Question

Justify the correctness of the following statements assuming that \(f(n)\) and \(g(n)\) are asymptotically positive functions. a. \(f(n)+g(n) \in O(f(n)), g(n))\) b. \(f^{2}(n) \in \Omega(f(n))\) c. \(f(n)+o(f(n)) \in \Theta(f(n), \text { where of } o(f(n))\) means any function \\[g(n) \in o(f(n))\\]

Short answer

All statements are correct. Statement a holds as \(f(n) + g(n)\) is less than or equal to some constant times the maximum of the two functions. Statement b is true because \(f^2(n)\) is always greater than \(f(n)\) for any sufficiently large \(n\), affirming it to be in \(\Omega(f(n))\). Statement c has been validated as \(f(n) + o(f(n))\) is both \(O(f(n))\) and \(\Omega(f(n))\), meaning \(f(n) + o(f(n))\) is \(\Theta(f(n))\).

Step-by-step solution

Statement a

To verify \((f(n)+g(n)) \in O(max(f(n), g(n)))\), check if there exists any positive constants \(c\) and \(n_0\) such that \(f(n) + g(n)\) is always less than or equal to \(c*\)max(\(f(n), g(n)\)) for all \(n > n_0\). By definition of the max function and properties of asymptotic functions, we can see that the statement holds true as \(f(n) + g(n)\) would be less than or equal to some constant times the maximum of the two functions.

Statement b

To verify \(f^{2}(n) \in \Omega(f(n))\), ensure there exists any positive constants \(c\) and \(n_0\) such that \(f^{2}(n)\) is always more than or equal to \(c*f(n)\)) for all \(n > n_0\). By definition of \(Omega\) notation and since \(f(n)\) is positive, \(f^2(n)\) is always greater than \(f(n)\) for any sufficiently large \(n\), thus the statement is correct.

Statement c

To verify \(f(n)+o(f(n)) \in \Theta(f(n))\), prove that the function \(f(n) + o(f(n))\) is both \(O(f(n))\) and \(\Omega(f(n))\). Since \(o(f(n))\) represents any function that is smaller in magnitude compared to \(f(n)\) for sufficiently large \(n\), it can be added to \(f(n)\) without changing the order of magnitude, proving the statement correct.

Key concepts

Big O Notation

Big O notation is a mathematical concept used to describe the upper bound of an algorithm's growth rate. In simpler terms, it tells you the maximum number of operations an algorithm might take to complete a task as the size of the input, usually denoted as n, increases. A function f(n) is said to be in Big O of g(n), written as \( f(n) \text{ is } O(g(n)) \), if f(n) does not grow faster than a constant multiple of g(n) as n approaches infinity. This constant multiple is represented as c in mathematical definitions.

In the provided exercise statement a, '[a. \( f(n)+g(n) \text{ is } O(f(n) \text{ or } g(n)) \)]', the concept of Big O notation is applied to the sum of two functions f(n) and g(n). We look for a constant c and a point of growth n0 such that for all n > n0, the sum of f(n) and g(n) is less than or equal to c times the maximum of f(n) or g(n). This concept holds true as the sum of two functions can't exceed the maximum of the two, multiplied by some constant, for large values of n.

Big Omega Notation

Big Omega notation, written as \( \text{Omega} \), serves as an asymptotic lower bound. It is similar to Big O notation, but instead of providing an upper limit, it assures a minimum growth rate of an algorithm. Function f(n) is in Big Omega of g(n), denoted by \( f(n) \text{ is } \text{Omega}(g(n)) \), if f(n) grows at least as fast as a constant multiple of g(n) when n gets large.

Referencing the exercise statement b, '[b. \( f^2(n) \text{ is } \text{Omega}(f(n)) \)]', the assertion is that when we square the function f(n), the growth rate is guaranteed not to be less than the original function f(n) times some constant, for sufficiently large values of n. Since squaring a positive function will always result in a larger or equal value as compared to the original, statement b correctly represents the application of Big Omega notation for asymptotic analysis.

Theta Notation

Theta notation, symbolized by \( \text{Theta} \), is used in asymptotic analysis to provide a tight bound on the growth rate of an algorithm. It essentially means that a function f(n) is sandwiched between two other functions that both scale similarly. To put it formally, f(n) is in Theta of g(n), noted as \( f(n) \text{ is } \text{Theta}(g(n)) \), if it is both O(g(n)) and Omega(g(n)), indicating that f(n) grows at the same rate as g(n) up to constant multiples as n becomes very large.

The exercise statement c, '[c. \( f(n)+o(f(n)) \text{ is } \text{Theta}(f(n)) \)]', uses Theta notation to express that the function f(n) added to a smaller magnitude function o(f(n)) is equivalent to f(n) itself in terms of growth rate. Here, o(f(n)) is known as 'little o' notation, which describes a function that becomes negligible compared to f(n) as n increases. Since the added o(f(n)) does not alter the significant order of magnitude for f(n), the statement correctly claims that the function with the added 'little o' part remains within the tight bounds described by Theta notation.