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)]}} | ||
Latest revision as of 23:34, 8 March 2017
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Contents
[hide]Definition
Let X be a set. We call X a singleton if[1]:
- ∃t[t∈X∧∀s(s∈X→s=t)]Caveat:See:[Note 1]
- In words: X is a singleton if: there exists a thing such that ( the thing is in X and for any stuff ( if that stuff is in X then the stuff is the thing ) )
More concisely this may be written:
- ∃t∈X∀s∈X[t=s][Note 2]
Notes
- Jump up ↑ Note that:
- ∃t[t∈X→∀s(s∈X→s=t)]
- Jump up ↑ see rewriting for-all and exists within set theory
