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Big-O notation

  1. Jun 20, 2008 #1
    1. The problem statement, all variables and given/known data

    1. Prove the following subset relationships:
    a. [tex]Constant \subseteq Logarithmic[/tex]
    b. [tex]Logarithmic \subseteq Linear[/tex]
    c. [tex]n log n \subseteq Polynomial[/tex]
    2. Relevant equations

    O(1) Constant
    O(log n) Logarithmic
    O(n) Linear
    O(n log n) n log n
    O(n^m), m > 1 Polynomial of order m

    3. The attempt at a solution
    a. There exists an c, k element of Z where f(n) =< C * 1, f(n) =< C log n for all n => k.

    So to show that Constant is a subset of Log, I need to show 1 =< C log n, And I find a C and k that satisfies that condition right? C = 1 k = 2

    b. C = 1 k = 1 then log n =< Cn = log 1 =< 1 * 1

    c. C = 1 k = 1 then n log n =< n^2 = 1 log 1 =< 1^2
     
    Last edited: Jun 20, 2008
  2. jcsd
  3. Jun 23, 2008 #2
    I'm not sure I can help but here goes.. for a, you want to show that after a while,
    f(x) = c <= log n.

    Well easiest way is imo to do lim n->\infty C/Log N = 0. Not sure if that is a proof, so you might consider:

    Log n = C -> 10^c = n -> for n >= 10^c, log n > c. // as log is an increasing function

    for b, you can look at lim n->\infty Log N / N = 0. (or opposite = infitity) but proof wise consider you want to prove log n <= n for all n > some number.

    If you look at the graph, this is true for all of them, maybe use induction?
     
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