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xander
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Postby xander » Thu Mar 14, 2013 4:06 pm

Let $(X,d)$ and $(\tilde X, \tilde d)$ be metric spaces. We say that a function $f\colon X\to\tilde X$ is \emph{continuous at $a\in X$} if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that if $x\in X$ and $d(x,a) < \delta$, then $\tilde d(f(x),f(a) < \varepsilon$. If $A\subset X$, then we say that $f$ is \emph{continuous on $A$} if $f$ is continuous at $a$ for all $a\in A$. Prove or disprove: this definition of continuity is equivalent to the topological definition, where the topologies on $X$ and $\tilde X$ are assumed to be those induced by the metric.

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Captain Jean-Luc Picard
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Postby Captain Jean-Luc Picard » Thu Mar 14, 2013 4:29 pm

Prove or disprove: this definition of continuity is equivalent to the topological definition, where the topologies on $X$ and $\tilde X$ are assumed to be those induced by the metric.


You know what to do.



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bert_the_turtle
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Postby bert_the_turtle » Thu Mar 14, 2013 4:50 pm

You forgot to tell us what the topological definition of a continuous function is, and what the topology induced by a metric is.

And you made me hate math with the pure legwork proof you made me go through in my head. Congratulations, that is quite an accomplishment! I'll need to read some Lie Algebra stuff now.
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Xocrates
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Postby Xocrates » Thu Mar 14, 2013 9:16 pm

Today in class we proved Morgan's laws... by using Morgan's laws. I did a double take at that.

Granted, the point was to practice a method that relied on Morgan's laws, so what we were proving was irrelevant. Still weird though.
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Postby NeatNit » Thu Mar 14, 2013 9:19 pm

Xocrates wrote:Today in class we proved Morgan's laws... by using Morgan's laws. I did a double take at that.

Granted, the point was to practice a method that relied on Morgan's laws, so what we were proving was irrelevant. Still weird though.
What? To prove De Morgan's Laws you just need a quick n' dirty truth table. How could you POSSIBLY mess that up?!
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Xocrates
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Postby Xocrates » Thu Mar 14, 2013 9:59 pm

Read the fine print ;)
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NeatNit
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Postby NeatNit » Thu Mar 14, 2013 10:36 pm

I have read said print and deliberately chosen to ignore it.
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Postby Mas Tnega » Thu Mar 14, 2013 10:47 pm

I had to check to see if there were some other laws to be sure I was at the appropriate level of confusion. I was at the appropriate level of confusion.

How do you even do that without just saying "If A, Q. A. Q. Q.E.D."?
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Postby NeatNit » Thu Mar 14, 2013 10:50 pm

I am guessing they were using Karnaugh maps or something like that. No wait, that wouldn't... Um.
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Xocrates
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Postby Xocrates » Fri Mar 15, 2013 12:11 am

It was actually an exercise in propositional logic. :P The Resolution method, specifically.

And like I said, the method (or more precisely, the conversion of the equation into canonical normal form) assumes the laws are true. So the goal of the exercise wasn't to prove the laws to be true, but to learn to apply the method.

But since you're curious:

Hipothesis:
~( A || B) -> ~A && ~B

proof by contradiction, so we'll deny it:

~(~( A || B) -> ~A && ~B)

Convert into CNF (skipping several steps here - which includes applying de morgan's laws)

~A && ~B && (A || B)

or, alternative representation:

{{~A},{~B},{A,B}} -- this essentially means that in order for the hipothesis to be true, all internal groups must be true


Proof:

1. {~A} Premise
2. {~B} Premise
3. {A,B} Premise
4. {B} Resolution, (1,3) -- this essentially means that for ~A to be true, the remaining symbols in {A,B} - in this case B - must be true
5. {} Resolution, (2,4) -- contradiction, since it means both ~B and B must be true.

So the hipothesis must be true.
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Postby NeatNit » Fri Mar 15, 2013 2:30 am

Just thought I'd throw this up here, there is a scary accurate comparison in that half-sentence mention: http://garry.tv/2013/03/12/office-progress/

Edit: Uh, does my avatar work? I think I may have messed something up, it doesn't appear on my screen :oops:
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xander
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Postby xander » Fri Mar 15, 2013 9:12 pm

bert_the_turtle wrote:You forgot to tell us what the topological definition of a continuous function is, and what the topology induced by a metric is.

And you made me hate math with the pure legwork proof you made me go through in my head. Congratulations, that is quite an accomplishment! I'll need to read some Lie Algebra stuff now.

It was assumed that you knew what those were. :P And yes, it is a pure legwork proof, and a clever undergrad should be able to figure it out. It was a nice break from covering r-balls with rho-balls.

xander
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Postby paktsardines » Sat Mar 16, 2013 3:37 am

It was a nice break from covering r-balls with rho-balls.

I think they can treat that these days.
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Postby MeatNit » Sun Mar 17, 2013 6:26 am

We apparently have a new Cyan.

Any ideas?
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Postby Feud » Sun Mar 17, 2013 1:47 pm

Reproductive rights portion in constitutional law. People often worry about conservative judges being appointed that will over turn abortion. They really should worry more about the fact that the cases that made abortion a federally protected right are poorly written legal messes. If I turned in a paper written like that I'd probably fail the assignment. The spend half the opinion saying what they can't and won't do, and the other half ignoring what they just said and doing exactly that. Woof.

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