MikeEnnen

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

$\exists x_{1},x_{2}\in \mathbb {A} :x_{1}\neq x_{2}\ \land \mathbb {A} \backslash \{x_{1},x_{2}\}=\emptyset$.

Example:

Notation:

Let $\mathbb {Z^{*}} =\{x\in \mathbb {Z} :x>=0\}$. That is, $\mathbb {Z^{*}}$ is the set of all non-negative integers.

Let ${\mathfrak {P}}=\{x^{2}:x\in \mathbb {Z} \}$ denote the set of all perfect squares.

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

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

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

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

Looking back I have no idea what I was doing here.

MikeEnnen (talk) 19:10, 14 September 2011 (UTC)