Def: Let f: N -> R be a function. 

   O(f) = {g: N->R: there exists C, N_0 such that
              g(n) <= C f(n) for all n >= N_0}


   \Theta(f) = {g: N->R: there exists c, C, N_0 such that
              c f(n) <= g(n) <= C f(n) for all n >= N_0}



We use O to give an upper bound:

   g \in O(f) means, informally, that g <= f, 
   if inequality is understood to ignore constants and
   small values of the input.


We use Theta to give an upper and lower bound:

   g \in Theta(f) means, informally, that g = f, 
   if equality is understood to ignore constants and
   small values of the input.