Definition:Structure (Set Theory)
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Definition
Let $A$ be a class.
Let $\RR$ be a relation.
The relational structure $\struct {A, \RR}$ satisfies well-formed formula $p$, denoted $\struct {A, \RR} \models p$, shall be defined on the well-formed parts of $p$:
| \(\ds \struct {A, \RR} \models x \in y\) | \(\iff\) | \(\ds \paren {x \in A \land y \in A \land x \mathrel \RR y}\) | ||||||||||||
| \(\ds \struct {A, \RR} \models \neg p\) | \(\iff\) | \(\ds \neg \struct {A, \RR} \models p\) | ||||||||||||
| \(\ds \struct {A, \RR} \models \paren {p \land q}\) | \(\iff\) | \(\ds \paren {\struct {A, \RR} \models p \land \struct {A, \RR} \models q}\) | ||||||||||||
| \(\ds \struct {A, \RR} \models \forall x: \map P x\) | \(\iff\) | \(\ds \forall x \in A: \struct {A, \RR} \models \map P x\) |
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 12.1$