Notation for "elementwise" set operations Let $\mathcal{C}_\alpha \subset \mathcal{P}X$ be subsets of powerset of $X$ and $f: \mathcal{P}X \to W$. Is there a standard unambiguous notation for sets like $\{A_1 \cap A_2 : A_i \in \mathcal{C_i}\} $ and $\{f^{-1}(w) : w \in W\}$?
 A: Given a function $f:X\to Y$ and $A\subseteq X$, the "image of the set $A$ (under $f$)" is defined to be $\left\{f(x):x\in A\right\}$. There are many competing notations for the image of $A$: "$f(A)$", "$f[A]$, and "$f``A$" are all in somewhat common use (though the latter is only common in set theory, I think). "$f_>(A)$" is a very rare notation I've seen, and Wikipedia mentions "$f^{\rightarrow}(A)$" and "$f_\star(A)$", too.
If $f$ is invertible, then its inverse $f^{-1}$ is a function, and if $B\subseteq Y$, we can similarly consider "the image of $B$ under $f^{-1}$": $\left\{f^{-1}(y):y\in B\right\}=\left\{x:f(x)\in B\right\}$, and write it as $f^{-1}[B]$, etc.
If $f$ is not necessarily invertible, we may still care about the set $\left\{x:f(x)\in B\right\}$. This is called the "preimage" or (confusingly) "inverse image" of $B$ under $f$. Even when $f$ is not invertible, this is commonly denoted by "$f^{-1}(B)$" or "$f^{-1}[B]$" or (in at least Convergence Foundations of Topology) "$f^-(B)$". I've also seen the rare $f^<(B)$, and Wikipedia mentions $f^{\leftarrow}(B)$ and $f^\star(B)$, too.

I don't know of a standard notation for something like "the set of all pairwise intersections" as in your first example. However, that is related in that it's essentially of the image of "the set of all pairs of sets $A_i$" under "the intersection function (that takes in a pair and outputs the intersection)". This sort of viewpoint and related constructions are common in the "functional programming" paradigm. There, code like "map(f)" refers to something like $f^{\rightarrow}$. And there is often something similar to handle two or more lists/sets, etc.
