Difference between revisions of "Singleton (set theory)"
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− | + | -->** For proof see '''Claim 1'''. | |
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==Significance== | ==Significance== | ||
Notice that we have manage to define a set containing one thing without any notion of the number 1. | Notice that we have manage to define a set containing one thing without any notion of the number 1. |
Latest revision as of 17:38, 8 March 2017
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
- [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.
- [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]
- By "rewriting for-all and exists within set theory" we see:
Notes
- ↑ Note that:
- [ilmath]\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)][/ilmath]
- ↑ see rewriting for-all and exists within set theory