Talk:Counting quantification

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 x ∈ A..." := ...

"$\exists$ exactly 3 x ∈ A..." := ...

"$\exists$ exactly k 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)