Let $M$ be an arbitrary set and let $\mathscr{F}$ be a family of subsets of $M$. Is there a known name for the following equivalence relation or its corresponding partition?

$\sim_{M,\mathscr{F}}\,=\bigl\{(x,y)\in M\times M\bigm|\forall A\in\mathscr{F}\,(x\in A\leftrightarrow y\in A)\bigr\}$.

Another question about notations: Is there a common used notation for the set of all injections from $A$ into $B$? Some set-theorists use $B^{(A)}$ but some combinatorists use $B^{\underline{A}}$ or $(B)_A$.