definition of union of set
Topics
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of set.
Target Context
- The reader will have a definition of union 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:
\( S\): \(\in \{\text{ the sets }\}\)
\(*\cup S\): \(\in \{\text{ the sets }\}\)
//
Conditions:
\(p \in \cup S\)
\(\iff\)
\(\exists p' \in S (p \in p')\)
//
2: Natural Language Description
For any set, \(S\), the set, \(\cup S\), such that \(p \in \cup S\) if and only if \(\exists p' \in S (p \in p')\)
3: Note
\(\cup S\) is indeed a set in the ZFC set theory by the union axiom.
Some frequently seen expressions like \(S_1 \cup S_2\) and \(\cup_{\alpha \in A} S_\alpha\) are indeed \(S_1 \cup S_2 := \cup \{S_1, S_2\}\) and \(\cup_{\alpha \in A} S_\alpha := \cup \{S_\alpha \vert \alpha \in A\}\): when \(A\) is a set, \(\{S_\alpha \vert \alpha \in A\}\) is a set by the replacement axiom.
The special case of \(\cup_{\alpha \in A} S_\alpha\) where \(A = \{p\}\)cannot be expressed as \(\cup S_p\), because \(\cup_{\alpha \in A} S_\alpha\) is \(\cup \{S_p\} = S_p\) but not \(\cup S_p\).
\(\cup \emptyset = \emptyset\), because no element, \(p\), cannot be in \(\cup \emptyset\), because there is no \(p' \in \emptyset\) such that \(p \in p'\).
