A description/proof of that intersection of set of transitive relations is transitive
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of transitive relation.
Target Context
- The reader will have a description and a proof of the proposition that the intersection of any set of any possibly uncountable number of transitive relations is transitive.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any set of any possibly uncountable number of transitive relations, \(S = \{R_\alpha\vert \alpha \in A\}\) where \(A\) is a possibly uncountable indexes set, the intersection, \(\cap_{\alpha \in A} R_\alpha\), is transitive.
2: Proof
For any \(\langle s_1, s_2 \rangle, \langle s_2, s_3 \rangle \in \cap_{\alpha \in A} R_\alpha\), \(\langle s_1, s_2 \rangle, \langle s_2, s_3 \rangle \in R_\alpha\) for each \(\alpha\). As each \(R_\alpha\) is transitive, \(\langle s_1, s_3 \rangle \in R_\alpha\) for each \(\alpha\), so, \(\langle s_1, s_3 \rangle \in \cap_{\alpha \in A} R_\alpha\).
