3.2 Asymptotics

On which side of the equal sign should the asymptotic notations \(O() \), \(o() \), and \(\Theta() \) be used?

Exercise 1

  left  

  right  

  either  

Why is the statement "\(f \) is at least \(O(n^2) \)" a blunder?

Exercise 2

 \(O(n^2) \) is not a quantity. 

 Big Oh gives an upper bound on the growth of a function. 

 \(O(n^2) \) is a relation. 

 If \(f \) is at least \(n^2 \), then \(n^2 = O(f) \). 

 

Why is the statement \(\sum_{i=1}^{n} i = O(n) \) a blunder?

Exercise 3

 \(\sum_{i=1}^{n} i = o(n) \) 

 The Big Oh relation on individual terms of a sum does not necessarily apply to the sum. 

 \(\sum_{i=1}^{n} i = \Theta(n^{2}) \) 

 \(O(1) \) should not be treated as a number.