Definition:Structure (Set Theory)
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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: $\S 12.1$
