Monday, November 15, 2010

Uniqueness of inverses for Groups, by contradiction.

In mathematical groups, the uniqueness of inverses is almost a direct consequence of the definition of a Group. Yet I could not prove it in 10 minutes, was too lazy and looked it up here: http://planetmath.org/encyclopedia/UniquenessOfInverseForGroups.html.

It bothered me that I could not come up with this proof, I tried to discover why. One of the attempts I tried was proving it by contradiction which also failed, but looking at the proof, I can see I gave up too early.
After peeking, I wanted to at least try to still prove it by contradiction (unlike the proposed proof) which intuitively seemed like the way to go for me, because the proposed proof 'constructs statements from nothing', starting with an expression, expanding it for no very obvious reason at that point (but of course obvious later), then collapsing it.

It is interesting how doing it by contradiction shows a slightly clearer path through the infinite 'ocean of statements' and the infinite number of 'valid paths' between each 2 statements, because you always have the 2 sides of the inequality to visually look at.

Here is my proof by contradiction:

Given other proofs me and Tom have attempted in the last months, I am sure I would have solved it if I gave it a bit more time, but I knew it had to be short so I wanted to solve it fast, but I failed. All in all this was a good insight into my problem solving technique.
This one falls into the category 'so obvious that it blocks you'.