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Comments:
<0> I'm confused again.
<0> This def of limit point doesn't make sense to me.
<0> p is a limit point of the set A if and only if each open interval containing p contains a number in A different from p.
<0> I know I have this wrong, but I read that as: in any set, there can only be a limit point if there is more then one number in the set.
<0> how could an open interval have 1 number...
<0> I guess I'm confusing myself. I should just know that if a set has an open interval then it is infinite.
<1> hellop: yeah, u got it a bit wrong there
<2> hellop - a point p in a set S is a limit point of S if every neighbourhood about p contains infinitely many points of S. This definition is equivalent to the one you gave above (why?).
<0> Cat, because p is contained in an open interval.
<0> which is somewhat confusing
<2> That doesn't really answer the question. =)
<0> to me contained can only be understood in the notation (0, 1] where 0 is a limit point. so, 0 is "contained" in the open interval
<0> However, a though the limit point is 0 the set IMHO does not contain 0
<0> whoops, an extra a there
<1> hellop: think of a limit point as a limit of a sequence from distinct points of A
<2> Certainly every neighbourhood about 0 contains infinitely many points of (0, 1].
<0> so if p is a limit point, how can an open interval contain p?
<3> isn't border point a better term ?
<2> If it happens to be the case that every cluster point of a set S is contained in S, then S is said to be closed.
<3> for the 0 i mean
<2> kmh - you're welcome to invent your own terminology if you like, but 'limit point' and 'cluster point' are standard. =)
<0> kmh, yes, instead of the open interval contains 0, say the open interval has a border point or limit of 0..
<2> (of course, 0 is also a boundary point for S = (0, 1] in R, meaning that every neighbourhood about 0 contains a point in S and a point not in S)
<0> but only a closed interval could really contain 0 like the sequence 1/x does not contain 0 though it's limit point is 0
<2> (-1, 1) is an open interval that contains 0.
<0> yes, but does the open interval (0,1] contain 0?
<2> (0, 1] isn't an open interval. =)
<0> oh
<0> You're right
<4> Catfive, depends ;)
<2> (relative to R with the usual topology)
<4> heh
<2> @quote fear
<5> There he goes. One of God's own prototypes. Some kind of high powered mutant never even considered for m*** production. Too weird to live, and too rare to die.
<3> Catfive : actually boundary point not border point, was the term i'm looking for (regarding 0)
<0> I guess my confusion comes from this. Show an open interval that contains a number p such that there is no other number then p in the interval.
<0> like you have a set with just one number: 1 then you draw some parantheses and call it an open interval (1)
<6> "number than" ?
<0> that doesn't make sense to me
<0> ok, let me give the other definition of limit point
<0> P is a limit point of A if and only if each open interval containing p contains a number in A different from p.
<3> i.e. is a pint an open interval ?
<3> pint=point
<6> ah
<6> a point couldn't be an open interval... it has no neighbourhood?
<3> yes
<3> i tried to understand/reformulate hellop's question
<6> ah
<7> On a different topic, minmaxing in GURPS Mecha is a heck of a lot of fun.
<7> Hey, it's mathematical! :D
<0> Yes, but why do you have to say that the open interval contains a number different then p?
<3> hellop : well so p cannot be a limit point in {p}
<0> [p]?
<3> with A:= {p}, the definition fails as you cannot pick points in A being different from p
<0> is { the same as either ( or [?
<3> {p} <--- brackets to denote a set not an interval
<0> k
<0> oh ok... gotcha
<6> ... is the interval (-delta,delta) where delta is infinitely close to p?
<0> oohh ok, so having an open interval does not mean that a set is infinite, just that it doesn't include that point..
<6> exactly
<8> are you talking about surreal numbers?
<6> hehe
<6> yes, n in S ;)
<8> but even if you are, open inbervals are surely going to be infinite
<8> heh
<0> I can say, "on the open interval (1,2,3)" where my set only contains 1,2,3 then that open interval only contains 2
<6> syntax error?
<3> indeed
<0> and thats why the definition says each open interval contining p contains a number in A different from p
<8> what is (1,2,3)?
<6> hellop, if that would be (1,3)
<2> hellop - an open interval of R is a subset of R of the form (a, b) = {x in R: a < x < b}.
<0> oh yes, sorry
<2> hellop - please bear this definition in mind, it's quite specific.
<6> nice, a formal definition =)
<8> hellop: the poit of an open set is that if you sit at any point in that set, you can move a little bit and still be in the set
<8> so fi you have (a,b), wherever you start, you can always move a bit to the left or right and still be between a and b. but in [a,b), if you start at a, you cant move left at all
<0> _llll_ as long as the set is infinite
<6> are there no open sets in N?
<2> a finite set is closed
<6> ah
<0> gotcha
<6> ...of course...
<9> hi
<8> hellop: well, not necessarily, as long as you define "move a bit" correctly, but in R all open sets are infinite *except for the empty set*
<10> _llll_: what's left and right? ;)
<6> "(2,4]"=[3,4] in N ?
<11> Volatile, now I got forward with my logic :P. A slice is (2/3)*tan(a)*R^3, for small a ~= (2/3)*a*R^3 or (2/3)*da*R^3. We've got 2*pi/da slices, so (2*pi/da) * ((2/3)*da*R^3) is.. ta-daa.. 4*pi*R^3/3 :P
<8> (2,4] contains 2.5
<0> integers = N
<8> oh, you mean as subsets of N, then yeah they'd be the same
<6> Aldaron, congrats ;)
<6> _llll_, 2.5 in N?
<11> Volatile: thanks ;). I'm still not convinced, but it's logical and gives the right answer so maybe I'm happy :)
<6> Aldaron, you got the right answer. Don't poke things that stands... ;p
<0> thanks for all the help tho guys
<6> hellop, thanks for giving me the oppurtunity to get away frome everyday non-mathematical things ;)
<12> is -2^2 -4 or 4? I'm confused?
<6> (-2)^2 = 4
<11> 4 surely
<13> forngren: It's not welldefined that way.
<6> -2^2 = -4
<6> I think
<13> -(2^2) = -4, (-2)^2 = 4.
<11> yeah. What Volatile said :)
<12> ok, thx
<6> =)
<13> -2^2 ends up being a question of your notation, and is bad form to leave that way.
<14> i thought -b^x was always ***umed to be -(b^x)
<12> Syzygy: yeah, but that's what my book said...
<11> chessguy: That's what I'd think it is too, but I'd still like to clarify sometimes
<14> must be a crappy book
<12> yeah
<11> chessguy: Actually I agree with you. I'd hate to see -(b^x) for stuff that is like a^3 - ba^2 + b^2a - b
<14> yeah
<9> can i calculate with scalars with a CAS like mupad?
<9> or e.g. scalar-products?
<8> most cas will support vector stuff i think
<9> how can i type it?
<8> depends on the cas, find the documentation, look up the concept you want
<15> Say I have a map tau which maps an nxn matrix entry wise to a vector of complex numbers that is of size n*(n+1)/2. Now say I have a normal matrix X and a diagonal matrix D. If tau(X)=tau(D), what can I say about X????
<0> ^ comes before - in unary operations but it's subjective
<8> Safrole: surely that depends quite a lot of tau
<0> wheres my damn calculator
<16> Safrole, it doesn't make a lot of sense this way
<6> hellop, wanna borrow one of mine?
<15> tau((x_ij) from 1 to n) = (x_11, ..., x_1n, x_22, ..., x_2n, ... , x_nn)
<0> whats it say for -2^2?
<15> you just map all the entries of the matrix
<15> into this vector
<8> tau is injective, so X=D ?
<16> Not all, Safrole
<15> the upper triangle part
<15> thank you
<17> Does anyone have the book: linear algebra 3rd edition by Fraleigh Beauregard?
<15> interesting
<8> the upper triangular part?
<3> SoerenW : yes of course
<8> perhaps i misunderstood what tau is then
<15> yeah I didn't specify originally
<15> tau((x_ij) from 1 to n) = (x_11, ..., x_1n, x_22, ..., x_2n, ... , x_nn)
<15> okay that makes more sense now
<8> if you are only picing out the upper triangular part then all you can say is that X is lower triangular with D's entries onthe diagonal
<15> thank you deltb
<15> yeah _IIII_
<15> that's the conclusion I wanted
<16> yw
<15> I wasn't paying attention to what tau really was though
<15> now that I noticed, it's obvious
<2> the fact that n^2 =/= n(n-1)/2 in general should have been a tipoff. =)
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