Definition:Structure (Set Theory)

From ProofWiki
Jump to: navigation, search


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 $p$:

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