Proof by Contraspositive

I remember the days when I used to boast and announce, "Proofs? Please, they come second nature to me!"  I can still recall that horrendously loud laugh with which I used to flood the rooms of my poor, confused friends.

It still haunts me... I mean them.

Only this week did I realize that in reality, I only knew how to write out the structure.  And not just that; I only knew how to write out direct proof structure.  My ego has been in hiding ever-since...

After a number of fruitless past trials that left me frustrated, I came to the realization that there could be times when a direct proof doesn't necessarily work.  In such cases, resolving to an alternative proof mechanism is required.  This is where proof by contraspositive comes into play.

To refresh your memory, a contrapositive of an implication is equivalent to the implication itself.

∀x ∈ L, P(x) => Q(x)   <====> ∀x ∈ L, ¬ Q(x) => ¬ P(x)

How does that help with proving something that doesn't 'respond' to a direct proof?  Proving the contrapositive instead of the implication will simultaneously prove the original implication!

The proof structure for the above contrapositive is:

Assume x ∈ L  # x is a generic L

     Assume ¬ Q (x)  # assuming antecedent

              Then ..... # finding a link between ¬ Q(x) and ¬ P(x) 
                              # using proven mathematical theories and logic

              Then ¬ P(x)   # then ¬ Q(x) => ¬ P(x)

      Then P(x) => Q(x) # since an implication is equivalent to its contrapositive

Then ∀x ∈ L, P(x) => Q(x) # Assumed x, got result.

In conclusion, if a direct proof doesn't work, try using the contrapositive (especially if it becomes easier to prove), before rejecting the claim.