Singleton (set theory)/Definition

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Definition

Let $X$ be a set. We call $X$ a singleton if^[1]:

∃t[t∈X∧∀s(s∈X→s=t)]^Caveat:See:^{[Note 1]}
- In words: $X$ is a singleton if: there exists a thing such that ( the thing is in $X$ for any stuff ( if that stuff is in $X$ then the stuff is the thing ) )

More concisely this may be written:

$\exists t\in X\forall s\in X[t\eq s]$ ^{[Note 2]}

Notes

Jump up ↑
Note that:
- $\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)]$
Does not work! As if t∉X by the nature of logical implication we do not care about the truth or falsity of the right hand side of the first →! Spotted when starting proof of "A pair of identical elements is a singleton"
Jump up ↑ see rewriting for-all and exists within set theory

References

Jump up ↑ Warwick lecture notes - Set Theory - 2011 - Adam Epstein - page 2.75.

[2] Jump up ↑ Note that:
$\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)]$
Does not work! As if $t\notin X$ by the nature of logical implication we do not care about the truth or falsity of the right hand side of the first $\rightarrow$ ! Spotted when starting proof of "A pair of identical elements is a singleton"

[2] $\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)]$

[3] Jump up ↑ see rewriting for-all and exists within set theory

[War2011-1] Jump up ↑ Warwick lecture notes - Set Theory - 2011 - Adam Epstein - page 2.75.

[1]

[Note 1]

[Note 2]

Singleton (set theory)/Definition

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