August 27th, 2017, 09:39 PM  #21  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
Since the epsilon interval $\displaystyle (01/2,0+1/2) \not\subset [0,1]$. We were talking about the set $\displaystyle (0,1)\cup ${$\displaystyle 1,2,...,n$}, why it is not an interval when $\displaystyle n \geq 2?$  
August 27th, 2017, 09:43 PM  #22  
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Quote:
Quote:
Do you need practice or explanation in negating quantifiers? In other words if I am required to prove that there exists an epsilon, trying one epsilon and finding it's not the one I wanted isn't enough to prove that NO epsilon will work. Last edited by Maschke; August 27th, 2017 at 09:47 PM.  
August 27th, 2017, 09:55 PM  #23 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 
But you have asked to prove that $\displaystyle [0,1]$ is not a neighbourhood of $\displaystyle 0$ only right ? $\displaystyle [0,1]$ is a neighbourhood of each of its points, excluding the two extreme ends 0,1. That is why I have taken an $\displaystyle \epsilon>0$ for which the epsilon interval is not a subset of $\displaystyle [0,1]$. Even if we take $\displaystyle \epsilon=0.1$, it is still $\displaystyle (0.1,0.1)\not\subset [0,1]$. 
August 27th, 2017, 10:06 PM  #24 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  I will workout for the proof of this. And let me know if the subset of a set is not a neighbourhood of that point then it's not always true that the superset is also not a neighbourhood of that point right ?

August 27th, 2017, 10:12 PM  #25 
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Are you unsure about the quantifiers? That's important to nail down. You understand that to satisfy an $\exists$ you only have to find one that works. So if you try one and it doesn't work ... that doesn't prove anything. Is that clear? I'm afraid I don't understand that question. The set {1/2} is not a neighborhood of 1/2 but the superset (0,1) is a neighborhood of 1/2. Yes? 
August 27th, 2017, 10:24 PM  #26  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
I'm asking if $\displaystyle (0,1)$ not being a neighbourhood of $\displaystyle 0$ effect the relation between $\displaystyle 0$ and $\displaystyle [0,1]$ ? Since we know that Every super set of the set which is a neighbourhood of x, is also a neighbourhood of that point. I'm asking if the converse true ?  
August 27th, 2017, 10:33 PM  #27  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
let $\displaystyle \epsilon = 0.2 > 0 $ and $\displaystyle x = 0 $. we need to show that $\displaystyle (x \epsilon, x+ \epsilon) \not\subset [0,1] $ $\displaystyle (00.2, 0+0.2)$ $\displaystyle (0.2,0.2) \not\subset [0,1] $ Hence $\displaystyle [0,1] $ is not a neighborhood of $\displaystyle 0$. Is this enough ? or Is it to be more generalized ?  
August 28th, 2017, 09:23 AM  #28  
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Quote:
What's wrong with that proof? You are still confused about quantifiers.  
August 28th, 2017, 09:30 AM  #29  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
You have asked to prove for closed set $\displaystyle [0,1]$ right ?  
August 28th, 2017, 09:41 AM  #30  
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Quote:
Quote:
Are you drawing pictures for yourself to try to understand what a neighborhood is? And working with the quantifiers to understand what you need to show?  

Tags 
neighbourhood, point 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Nearest distance from point to point on line  abcdefgh123  Algebra  2  January 4th, 2014 04:08 PM 
the measure of 0neighbourhood for functions with large grad  medvedev_ag  Real Analysis  0  June 5th, 2013 02:38 AM 
Drawing a line from point a,b .. coords point b unkown  Vibonacci  Algebra  5  September 17th, 2012 01:28 AM 
calculating point of line given starting point, slope  mathsiseverything  Algebra  1  March 4th, 2008 07:41 AM 
the measure of 0neighbourhood for functions with large grad  medvedev_ag  Calculus  0  December 31st, 1969 04:00 PM 