Talk:Counting quantification

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== Explicit Statements in First-Order Logic ==

I believe that it would be a very useful addition to this page if it listed the explicit logical statements that correspond to the following:

" $\exists$ exactly 2 {{math|x ∈ A}}..." := ...

" $\exists$ exactly 3 {{math|x ∈ A}}..." := ...

" $\exists$ exactly k {{math|x ∈ A}}..." := ...

I will attempt to do a bit of research and see if I can't do this myself - but if this talk page or the article does not contain these additions within a couple days from me posting this - I probably won't do it (I'll forget!)...so if anyone feels that this is a relevant and good addition to this article, maybe you could give it a try!

MikeEnnen (talk) 10:50, 28 May 2011 (UTC)

"There exists exactly two things in an (arbitrary) set":

$\exists x_{1}, x_{2} \in \mathbb{A} : x_{1} \neq x_{2} \ \land \ \forall y,z \in \mathbb{A}\backslash \{x_{1}, x_{2}\} : y \neq x_{1} \land z \neq x_{2}$

Example:

Let $\mathfrak{M}$ denote the set of magic squares, explicitly:

$\{x^2 : x \in \mathbb{Z}\}$ , then:

$\exists x_{1}, x_{2} \in \mathbb{Z} : x_{1} \neq x_{2} \ \land \ \forall y,z \in \mathbb{Z}\backslash \{x_{1}, x_{2}\} : y \neq x_{1} \ \land \ z \neq x_{2} : \forall p \in \mathfrak{M} \ \ \forall x \in \mathbb{Z^*} \ x^2 = p \Rightarrow x_{1} = \sqrt{p} \in \mathbb{Z} \ \land \ x_{2} = -\sqrt{p} \in \mathbb{Z}$

where $x_{1}$ and $x_{2}$ are the only 2 solutions in $\mathbb{Z}$ to $x^2=p$ , where $p \in \mathfrak{M}$ .

I realize this may not be the best example...tell me what ya'll think!

MikeEnnen (talk) 12:07, 28 May 2011 (UTC)