My Math Forum  

Go Back   My Math Forum > College Math Forum > Applied Math

Applied Math Applied Math Forum


Reply
 
LinkBack Thread Tools Display Modes
April 24th, 2010, 05:27 PM   #1
Senior Member
 
Joined: Apr 2010

Posts: 451
Thanks: 1

empty set

For the proof that the empty set is closed i was given the following series of arguments ,and i was asked to justify the correctness or incorrectness of each argument,by citing laws of logic , theorems.axioms,or definitions involved in each argument.

The arguments are:

1)Since,closed <=> closure,it suffices to prove: if x?(closure),then x?.

2) Let x?(closure)

3)Since by definition,x?(closure) <=> there exists a ball B(x,h) such that we have: .

4) Since ,then we have : .

5)Since,if then ,we have from argument (3) :

6) Hence ,x?(closure) =>
outsos is offline  
 
April 25th, 2010, 09:09 AM   #2
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: empty set

It seems reasonable, but I'm not sure I understand the point of it all.
cknapp is offline  
April 25th, 2010, 02:26 PM   #3
Senior Member
 
Joined: Apr 2010

Posts: 451
Thanks: 1

Re: empty set

Quote:
Originally Posted by cknapp
It seems reasonable, but I'm not sure I understand the point of it all.
Meaning you are not sure 100% whether the arguments are right or wrong.

What do we do in this case? . I think that is the idea of the problem when they ask me to analyze ???,each argument.

But how do we do that??,by citing axioms ,laws of logic e.t.c........e.t.c
outsos is offline  
April 26th, 2010, 08:56 AM   #4
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: empty set

Quote:
Originally Posted by outsos
Meaning you are not sure 100% whether the arguments are right or wrong.
No. Meaning, "what is the point of the exercise?"

Quote:
What do we do in this case? . I think that is the idea of the problem when they ask me to analyze ???,each argument.

But how do we do that??,by citing axioms ,laws of logic e.t.c........e.t.c
Ok. Then, yes: Cite axioms etc.

I'll give you the first couple of steps (I'll put a bar over something to denote the closure)
Quote:
Originally Posted by outsos
1)Since,closed <=> closure,it suffices to prove: if x?(closure),then x?.
The "<=>" that's cited is a characterization of closure.

Quote:
2) Let x?(closure)

3)Since by definition,x?(closure) <=> there exists a ball B(x,h) such that we have: .
Again. This is just the definition... I'm guessing we're taking that the whole space is R, rather than an arbitrary topological space?

4 and 5 are the tricky ones. Do you want to give them a try?
cknapp is offline  
April 26th, 2010, 04:43 PM   #5
Senior Member
 
Joined: Apr 2010

Posts: 451
Thanks: 1

Re: empty set

Thank you very much .

But i really do not understand step 2. Is that step an argument??

For steps 4 and 5 i have no idea (not that i had any idea for the other steps).
outsos is offline  
April 26th, 2010, 05:10 PM   #6
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: empty set

Quote:
Originally Posted by outsos
But i really do not understand step 2. Is that step an argument??
In every step we either:
*apply a definition
*apply an inference rule
or
*state an assumption.

In step 2, we're doing the last step.
In order to show that
we want to show "if then "
I.e. we are assuming x is in , and trying to show that it must be in , so in step two we are stating the assumption that
cknapp is offline  
April 26th, 2010, 06:21 PM   #7
Senior Member
 
Joined: Apr 2010

Posts: 451
Thanks: 1

Re: empty set

Thank you again ,but what do we apply for step (6)??
outsos is offline  
April 27th, 2010, 08:07 AM   #8
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: empty set

Step 6 is just restating what we just proved. It isn't really a step at all.
cknapp is offline  
April 27th, 2010, 08:37 AM   #9
Senior Member
 
Joined: Apr 2010

Posts: 451
Thanks: 1

Re: empty set

Thank you again ,but i still do not understand ,why when we want to prove for example : p =>q ,if we assume p and then after many steps in a proof we come up with q ,we automatically say that we have proved =>q.

I have tried steps 4 and 5 but i cannot justify there existence . Can you help me please . Sorry for my questions .I just want to clear foggy things in my mind
outsos is offline  
April 27th, 2010, 08:48 AM   #10
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 937

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: empty set

Quote:
Originally Posted by outsos
Thank you again ,but i still do not understand ,why when we want to prove for example : p =>q ,if we assume p and then after many steps in a proof we come up with q ,we automatically say that we have proved =>q.
For technical reasons, proving q given p is different from proving (unconditionally) that p => q. See
http://us.metamath.org/mpeuni/mmdeduction.html
for information if you like -- but you might prefer to remember to throw in this extra step whenever you're asked for a literal statement like "p => q".
CRGreathouse is offline  
Reply

  My Math Forum > College Math Forum > Applied Math

Tags
empty, set



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
The empty set barokas Applied Math 4 September 25th, 2013 04:47 PM
Family of empty sets Vasily Applied Math 2 August 19th, 2012 01:31 PM
Empty Set proof jstarks4444 Applied Math 1 October 12th, 2011 06:40 PM
Empty set question z0r Applied Math 2 December 6th, 2008 09:58 PM
Empty Set proof jstarks4444 Number Theory 0 December 31st, 1969 04:00 PM





Copyright © 2017 My Math Forum. All rights reserved.