A description/proof of that formula that uniquely maps each element of set into set constitutes function
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set.
Target Context
- The reader will have a description and a proof of the proposition that any formula that uniquely maps each element of any set into another set constitutes a function.
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: Note
This proposition may seem obvious, but its motivation is that the map has to be proved to be a set to be called a function, in the ZFC set theory. Being a part of the product set is not enough, because the subset axiom requires a legitimate formula. So, the formula is required: just knowing that each element maps to the image uniquely somehow is not enough.
2: Description
For any sets, \(S_1, S_2\), and any formula, \(\phi (s_1, s_2, . . .)\), where \(s_1 \in S_1\) and \(s_2 \in S_2\), that maps each element of \(S_1\) uniquely to an element of \(S_2\), \(f: S_1 \rightarrow S_2, s_1 \mapsto s_2 \text{ if and only if } \phi (s_1, s_2, . . .)\) is a function.
3: Proof
\(f \subseteq S_1 \times S_2\), but we need a legitimate formula to declare \(f\) to be a set by the subset axiom. \(f = \{p \in S_1 \times S_2\vert \exists s_1 \in S_1, \exists s_2 \in S_2, \phi (s_1, s_2, . . .), p = \langle s_1, s_2 \rangle\}\). There is a unique element, \(p = \langle s_1, s_2 \rangle \in f\) for each \(s_1 \in S_1\), by the supposition for \(\phi\). So, \(f\) is a function.
