For each assertion in (a)-(c), prove the assertion directly from the definition of the big-asymptotic notation if it is true by finding values for the constants c and no. On the other hand, if the assertion is false, give a counter-example. Then answer the question in (d). F denotes the set of all functions from Z+ to R+.

(a) Let f(n) : Z+ R+. DEFINITION 1. A relation on a set is reflexive if each element is related to itself. Assertion: The relation "is big-2 of" is reflexive over F In other words, f(n) E S(f(n). [5 points] (b) Let f(n) : Z+ R+ and g(n) : Z+ R+. DEFINITION 2. A relation on a set is antisymmetric if whenever an element X is related to an element Y and Y is related X, then X-Y Assertion: The relation "is big-2 of" is antisymmetric over F In other words, if f(n) E (g(n)) and g(n) 62(f(n), then f(n) = g(n). [5 points] (c) Let e(n) : Z+ R", f(n): + R" and g(n) : Z+-R+. DEFINITION 3. A relation on a set is transitive if whenever an element X is related to Y and Y is related Z, then X is related to Z. Assertion: The relation "is big-2 of" is transitive over F. In other words, if e(nje (f(n)) and f(nje (g(n), then e(n) 62(g(n)). [10 points] (d) Is "is big-92 of" a partial order on F? [5 points] DEFINITION 4. A relation is a partial order on a set if it is reflexive, antisymmetric and transitive