I hate having to cater to the lowest common denominator, I had to struggle with un-engaging classes all throughout elementary and middle school. I've seriously thought about becoming a teacher so I'd like to ask, in your experience, what happens to the children that are able to process more advanced information? Can something be done to keep them engaged and nurture their development too?
Edited an unfortunate typo
No, it's correct. You define the operation by it's properties. It's not saying that "a plus 0 = a" but "the result of applying the binary operation '+' to any number with 0 should give the original number."
You have to have previously defined 1=S(0), 2=S(1), 3=S(2), and so on.