A definition of product set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of function.
- The reader knows a definition of tuple.
Target Context
- The reader will have a definition of product 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: Definition 1
For any possibly uncountable indices set, \(A\), and any sets, \(\{S_\alpha \vert \alpha \in A\}\), indexed with \(A\), the set of all the functions, \(\{f: A \to \cup_{\alpha \in A} S_\alpha \vert f (\alpha) \in S_\alpha \text{ for each } \alpha \in A\}\), denoted as \(\times_{\alpha \in A} S_\alpha\)
2: Definition 2
For any finite number of sets, \(S_1, S_2, ..., S_n\), the set of all the \(n\)-tuples, \(\{\langle p_1, p_2, ..., p_n \rangle \vert p_j \in S_j \text{ for each } j = 1 \sim n\}\), denoted as \(S_1 \times S_2 \times ... \times S_n\)
3: Note
Definition 1 is indeed a set: the set of all the functions from \(A\) into \(\cup_{\alpha \in A} S_\alpha\) is a set (see Proof 8 of the proposition that some expressions can be parts of legitimate formulas for the ZFC set theory), and the formula for the subset axiom is legitimate.
Definition 2 is indeed a set, by the proposition that the product of any finite number of sets is a set.
