anonymous
  • anonymous
please can someone explain what open balls, closed balls and spares are in a metric space
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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zzr0ck3r
  • zzr0ck3r
The best way to think about it, so that it makes sense with the name, is with the standard metric on \(\mathbb{R}^2\). Give me a radius \(\delta\) and some point \(a\) and the open ball about \(a\) is all the points within \(\delta\) of \(a\). So its like you surround \(a\) with an open ball.
zzr0ck3r
  • zzr0ck3r
or circle...
ikram002p
  • ikram002p
They are two important conditions in metric I think we should mention them right ?

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zzr0ck3r
  • zzr0ck3r
3, he didnt ask about metrics...
zzr0ck3r
  • zzr0ck3r
The concept stays the same through different metrics but they no longer match the name. The taxi cab metric will gave an open ball that looks like a diamond, and the infinity metric a square.... Also note that the actual ball is not the border, its all the stuff inside(for an open ball). So if you take a basket ball and fill it with air, then the open ball is the air(in R^3 with euclid metric). I hope that makes sense.
zzr0ck3r
  • zzr0ck3r
I do not know what a spare is
zzr0ck3r
  • zzr0ck3r
repost that
anonymous
  • anonymous
let E =R endowed with the metric Do defined by Do(x,y)= 1, if x is not y and 0 if x =y for arbitrary x,y element of R . compute the ball B(1;1/2)
zzr0ck3r
  • zzr0ck3r
well that says that everything but 1, has a distance of 1 from 1. And we want to know about the points within 1/2 of 1. So there is only one. what is it?
zzr0ck3r
  • zzr0ck3r
Do(1,1.4)=1 Do(1,1.2)=1 Do(1, 0.8)=1 Do(1,1)=0 Do(1,0.6)=1
ganeshie8
  • ganeshie8
lol it is a weird metric, everything is at 1 unit away from everything
zzr0ck3r
  • zzr0ck3r
how many points have a distance of less than 1?
zzr0ck3r
  • zzr0ck3r
not everything :)
ganeshie8
  • ganeshie8
Ah except the self
zzr0ck3r
  • zzr0ck3r
Right, so here is a metric that gives an open ball that is a singleton. Does this make sense? @GIL.ojei ?
anonymous
  • anonymous
So Sir , what is the question asking us to find and how did u get all those point s and equate them to 1 and how was tour conclusion made??
zzr0ck3r
  • zzr0ck3r
\(B_{Do}(1;1/2)=\{x\in R \mid Do(1,x)<\frac{1}{2}\}=\{1\}\)
zzr0ck3r
  • zzr0ck3r
The question wants the set of all points that are within distance 1/2 of 1. But with this metric, everything, except 1, is distance 1 from 1. So the only point in the set is 1 itself.
zzr0ck3r
  • zzr0ck3r
because the distance from 1 to 1 is 0.
zzr0ck3r
  • zzr0ck3r
if this was the euclidean metric we would have the interval (0.5, 1.5)
ganeshie8
  • ganeshie8
i think "n" points require "n-1" dimensions for this metric to be valid/used
zzr0ck3r
  • zzr0ck3r
I don't know what you mean.
zzr0ck3r
  • zzr0ck3r
It passes all the rules of a metric on a set.
ganeshie8
  • ganeshie8
at least in euclidean metric in \(\mathbb{R}^n\)... if we have 3 points, then they can be at 1 unit away from each other only if they are at corners of an equilateral triangle - two dimensions
ganeshie8
  • ganeshie8
similarly if we have 4 points, we must go to 3-space where the points can be at vertices of a tetrahedron or something .. its hard to visualize for more points idk lol
zzr0ck3r
  • zzr0ck3r
A metric is a binary operation on the set. It takes only two elements as an argument .
zzr0ck3r
  • zzr0ck3r
err not a binary operation but from XxX to R.
ganeshie8
  • ganeshie8
XxX to R is a binary operation which takes two operands as input and spits out one real number as output right
zzr0ck3r
  • zzr0ck3r
I think a binary operation on X has to have X itself as the codomain
zzr0ck3r
  • zzr0ck3r
But yes.
zzr0ck3r
  • zzr0ck3r
\(\circ : X \times X \rightarrow X\)
zzr0ck3r
  • zzr0ck3r
That's a binary operation... but anyway.
zzr0ck3r
  • zzr0ck3r
But I think I see what you are trying to do and that is think about shapes with this metric. I am not willing to take that jump tonight :)
ganeshie8
  • ganeshie8
Exactly! I am trying to visualize, which is forbidden sometimes in real analysis haha!
zzr0ck3r
  • zzr0ck3r
How would we define a square with a normal metric?
ganeshie8
  • ganeshie8
I see your point, taxicab metric works well i think ?
zzr0ck3r
  • zzr0ck3r
I want to think about a square in this metric with that definition.
zzr0ck3r
  • zzr0ck3r
Well not even that. I am just saying how ever we define a square with the normal distance function on R^2, lets use that definition on this metric and try and think of what a square looks like.
ganeshie8
  • ganeshie8
there are only two possible values for distances here : {0, 1}
zzr0ck3r
  • zzr0ck3r
So a square with side length 1, lets say the unit square and look at the point (0,1/2) normally we would have all the points that are 1 unit away in one direction and we get only one point (1,1/2) But with this metric we get EVERYTHING... lol
zzr0ck3r
  • zzr0ck3r
So most shapes will give everything.
zzr0ck3r
  • zzr0ck3r
I think the only purpose of this metric is to ask this question :) ok 5am good night
ganeshie8
  • ganeshie8
Haha that is really weird to visualize! xD
anonymous
  • anonymous
the metric is just the one that induces the discrete topology, so balls of radius \(r<1\) only contain one point: \(B_{r\,<\,1}(p)=\{p\}\)
anonymous
  • anonymous
https://en.wikipedia.org/wiki/Discrete_space#Definitions this is because all the points are isolated
anonymous
  • anonymous
please guys, you have been arguing and i do not understand one bit please, what are the steps in solving the quation i gave and what would be the final answer
anonymous
  • anonymous
@ikram002p
anonymous
  • anonymous
@oldrin.bataku
ikram002p
  • ikram002p
i need to see the definition in your book of metric space in your book + which class is this to help you more ^_^
anonymous
  • anonymous
i am in my finals in national open university. here is a link to the book
ikram002p
  • ikram002p
where is it ? :D
anonymous
  • anonymous
http://www.nou.edu.ng/uploads/NOUN_OCL/pdf/SST/MTH%20401.pdf
anonymous
  • anonymous
@zzr0ck3r
anonymous
  • anonymous
@triciaal
anonymous
  • anonymous
@dan815
triciaal
  • triciaal
sorry can't help on this I don't know
anonymous
  • anonymous
let E =R endowed with the metric Do defined by Do(x,y)= 1, if x is not y and 0 if x =y for arbitrary x,y element of R . compute the ball B(1;1/2)
dan815
  • dan815
what is the definition of a ball
anonymous
  • anonymous
the open ball of radius \(r\) about \(p\) in a metric space \((X,d)\) is defined as $$B(p;r)=\{x\in X:d(p,x)
anonymous
  • anonymous
yes that was the definition in my book, they gave just open balls, closed balls and sphare
anonymous
  • anonymous
but did not define a ball
anonymous
  • anonymous
in this case, it doesn't matter whether they ask for open or closed balls, because there are no points other than \(p\) that are up to or within distance \(1/2\) of \(p\), so the ball in our discrete metric is a singleton: $$B(p;1/2)=\{p\}$$
anonymous
  • anonymous
ok, so what nxt
anonymous
  • anonymous
remember the definition of the metric here: $$d(x,y)=\left\{\begin{matrix}0&\text{if }x=y\\1&\text{if }x\ne y \end{matrix}\right.$$
Kainui
  • Kainui
@GIL.ojei Explain what you understand.
anonymous
  • anonymous
yes i remember
anonymous
  • anonymous
for example, suppose our space consisted of the following points \(p,q,r\). we know: $$d(p,p)=0\\d(p,q)=1\\d(p,r)=1$$ so the only thing within a distance of \(1/2\) is \(p\), since \(d(p,q)=d(p,r)=1>1/2\) and \(d(p,p)=0<1/2\)
anonymous
  • anonymous
yes, i know that
anonymous
  • anonymous
okay, and that's it
anonymous
  • anonymous
that's the problem you asked about, @ganeshie8 answered it hours ago
anonymous
  • anonymous
and @zzr0ck3r
anonymous
  • anonymous
so, how did he get does points like d(1;0.8)=1 I MEAN THE 0.8
anonymous
  • anonymous
OK, WHAT ABOUT THE COMPUTATION OF THIS AND SOLVE COMPLETELY WITH STEPS, PLEASE
anonymous
  • anonymous
OK, WHAT ABOUT THE COMPUTATION OF THIS AND SOLVE COMPLETELY WITH STEPS, PLEASE
anonymous
  • anonymous
let E =R endowed with the metric Do defined by Do(x,y)= 1, if x is not y and 0 if x =y for arbitrary x,y element of R . compute the ball B(1;5)
anonymous
  • anonymous
HELLO
anonymous
  • anonymous
can some one please answer
anonymous
  • anonymous
hello
zzr0ck3r
  • zzr0ck3r
You need to tell us when you don't understand us. We were not arguing we were discussing math. What do you not understand?
anonymous
  • anonymous
does it mean that hat they told us to do is to find points from 1 to <5?
zzr0ck3r
  • zzr0ck3r
Do you understand that \(B(1, \frac{1}{2})\) is the set of all points that are within one half of 1?
anonymous
  • anonymous
yes that is 1
zzr0ck3r
  • zzr0ck3r
So with the normal metric we would get stuff like 0.6,0.7,0.8,0.9,1.1,1.2,1.3
zzr0ck3r
  • zzr0ck3r
no with the standard metric it would be 0.5
zzr0ck3r
  • zzr0ck3r
Do you see that? \(1\pm0.5\)
anonymous
  • anonymous
yes, e - neighborhood of 1
zzr0ck3r
  • zzr0ck3r
So this is with the standard metric, but we are not in that metric. In this metric distance works differently than you are used to. The distance between any two different points is 1
zzr0ck3r
  • zzr0ck3r
So now there are no points within 1/2 of 1 (except 1 itself)
zzr0ck3r
  • zzr0ck3r
because everything has distance 1
anonymous
  • anonymous
So now there are no points within 1/2 of 1 (except 1 itself) ,, please give mare example on it
zzr0ck3r
  • zzr0ck3r
yes exactly
zzr0ck3r
  • zzr0ck3r
the distance between 1 and 3 is 1 the distance between 1 and 7 is 1 the distance between 1 and 900000000000 is 1
anonymous
  • anonymous
waw
zzr0ck3r
  • zzr0ck3r
So when you ask me what are all the points within 1/2 of 1, I tell you there is only 1 and that is 1 itself
anonymous
  • anonymous
ok
zzr0ck3r
  • zzr0ck3r
So now you tell me the answer to this question and I will know you understand \(B(56, 0.7)=?\)
anonymous
  • anonymous
1 following my definition of d(x,y)
zzr0ck3r
  • zzr0ck3r
I am asking for all the points within 0.7 of 56.
zzr0ck3r
  • zzr0ck3r
not 1 but 56
zzr0ck3r
  • zzr0ck3r
every open ball contains only its center point in this metric
zzr0ck3r
  • zzr0ck3r
if the radius is less than 1
anonymous
  • anonymous
ok
zzr0ck3r
  • zzr0ck3r
ok what about \(B(1, 3)\)?
anonymous
  • anonymous
1
zzr0ck3r
  • zzr0ck3r
no, we want to know all the things with less than distance 3 of 1, and everything has distance 1 from 1.
zzr0ck3r
  • zzr0ck3r
1<3, so it contains every point
zzr0ck3r
  • zzr0ck3r
If I tell you everything is 0 or 1, and then I ask you what is < 3. you say everything.
zzr0ck3r
  • zzr0ck3r
Sorry if I sound condescending in the way I explain things, I am not trying to. :)
anonymous
  • anonymous
no sir, its ok . as far as i understand it. you are great. please continue
zzr0ck3r
  • zzr0ck3r
That is about it. Everything has distance 1 from each other. So there will be only two outcomes for open balls \(B(x, r)=\{x\}\) if \(0
zzr0ck3r
  • zzr0ck3r
So what is \(B(a, 3)\)?
anonymous
  • anonymous
waw, am thinking
anonymous
  • anonymous
i don't know because the first condition seems not to be the answer because r>1 so it is not a
anonymous
  • anonymous
is it 3? please don't be angry
zzr0ck3r
  • zzr0ck3r
lol I would never get angry. Ok so the question is this. What is the set of points that are < 3 distance from a Everything has distance 1 from a, and 1<3. So everything has distance <3. So the answer is ?
anonymous
  • anonymous
everything
anonymous
  • anonymous
or 1
zzr0ck3r
  • zzr0ck3r
everything.
anonymous
  • anonymous
B(1,5) will be what?
anonymous
  • anonymous
you there?
zzr0ck3r
  • zzr0ck3r
everything has distance 1 from 1, we want all the things with distance less than 5 what is the answer?
anonymous
  • anonymous
everything
zzr0ck3r
  • zzr0ck3r
correct, if you change the radius to something smaller than 1, you will get only the center, which in this case is 1
anonymous
  • anonymous
so, what do you think are some important important points to note down about the balls?
zzr0ck3r
  • zzr0ck3r
If you understand the concept, the rest will follow.
zzr0ck3r
  • zzr0ck3r
What class is this for?
zzr0ck3r
  • zzr0ck3r
Have you dealt with the \(\epsilon -\delta\) definition of a limit?
anonymous
  • anonymous
finals in my university. i am facing a very big challenge
anonymous
  • anonymous
i have done limit but i don't know if i did indept
zzr0ck3r
  • zzr0ck3r
This is a hard topic and most people struggle with it.
anonymous
  • anonymous
let E =R endowed with the metric Do defined by Do(x,y)= 1, if x is not y and 0 if x =y for arbitrary x,y element of R . compute the ball B(1;5)
anonymous
  • anonymous
so, if i was to see that question in exam, what will be my steps to solving it?
zzr0ck3r
  • zzr0ck3r
First what is the ball asking for?
zzr0ck3r
  • zzr0ck3r
all the points that are at least distance ? from point ?
anonymous
  • anonymous
point 5 and 1
anonymous
  • anonymous
so what next?
zzr0ck3r
  • zzr0ck3r
We want all the points that within \(5\) of \(1\) but everything is distance \(0\) or \(1\) from \(1\) So EVERYTHING is distance less than \(5\).
anonymous
  • anonymous
so which means that there are three points to note, if r<1, the point becomes 1 and if x=r, the point becomes zero but if r>1, then the point becomes everything , right?
anonymous
  • anonymous
hello sir
anonymous
  • anonymous
hello
anonymous
  • anonymous
hello

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