mathbb{R}\rightarrow \mathbb{R} \mid k\in [1,\dots, K]\subset \mathbb{Z}\}$, for some $K$ and some given functions $x_k$.
I want to define a set $R$ which contain functions $g:\mathbb{Z}\rightarrow \mathbb{Z}, g(n)= i$ if $x(n) \in f^{-1}(i)$, where $x\in O$.
I believe that I can write the set $R$ like this
$$ R = \left\{g:\mathbb{Z} \rightarrow \mathbb{Z}, g(n)= i \text{ if } x(n)\in f^{-1}(i) \mid x\in O\right\}. $$ but I'm also considering to write the set like this $$ R = \left\{n \in \mathbb{Z} \mapsto i \in \mathbb{Z} \text{ if } x(n)\in f^{-1}(i) \mid x\in O \right\}, $$ avoiding the introduction of the symbol $g$. Are these ways of writing understandable? Do you have any other suggestions?
Formally, a set is written in the form $\{x \:|\: P(x) \}$ with $P(x)$ describing the properties of a set $x$ to be contained in the set in question. Of course, this is relaxed a bit in practice (for example writing $\{x \in \mathbb{R} \:|\: x^2 \geq 2\}$ instead of $\{x \:|\: x \in \mathbb{R} \text{ and } x^2 \geq 2\}$ or even $\{x^2 \:|\: x \in \mathbb{R}\}$ instead of $\{y \:|\: \text{ there exists } x \in \mathbb{R} \text{ such that } y = x^2\}$ ), but in general, the properties your elements have to satisfy should still all come after the bar, i.e. $R = \{g : \mathbb{Z} \to \mathbb{Z} \:|\: ...\}$.
Your description of $R$ is not completely clear, there are two possible in terpretations coming to my mind:
a) We have $g \in R$ if there exists $x \in O$ such that for all $n \in \mathbb{Z}$ and $i \in \mathbb{Z}$ if we have $g(n) = i$, then $x(n) \in f^{-1}(i)$.
b) We have $g \in R$ if there exists $x \in O$ such that for all $n \in \mathbb{Z}$ we have $x(n) \in f^{-1}(g(n))$.
a) We have $g \in R$ if for all $n \in \mathbb{Z}$ and $i \in \mathbb{Z}$ if we have $g(n) = i$, then there exists $x \in O$ such that we have $x(n) \in f^{-1}(i)$.
b) We have $g \in R$ if for all $n \in \mathbb{Z}$ there exists $x \in O$ such that we have $x(n) \in f^{-1}(g(n))$.
(the descriptions 1 a) and b) describe the same set as do 2 a) and 2 b), the a) parts are more in the flavor you used in your question, whereas the b) parts are a bit shorter)
If one of these two adequately describes what you want, then you can just take the part of the sentence beginning after if and write the part behind the bar, i.e. $R = \{g : \mathbb{Z} \to \mathbb{Z} \:|\: \text{ there exists } x \in O ...\}$.