Definition:Structure (Set Theory)

Definition
Let $A$ be a class.

Let $\mathcal R$ be a relation.


 * $[A,\mathcal R] \models p$, or "the relational structure $[A,\mathcal R]$ satisfies well-formed formula $p$" shall be defined on the well-formed parts of $p$:


 * $[A,\mathcal R] \models x \in y \iff \left({ x \in A \land y \in A \land x \mathcal R y }\right)$


 * $[A,\mathcal R] \models \neg p \iff \neg [A,\mathcal R] \models p$


 * $[A,\mathcal R] \models \left({ p \land q }\right) \iff \left({ [A,\mathcal R] \models p \land [A,\mathcal R] \models q }\right)$


 * $[A,\mathcal R] \models \forall x: P\left({x}\right) \iff \forall x \in A: [A,\mathcal R] \models P\left({x}\right)$