Asymptotic analysis is foundational in many disciplines in Computer Science. At the heart of asymp- totic analysis is big-O notation. Definition 1 on page 205 of big O notation can be written formally using quantifiers in the following way. Let f and g be functions from the set of integers to the set of real numbers. We say that f(x) is O(g(x)) if ∃C∃k∀x (x > k → |f(x)| ≤ C|g(x)|) (where the variables x and k are assumed to be over the set of integers and the variable C is over the real numbers). (a) Using this definition and our proof strategies for quantified statements, prove that 2x is O(x 2 ). That is, show that the proposition above evaluates to true when f(x) = 2x and g(x) = x 2 . (b) Using this definition and our proof strategies for quantified statements, prove that x 2 is not O(2x). That is, show that the proposition above evaluates to false when f(x) = x 2 and g(x) = 2x.