My Math Forum  

Go Back   My Math Forum > College Math Forum > Advanced Statistics

Advanced Statistics Advanced Probability and Statistics Math Forum


Reply
 
LinkBack Thread Tools Display Modes
June 8th, 2009, 07:17 PM   #1
Newbie
 
Joined: Jun 2009

Posts: 3
Thanks: 0

Help needed

Help needed with this question:

Let Omega(Universe) be an infinite set, and A(Algebra) the class of all subsets B of Omega such that either B or its complement is finite:
(a) show that A is a field.
(b) is A a sigma-field?why?

Thanks.
mohit.choudhary is offline  
 
June 9th, 2009, 01:53 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,607
Thanks: 616

Re: Help needed

Quote:
Originally Posted by mohit.choudhary
Help needed with this question:

Let Omega(Universe) be an infinite set, and A(Algebra) the class of all subsets B of Omega such that either B or its complement is finite:
(a) show that A is a field.
(b) is A a sigma-field?why?

Thanks.
(a) looks straightforward, but tedious.

(b) answer is no. Let a1, a2, a3, ... be a countable set of points in Omega. The following are members of A, {a1}, {a3}, {a5}, .... However their (countable) union is not. It is an infinite set {a1,a3,a5,..} and its complement is also infinite since it contains {a2,a4,a6,...}.
mathman is offline  
June 10th, 2009, 08:42 PM   #3
Newbie
 
Joined: Jun 2009

Posts: 3
Thanks: 0

Re: Help needed

Hi mathamn,
Thank you for taking out time and looking at this query. I worked on your reply and the model you have used is good. I would just like to put forward some thing for part b:

Your answer:
answer is no. Let a1, a2, a3, ... be a countable set of points in Omega. The following are members of A, {a1}, {a3}, {a5}, .... However their (countable) union is not. It is an infinite set {a1,a3,a5,..} and its complement is also infinite since it contains {a2,a4,a6,...}.

My analysis:
answer is yes. Let a1, a2, a3, ... be a countable set of points in Omega. The following are members of A, {a1}, {a3}, {a5}, .... However their (countable) union is not. It is an infinite set {a1,a3,a5,..} and its complement is finite (which is countable intersection = 'empty set' now) hence it is a sigma field.
[Union(A)]^c=Intersection(A^c) = empty set
mohit.choudhary is offline  
June 11th, 2009, 01:52 PM   #4
Global Moderator
 
Joined: May 2007

Posts: 6,607
Thanks: 616

Re: Help needed

Quote:
Originally Posted by mohit.choudhary
Hi mathamn,
Thank you for taking out time and looking at this query. I worked on your reply and the model you have used is good. I would just like to put forward some thing for part b:

Your answer:
answer is no. Let a1, a2, a3, ... be a countable set of points in Omega. The following are members of A, {a1}, {a3}, {a5}, .... However their (countable) union is not. It is an infinite set {a1,a3,a5,..} and its complement is also infinite since it contains {a2,a4,a6,...}.

My analysis:
answer is yes. Let a1, a2, a3, ... be a countable set of points in Omega. The following are members of A, {a1}, {a3}, {a5}, .... However their (countable) union is not. It is an infinite set {a1,a3,a5,..} and its complement is finite (which is countable intersection = 'empty set' now) hence it is a sigma field.
[Union(A)]^c=Intersection(A^c) = empty set
You are dead wrong!
The complement is the countable intersection of the complements (Omega - {ai} for i odd) which as I stated contains all ai for i even.

In case you haven't solved part a, show that it is true for union or intersection of two sets and then use mathematical induction.
mathman is offline  
Reply

  My Math Forum > College Math Forum > Advanced Statistics

Tags
needed



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Help needed - UFD al-mahed Number Theory 5 February 16th, 2011 02:31 PM
Help needed plz!!! mathnoob99 Calculus 4 June 10th, 2010 06:40 PM
Help needed geeth416 Algebra 1 April 28th, 2010 08:38 AM
Help Needed CutieCutiePi Algebra 2 October 28th, 2009 07:34 PM
Help needed mikeportnoy Number Theory 3 September 26th, 2008 03:34 AM





Copyright © 2018 My Math Forum. All rights reserved.