P2.3.N (10) In CS 187, 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, no range over the naturals: P:3c: 3no: Vn:n2 nof(n) cg(n)(this is the definition of "f(n) is O(g(n))") Q Vn R:Vc : 3r0 : Yn : n no f(n) cg(n) 3c: Vn : n2 no-f(n) S 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? P2.3.N (10) In CS 187, 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, no range over the naturals: P:3c: 3no: Vn:n2 nof(n) cg(n)(this is the definition of "f(n) is O(g(n))") Q Vn R:Vc : 3r0 : Yn : n no f(n) cg(n) 3c: Vn : n2 no-f(n) S 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