Discussion Overview
The discussion revolves around the properties of relations in set theory, specifically examining the inclusion of a relation \( R \) within the Cartesian product of its domain and range. Participants explore the implications of this relationship and seek examples to illustrate their points.
Discussion Character
- Technical explanation
- Mathematical reasoning
Main Points Raised
- Some participants propose that if \( \in R \), then \( x \in dom(R) \) and \( y \in rng(R) \), leading to the conclusion that \( R \subset dom(R) \times rng(R) \).
- Others suggest that this inclusion holds for any relation \( R \), providing an example \( R=\{\langle0,2\rangle,\langle1,3\rangle,\langle0,3\rangle\} \) to illustrate the point.
- One participant confirms the domain and range for the provided example, stating \( dom(R)=\{0,1\} \) and \( rng(R)=\{2,3\} \).
- Another participant calculates \( dom(R) \times rng(R) \) and shows that \( R \) is indeed a subset of this product.
Areas of Agreement / Disagreement
Participants generally agree on the inclusion of \( R \) within \( dom(R) \times rng(R) \) and confirm the calculations related to the example provided. However, there is no explicit consensus on the necessity of examples beyond the initial proof.
Contextual Notes
The discussion does not resolve the broader implications of these relationships or the potential limitations of the examples provided. There is an implicit assumption that the properties discussed hold universally for all relations.