definition of intersection of set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of union of set.
Target Context
- The reader will have a definition of intersection of 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: Structured Description
Here is the rules of Structured Description.
Entities:
\( S\): \(\in \{\text{ the sets }\}\)
\(*\cap S\): \(= \{p \in \cup S \vert \forall s \in S (p \in s)\}\), \(\in \{\text{ the sets }\}\)
//
Conditions:
//
2: Note
\(\cap S\) is indeed a set in the ZFC set theory, by the union axiom and the subset axiom.
Some frequently seen expressions like \(S_1 \cap S_2\) and \(\cap_{\alpha \in A} S_\alpha\) are indeed \(S_1 \cap S_2 := \cap \{S_1, S_2\}\) and \(\cap_{\alpha \in A} S_\alpha := \cap \{S_\alpha \vert \alpha \in A\}\): when \(A\) is a set, \(\{S_\alpha \vert \alpha \in A\}\) is a set by the replacement axiom.
The special case of \(\cap_{\alpha \in A} S_\alpha\) where \(A = \{p\}\) cannot be expressed as \(\cap S_p\), because \(\cap_{\alpha \in A} S_\alpha\) is \(\cap \{S_p\} = S_p\) but not \(\cap S_p\).
\(\cap \emptyset = \emptyset\), because \(\cup \emptyset = \emptyset\).
We could not define like \(p \in \cap S \iff \forall p' \in S (p \in p')\), because when \(S = \emptyset\), \(\forall p' \in S (p \in p')\) would be vacuously satisfied, but where would \(p\) come from?
