## 2013-12-23

### Remedial Study of Proofs

2013-12-23

Disproportionately proud of myself. This is very, very late homework. Like 10 years late. (I think I got it right - let me know where the mistakes are, if you see any.)

Theorem:
$\forall x: (\frac{x}{2} \in \mathbb{Z}) \implies (\frac{x^2}{2} \in \mathbb{Z})$

"The square of an even integer, is always an even integer."
Proof:
1. axiom $A$:
$\frac{x}{2} \in \mathbb{Z}$

"x is an even integer."
2. axiom $B$:
$\times$ (multiplication), an operation, is closed over the set of integers.
3. lemma $C$ $\Longleftarrow B$:
$\forall y: (\frac{y}{2}\in\mathbb{Z})\Longrightarrow\left((\frac{y}{2} \times 2) = y\right)\in\mathbb{Z})$

"If y/2 is an integer, then y is an integer."
4. lemma $D$ $\Longleftarrow B$:
$\forall z: (z\in \mathbb{Z})\implies(z^2 \in \mathbb{Z})$

"If z is an integer, then z squared is an integer."
5. lemma $E$ $\Longleftarrow (A \wedge D)$:
$\left((\frac{x}{2})^2 = (\frac{x^2}{4}) = \frac{(\frac{x^2}{2})}{2}\right) \in \mathbb{Z}$

"(x^2)/4 is an integer."
6. Theorem $\Longleftarrow (C \wedge E)$:
$\left(\frac{(\frac{x^2}{2})}{2} \in \mathbb{Z}\right) \implies \left(\frac{x^2}{2} \in \mathbb{Z}\right)$