Definition:Structure (Set Theory)

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Let $A$ be a class.

Let $\mathcal R$ be a relation.

The relational structure $\left[{A, \mathcal R}\right]$ satisfies well-formed formula $p$, denoted $\left[{A, \mathcal R}\right] \models p$, shall be defined on the well-formed parts of $p$:

\(\ds \left[{A, \mathcal R}\right] \models x \in y\) \(\iff\) \(\ds \left({x \in A \land y \in A \land x \mathrel {\mathcal R} y}\right)\)
\(\ds \left[{A, \mathcal R}\right] \models \neg p\) \(\iff\) \(\ds \neg \left[{A, \mathcal R}\right] \models p\)
\(\ds \left[{A, \mathcal R}\right] \models \left({p \land q}\right)\) \(\iff\) \(\ds \left({\left[{A, \mathcal R}\right] \models p \land \left[{A, \mathcal R}\right] \models q}\right)\)
\(\ds \left[{A, \mathcal R}\right] \models \forall x: P \left({x}\right)\) \(\iff\) \(\ds \forall x \in A: \left[{A, \mathcal R}\right] \models P \left({x}\right)\)