# Definition:Structure (Set Theory)

## 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 $p$:

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

\(\displaystyle \left[{A, \mathcal R}\right] \models \neg p\) | \(\iff\) | \(\displaystyle \neg \left[{A, \mathcal R}\right] \models p\) | $\quad$ | $\quad$ | |||||||||

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

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

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 12.1$