Consider two functions with natural arguments and positive real values: f,g : N R+. Consider the following propositions, where c ranges over positive reals, and n, n0 range over the naturals:

P :c:n0 :n:nn0 f(n)cg(n) (this is the definition of f(n)isO(g(n)))Q:n0 :c:n:nn0 f(n)cg(n) R:c:n0 :n:nn0 f(n)cg(n)

a) Are any two of these propositions equivalent, for arbitrary choices of f and g? (prove it, or give examples of f and g for which they differ). b) Is any of the propositions always true, whatever f and g? c) Can you find two functions f and g for which R is true ?