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YCor
YCor
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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 $(B)_A$.

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 $(B)_A$.

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?

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

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Guozhen Shen
Guozhen Shen
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Is there a name for this equivalence relation?

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 $(B)_A$.