In CS187, you’ve encountered big-O notation. 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 : n ≥ n0 → f(n) ≤ cg(n) (this is the definition of "f(n) is O(g(n))") Q : ∀n0 : ∃c : ∀n : n ≥ n0 → f(n) ≤ cg(n) R : ∀c : ∃n0 : ∀n : n ≥ n0 → 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 ?