Singleton (set theory)/Definition

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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 ) )

Notes

↑
Note that:
- [ilmath]\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)][/ilmath]
Does not work! As if [ilmath]t\notin X[/ilmath] by the nature of logical implication we do not care about the truth or falsity of the right hand side of the first [ilmath]\rightarrow[/ilmath]! Spotted when starting proof of "A pair of identical elements is a singleton"

References

↑ Warwick lecture notes - Set Theory - 2011 - Adam Epstein - page 2.75.

[2] Note that:
[ilmath]\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)][/ilmath]
Does not work! As if [ilmath]t\notin X[/ilmath] by the nature of logical implication we do not care about the truth or falsity of the right hand side of the first [ilmath]\rightarrow[/ilmath]! Spotted when starting proof of "A pair of identical elements is a singleton"

[2] [ilmath]\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)][/ilmath]

[War2011-1] Warwick lecture notes - Set Theory - 2011 - Adam Epstein - page 2.75.

[1]

[Note 1]

Singleton (set theory)/Definition

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Definition

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