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 $x &isin; A$..." := ...

"$$\exists$$ exactly 3 $x &isin; A$..." := ...

"$$\exists$$ exactly k $x &isin; A$..." := ...

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)