A description/proof of that for 2 sets, collection of relations between sets is set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set.
- The reader knows a definition of relation.
- The reader knows a definition of function.
- The reader admits the proposition that the product of any finite number of sets is a set.
Target Context
- The reader will have a description and a proof of the proposition that for any 2 sets, the collection of all the relations between the sets is a set.
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 sets, \(S_1, S_2\), the collection of all the relations from \(S_1\) to \(S_2\), \(R: = Pow (S_1 \times S_2)\), is a set.
2: Proof
By the proposition that the product of any finite number of sets is a set, \(S_1 \times S_2\) is a set. By the power set axiom, \(Pow (S_1 \times S_2)\) is a set.
