This question arose while learning nonstandard analysis.
The superstructure $V(X)$ of a nonempty set $X$ is defined recursively:
$\begin{align*}V_0(X) &= X \\ V_{i+1}(X) &= V_i(X) \cup P(V_i(X)) \\ V(X) &= \bigcup_{i=0}^\infty V_i(X)\;, \end{align*}$
where $i \in \mathbb{N}$ and $P$ is the powerset function. So it's a way to get every relation and function on $X$ that you could possibly want, by identifying the ordered pair $(a,b)$ with the set $\{\{a\},\{a,b\}\}$ or such. I'm great with this.
BUT what happens when there are relations in $X$ (as members of $X$)? I think that I need to distinguish between the relations that are formed by the superstructure construction and any relations that I might have started with in $X$. As an example problem, I want to prove (my book says it's true) that a relation $R$ is in $V(X)$ iff the domain and range of $R$ are in $V(X)$. This is not necessarily true if $R$ is in $X$. The members of $X$ that have members themselves don't necessarily empty out.
So are we supposed to distinguish between relations in $X$ and the relations that are formed from $X$?
Cheers, Rachel