# Singleton (set theory)/Definition

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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
*t*hing such that ( the thing is in [ilmath]X[/ilmath]*and*for any*s*tuff ( if that stuff is in [ilmath]X[/ilmath] then the stuff is the thing ) )

- In words: [ilmath]X[/ilmath] is a singleton if: there exists a

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]

*A pair of identical elements is a singleton*" - ↑ see rewriting for-all and exists within set theory