Difference between revisions of "Singleton (set theory)/Definition"
From Maths
(Silly mistake, good job I spotted it, now have a reference. Spotted during proof.) |
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Does not work! As if {{M|t\notin X}} by the nature of [[logical implication]] we do not care about the truth or falsity of the right hand side of the first {{M|\rightarrow}}! Spotted when starting proof of "''[[A pair of identical elements is a singleton]]''"</ref> | Does not work! As if {{M|t\notin X}} by the nature of [[logical implication]] we do not care about the truth or falsity of the right hand side of the first {{M|\rightarrow}}! Spotted when starting proof of "''[[A pair of identical elements is a singleton]]''"</ref> | ||
** In words: {{M|X}} is a singleton if: there exists a ''t''hing such that ( the thing is in {{M|X}} {{underline|''and''}} for any ''s''tuff ( if that stuff is in {{M|X}} then the stuff is the thing ) ) | ** In words: {{M|X}} is a singleton if: there exists a ''t''hing such that ( the thing is in {{M|X}} {{underline|''and''}} for any ''s''tuff ( if that stuff is in {{M|X}} then the stuff is the thing ) ) | ||
| + | More concisely this may be written: | ||
| + | * {{M|\exists t\in X\forall s\in X[t\eq s]}}<ref group="Note">see [[rewriting for-all and exists within set theory]]</ref> | ||
<noinclude> | <noinclude> | ||
==Notes== | ==Notes== | ||
Revision as of 17:38, 8 March 2017
Grade: B
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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 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]
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
