Ah! Crazyguy is a guy who knows what he talks about!
i forgot Combinatorics in the resume...
math isn't my field though it's background to computer eng. (that's why i didn't mention Digital Systems and the sort...)
but i believe you're wrong if you solve the same question in different way you might not get the same answer (if you're using boolean algebra or in finite fields like Z5 for example...)
actually if you combine these theories you get that:
1+1=2 (of course)
1+1=1 (Radiohead wrote on their OK computer that they hope you choked but its correct in switching algebra)
1+1=0 (using finite field Z2)
you can inductively prove that the natural numbers equal 0, and the integers, and the non-rational (of course you dont have rationals cause you cant divide by 0)
think about it...