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 April 24th, 2010, 04:27 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,$\emptyset$closed <=> closure$\emptyset\subset\emptyset$,it suffices to prove: if x?(closure$\emptyset$),then x?$\emptyset$. 2) Let x?(closure$\emptyset$) 3)Since by definition,x?(closure$\emptyset$) <=> $\forall h>o$ there exists a ball B(x,h) such that $B(x,h)\cap\emptyset\neq\emptyset$ we have: $B(x,h)\cap\emptyset\neq\emptyset$. 4) Since $B(x,h)\cap\emptyset= \emptyset$,then we have : $(B(x,h)\cap\emptyset= \emptyset)\vee (x\in\emptyset)$. 5)Since,if $(B(x,h)\cap\emptyset= \emptyset)\vee (x\in\emptyset)$ then $B(x,h)\cap\emptyset\neq\emptyset\Longrightarrow x\in\emptyset$ ,we have from argument (3) : $x\in\emptyset$ 6) Hence ,x?(closure$\emptyset$) => $x\in\emptyset$
 April 25th, 2010, 08:09 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, 01:26 PM   #3
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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

April 26th, 2010, 07:56 AM   #4
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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,$\emptyset$closed <=> closure$\emptyset\subset\emptyset$,it suffices to prove: if x?(closure$\emptyset$),then x?$\emptyset$.
The "<=>" that's cited is a characterization of closure.

Quote:
 2) Let x?(closure$\emptyset$) 3)Since by definition,x?(closure$\emptyset$) <=> $\forall h>o$ there exists a ball B(x,h) such that $B(x,h)\cap\emptyset\neq\emptyset$ we have: $B(x,h)\cap\emptyset\neq\emptyset$.
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?

 April 26th, 2010, 03:43 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, 04:10 PM   #6
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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 $\overline\emptyset\subseteq \emptyset$
we want to show "if $x\in\overline\emptyset$ then $x\in\emptyset$"
I.e. we are assuming x is in $\overline\emptyset$, and trying to show that it must be in $\emptyset$, so in step two we are stating the assumption that $x\in\overline\emptyset$

 April 26th, 2010, 05:21 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, 07:07 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, 07:37 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, 07:48 AM   #10
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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".

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