Proofs!

With our first midterm in the books, I can finally commence my study of proofs.  Immediately following the midterm test, we began our study of proof techniques.  These techniques range from direct proof, proof by contradiction and proof by contrapositive.

The structure of these proof methods is as follows:

Consider some arbitrary statement P -> Q.  To prove this, we can do the following:

1. Proof by Contradiction:

Proof: Assume P ∧ ∼ Q.  Show some sort of contradiction.

2. Proof by Contrapositive:

Proof: Assume ∼ Q.  Show ∼ Q  ->  ∼ P.

3. Direct Proof:

Proof: Assume P.  Show  P  ->  Q.

I am confident that I understood the majority of content covered in lecture.  However, as with most academic topics, there are aspects that are incredibly easy, and others that are laborious and demanding.  The same applies to proofs.  While there may be ridiculously simple proofs such as showing that for all n∈ℕ, n is odd → n^2 is odd, there are also ridiculously difficult proofs such as 
Euclid's Theorem, which states there are infinitely many primes or something almost near
 impossible (even for the mathematicians), Fermat's Last Theorem.  For those interested, 
heres a nice and simple explanation of Fermat's Last Theorem: 
http://mathforum.org/library/drmath/view/52464.html 

Although, it should be mentioned that the proof is not included, as 
its far to advanced for non-mathematicians.

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