Ok, the problem is this:
Provide a specific counterexample that shows the statement is false.
Statement: for all function f and g on N, if , then .
I've chosen and as this seems fairly obvious.
It is easy to show that is in ...though this is not accurate, it still holds. I much pick values for N and K such that:
so I choose K = 1 and N = 2
The second one, I am trying a proof by contradiction (though I was always told to assume the negation of the hypothesis). If we simply do the same thing we get:
, choose n = 2K and we get
which is false, so the original statement must be false. Is this a correct way to do this?