I'm trying to prove that this is a group. I already established elsewhere that it is a binary operation, so now I am onto proving associativity. I've tried many examples and so I'm confident it is associative, but now I just have to prove that.
The Attempt at a Solution
Let [itex]x, y, z \in G[/itex]. Then [itex]x*(y*z) = x*(y + z - [y + z]) = x + y + z - [y + z] - [x + y + z - [y + z]][/itex]
Then [itex](x*y)*z = (x + y - [x + y])*z = x + y + z - [x + y] - [x + y + z - [x + y]] [/itex]
Now the problem is coming up with an equality to show this. We would have to show that [y + z] = [x + y]. I guess you could do a ton of cases where you show what happens when [itex]y + z \geq 1[/itex] and [itex]y + z < 1[/itex], same goes for [itex]x + y[/itex], but I'm now sure that even that would work.