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) => 
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.

April 25th, 2010, 02:26 PM  #3  
Senior Member Joined: Apr 2010 Posts: 451 Thanks: 1  Re: empty set 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  
April 26th, 2010, 08:56 AM  #4  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: empty set Quote:
Quote:
I'll give you the first couple of steps (I'll put a bar over something to denote the closure) Quote:
Quote:
4 and 5 are the tricky ones. Do you want to give them a try?  
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). 
April 26th, 2010, 05:10 PM  #6  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: empty set Quote:
*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  
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)??

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.

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 
April 27th, 2010, 08:48 AM  #10  
Global Moderator 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:
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".  

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