# Definition:Structure (Set Theory)

 It has been suggested that this page or section be merged into Definition:Relational Structure. (Discuss)

## Definition

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 $p$:

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