Difference between revisions of "Singleton (set theory)/Definition"
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(Created page with "<noinclude> {{Requires references|grade=B|msg=Would be good to get this confirmed.}} __TOC__ ==Definition== </noinclude>Let {{M|X}} be a set. We call {{M|X}} a ''singleton...") |
(Silly mistake, good job I spotted it, now have a reference. Spotted during proof.) |
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| Line 1: | Line 1: | ||
<noinclude> | <noinclude> | ||
| − | {{Requires references|grade=B|msg= | + | {{Requires references|grade=B|msg=Book reference would be great!}} |
__TOC__ | __TOC__ | ||
==Definition== | ==Definition== | ||
| − | </noinclude>Let {{M|X}} be a [[set]]. We call {{M|X}} a ''singleton'' if: | + | </noinclude>Let {{M|X}} be a [[set]]. We call {{M|X}} a ''singleton'' if<ref name="War2011">[[Media:WarwcikSetTheoryLectureNotes2011.pdf|Warwick lecture notes - Set Theory - 2011 - Adam Epstein]] - page 2.75.</ref>: |
| + | * {{M|\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]}}<sup>{{Caveat|See:}}</sup><ref group="Note">Note that: | ||
* {{M|\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)]}} | * {{M|\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)]}} | ||
| − | ** In words: {{M|X}} is a singleton if: there exists a ''t''hing such that ( | + | 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 ) ) | ||
<noinclude> | <noinclude> | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Set Theory|Elementary Set Theory}} | {{Definition|Set Theory|Elementary Set Theory}} | ||
</noinclude> | </noinclude> | ||
Revision as of 16:14, 8 March 2017
Grade: B
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
The message provided is:
Book reference would be great!
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 ) )
Notes
- ↑ Note that:
- [ilmath]\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)][/ilmath]
