Singleton (set theory)

Definition

Let [ilmath]X[/ilmath] be a set. We call [ilmath]X[/ilmath] a singleton if:

• [ilmath]\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)][/ilmath]^Caveat:See:
  • 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]
  • 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

• A pair of identical elements is a singleton

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.