Difference between revisions of "Singleton (set theory)/Definition"

From Maths
Jump to: navigation, search
(Silly mistake, good job I spotted it, now have a reference. Spotted during proof.)
(Concise form added)
Line 8: Line 8:
 
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
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:
Book reference would be great!

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

  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.