September 18th, 2008, 03:55 PM  #1 
Newbie Joined: Sep 2008 Posts: 2 Thanks: 0  Big Oh Notations Help
I was wondering if I could get help on proving one of the Big Oh rules, specifically the Transitive Rule which states: If f(n) = O(g(n)) and g(n) = O(h(n)), then f(n) = O(h(n)) So far this is what I have: there exists c1 and n1 f(n) <= c1 * g(n) for all n >= n1 g(n) <= c1 * h(n) for all n>= n1 therefore f(n) = h(n) and therefore O(h(n)) Would this be correct? 
September 19th, 2008, 04:54 AM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Big Oh Notations Help Quote:
f(n) <= c1 * g(n) for all n >= n1 g(n) <= c2 * h(n) for all n>= n2 so f(n) <= c1 * c2 * h(n) for all n >= max(n1, n2)  
September 19th, 2008, 11:28 AM  #3 
Newbie Joined: Sep 2008 Posts: 2 Thanks: 0  Re: Big Oh Notations Help
Thanks CRGreathouse. But I was wonder where you say max(n1, n2), should you not do the same thing with c1 and c2?

September 19th, 2008, 12:10 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Big Oh Notations Help Quote:
 
September 20th, 2008, 08:00 AM  #5 
Newbie Joined: Sep 2008 Posts: 4 Thanks: 0  Re: Big Oh Notations Help
I was wondering how would you go about doing it with the Log of a Power Rule: log(n^k) is O(log(n)) for any constant k > 0.

September 21st, 2008, 07:06 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Big Oh Notations Help Quote:
And I don't use these rules you use (Transitive Rule, Power Rule), I just use the definition of Big O. log(n^k) = klog(n), and since the O is insensitive to multiplication by a constant, O(log(n^k)) = O(log(n)).  

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