Singleton (set theory)

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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 ) )

More concisely this may be written:

  • [ilmath]\exists t\in X\forall s\in X[t\eq s][/ilmath][Note 2]
    • For proof see Claim 1.

Significance

Notice that we have manage to define a set containing one thing without any notion of the number 1.

See next

Proof of claims

  1. [ilmath]\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X\forall s\in X[t\eq s]\big)[/ilmath]
    • By "rewriting for-all and exists within set theory" we see:
      • [ilmath]\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X[\forall s(s\in X\rightarrow s\eq t)]\big)[/ilmath]
        [ilmath]\iff\big(\exists t\in X\forall s(s\in X\rightarrow s\eq t)\big)[/ilmath] by simplification
        [ilmath]\iff\big(\exists t\in X\forall s[s\in X\rightarrow s\eq t]\big)[/ilmath] by changing the bracket style
        [ilmath]\iff\big(\exists t\in X\forall s\in X[s\eq t]\big)[/ilmath] by re-writing again.

Notes

  1. 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. see rewriting for-all and exists within set theory

References

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