I have a sentence I am trying to translate as a mathematical expression but I am not sure if the expression matches what I mean.
The mathematical expression I came up with:
What I actually mean:
F is a function that takes an element from set scriptP and set scriptB and returns either 0 or 1.
B_acc is a set depending on set P_s. It is the set of all the elements B belonging to scriptB such that there is at least one element P in the set P_s such that F(P,B) = 1.
P_s* = is the set P_s for which the cardinal of set B_acc is minimum.
I am not sure about how to express the double "such that" in the first equation.
I like the words version $\{B\in\mathcal B\colon \text{there exists } P\in\mathcal P_s \text{ such that } F(P,B)=1\}$ the best.
While I think that's already fine, if you wanted you could write $\mathcal B_{acc}(\mathcal P_s) = \pi_2(F^{-1}(1))$, where $\pi_2$ (or $\pi_{\mathcal B}$) is the projection map from $\mathcal P\times \mathcal B$ to the second coordinate $\mathcal B$. (By the way, it's better to write the domain of $F$ as $\mathcal P\times \mathcal B$ rather than $(\mathcal P,\mathcal B)$.) There's a nuance between $\mathcal P$ and $\mathcal P_s$ that I'm probably missing here, so this might need some tweaking.
Of course, any function to the set $\{0,1\}$ can be naturally identified with a subset of the domain (namely the set of elements sent to $1$). If we write $F$ as a subset of $\mathcal P\times \mathcal B$ rather than a function on $\mathcal P\times \mathcal B$, then it becomes even simpler: $\mathcal B_{acc}(\mathcal P_s) = \pi_2\big( F \cap (\mathcal P_s\times\mathcal B) \big)$ (assuming $\mathcal P_s$ is a subset of $\mathcal P$).