Question

Use the properties of Big - Oh, Big - Omega, and Big - Theta to prove that if f (n) = theta (3 Squareroot n) and g (n) = Ohm (f (n) + 7 f (n)^2 + 49 Squareroot n), then g (n)^3 = Ohm (n^2). You may use the fact that n^a = 0 (n^b) if and only if a lessthanorequalto b, where a and b are constants.

Answer 1

importat fact is that the base of logarithms is not important. Any two log functions are related by multiplication by a constant. The formula is loga(n) = logb(n) loga(b) (so the constant is loga(b)). To prove this, start with aloga(n) = blogb(n) (both of these equal n), take logs to base a of both sides and simplify using the rules of logarithms.Fact B1. If f is O(g) and g is O(h), then f is O(h). You should know how to prove this Fact. It implies that if f is O(g), then it is also Big-O of any function “bigger” than g. This is why one is typically interested in finding a “best possible” big-O expression for f (i.e., one that can not be replaced by a “smaller” function). Usually one can guess a “best possible” big-O estimate for a function by first throwing away all constants, and second keeping only the biggest term in the expression. (You still need to prove that your guess is correct.) For example applying these guidelines to f(n) = 10 · 2nn2 + 17n3 log(n) − 500 suggests that a best possible big-O form is O(2nn2). A quick way to decide if f is O(g) is to use limits. It turns out that f is O(g) if

limn→∞ |f(n)| |g(n)|

exists and is finite. The proof that this works involves the definition of the limit of a function, which we have not studied. Most often you will be asked to prove f is O(g) using the definition, so this method will not be accepted. Memorize: Suppose f : Z → R and g : Z → R are functions. We say f is Ω(g) if there exists constants C and k so that |f(n)| ≥ C|g(n)| for all n>k. Big-Ω is just like big-O, except that Cg(n) is now a lower bound for f(n) for all large values of n. All of the same comments and proof techniques as above apply except the inequalities are in the other direction. Fact B2. A function f is Ω(g) if and only if g is O(f). You should know how to prove this Fact, and should also be able to use it in arguments involving big-Ω. Memorize: Suppose f : Z → R and g : Z → R are functions. We say f is Θ(g) if f is O(g) and f is Ω(g). In other words, a function f is Θ(g) if and only if there are constants C1 and C2 so that C1g(n) ≤ f(n) ≤ C2g(n) for all large vales of n. Fact B3. A function f is Θ(g) if and only if f is O(g) and g is O(f). You should know how to prove this Fact, and should also be able to use it in arguments involving big-Θ. It turns out that f is Θ(g) if

limn→∞ |f(n)| |g(n)|