Difference between revisions of "Singleton (set theory)/Definition"
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==Definition== | ==Definition== | ||
| − | </noinclude>Let {{M|X}} be a [[set]]. We call {{M|X}} a ''singleton'' if<ref name="War2011">[[Media: | + | </noinclude>Let {{M|X}} be a [[set]]. We call {{M|X}} a ''singleton'' if<ref name="War2011">[[Media:WarwickSetTheoryLectureNotes2011.pdf|Warwick lecture notes - Set Theory - 2011 - Adam Epstein]] - page 2.75.</ref>: |
* {{M|\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]}}<sup>{{Caveat|See:}}</sup><ref group="Note">Note that: | * {{M|\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]}}<sup>{{Caveat|See:}}</sup><ref group="Note">Note that: | ||
* {{M|\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)]}} | * {{M|\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)]}} | ||
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Contents
Definition
Let [ilmath]X[/ilmath] be a set. We call [ilmath]X[/ilmath] a singleton if[1]:
- [ilmath]\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)][/ilmath]Caveat:See:[Note 1]
- In words: [ilmath]X[/ilmath] is a singleton if: there exists a thing such that ( the thing is in [ilmath]X[/ilmath] and for any stuff ( if that stuff is in [ilmath]X[/ilmath] then the stuff is the thing ) )
More concisely this may be written:
- [ilmath]\exists t\in X\forall s\in X[t\eq s][/ilmath][Note 2]
Notes
- ↑ Note that:
- [ilmath]\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)][/ilmath]
- ↑ see rewriting for-all and exists within set theory
