Definition:Structure (Set Theory)
Jump to navigation
Jump to search
This page has been identified as a candidate for refactoring. In particular:
The bulk of this page needs to be put into separate pages.
 It has been suggested that this page or section be merged into Definition:Relational Structure. (Discuss) |
The bulk of this page needs to be put into separate pages.
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
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 $p$:
| \(\ds \left[{A, \mathcal R}\right] \models x \in y\) | \(\iff\) | \(\ds \left({x \in A \land y \in A \land x \mathrel {\mathcal R} y}\right)\) | ||||||||||||
| \(\ds \left[{A, \mathcal R}\right] \models \neg p\) | \(\iff\) | \(\ds \neg \left[{A, \mathcal R}\right] \models p\) | ||||||||||||
| \(\ds \left[{A, \mathcal R}\right] \models \left({p \land q}\right)\) | \(\iff\) | \(\ds \left({\left[{A, \mathcal R}\right] \models p \land \left[{A, \mathcal R}\right] \models q}\right)\) | ||||||||||||
| \(\ds \left[{A, \mathcal R}\right] \models \forall x: P \left({x}\right)\) | \(\iff\) | \(\ds \forall x \in A: \left[{A, \mathcal R}\right] \models P \left({x}\right)\) |
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 12.1$