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.)

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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 \(\Longleftarrow (C \wedge E)\):

   \( \left(\frac{(\frac{x^2}{2})}{2} \in \mathbb{Z}\right) \implies \left(\frac{x^2}{2} \in \mathbb{Z}\right) \)