Studying I'm doing EVERY exercise in munkres' topology textbook

[Tom1992]

please understand that i am only the youngest student in my first year classes, but probably not the smartest.

 

[verty]

please understand that i am only the youngest student in my classes, not necessarily the smartest, otherwise i might actually speak out in class.

My point is true independently of whether you are the smartest or dumbest or somewhere inbetween. And why should you need to be the smartest?

Oh, sorry, you meant you should need to not be the youngest (or be the smartest). Is age that important?

 

[Tom1992]

here is another interesting munkres problem from chapter 1. i will repeat once more that all problems i post in this thread do not already have a solution posted in the web.

prove that the set A = {a,b}x{a,b}x{a,b}x... is uncountable.

the solution is very short, but requires deep thinking.


proof:
suppose that A is countable. then there is a surjective map f:N -> A, where N is the set of positive integers. define x as the unique element in A whose nth coordinate is different from f(n) for all n in N (this construction is possible since each component has two possible values). since f is surjective, then there exists m in N such that f(m)=x. but this is impossible since by construction x differs from f(m) in the mth component. this contradiction means that A cannot be countable.

this same trick can be used to prove that the real numbers is not a countable set, using the infinite decimal repesentation of a real number. try it!

 

[verty]

If P is a countable subset of A with cardinality omega (equal to the cardinality of omega), I'm wondering what the cardinality is of the set of surjective maps from N onto P...

 

[Tom1992]

my reading has now actually slowed down. whenever i read a theorem i glance at how long the proof is. if it is short, i try to come up with the proof myself. usually such short proofs to theorems serve as valid a problem as the textbook problems. so i guess now it will take me longer to finish this book than i expected, but time is on my side. however, i don't always manage to come up with the proof: some short proofs are sooooo hard! for example:


prove that there is no surjective map f: A-> P(A), where P(A) is the set of all subsets of A.


proof: define B={a in A: a belongs to A-f(a)}. suppose B=f(c). then [c belongs to B] iff [c belongs to A-f(c)] iff [c belongs to A-B], a self-contradiction.

wow!

 

[mathwonk]

these arguments are all due to cantor. you might enjoy reading his own work, contributions to the founding of the theory of transfinite numbers.the cardinality of the set of all maps from S to T is #(T)^[#(S)].

thus the cardinality of the maps from a set say Z to {0,1} is 2^alephnull.

the basic argument shows that this is always larger than the cardinality of S, if #T > 1. I guess.

Equivalently, since a subset of a set S is equivalent to a map from S to {0,1}, the set of subsets of S always has greater cardinality than does S.

It was knowing these arguments that got me into honors calc as a freshman in college, since it showed my interest in math. I read them in high school.

 

[Tom1992]

munkres doesn't teach cantor's result in his book, so i couldn't use it. my original attempt at a solution to the above problem (in my previous post) was:

every member of A is also a subset of A, and hence is a member of P(A). thus there is a bijective map from A to a proper subset of P(A), which means that there cannot be a surjective map from A onto P(A).

but i wasn't convinced of this proof of mine. the above proof is much more elegant and indisputable.

 

[tehno]

If he is able to understand and work through these abstract mathematics at such a young age, why would you discourage him? All of the idiots that I hung out with in high school that were 'popular' never left the town we grew up in and will never do anything beyond high school.

Don't get an impression I discourage young Tom from doing math.
Just on the contrary:I encourage him to do it but in a different way.
The way he is doing it now seems to me completely unnecessary and premature no matter how advanced he may be.
If he wants to excell in math and be a successful & productive mathematician there is much better route to take than to study ton of the books and doing every single exercise from them.
I bet there is still a lot stuff from areas of so called elementary math he needs to work on before he starts to study Riemann geometry.
For example : https://www.physicsforums.com/showthread.php?t=145089" is a sort of problem I would expect from a talented 14 year old to deal with.Problem in a common Euclidean plane.Nothing less and nothing more.Not trivial though.
Nobody will stop young Tom to learn about Riemann geometry .However,as they say the science isn't a rabbit ,won't flee from you..
Except advanced course in Linear algebra and maybe first course /introduction to group theory I would rather recommend him title like:
Arthur Engel :"Problem Solving Strategies"
I think he may benefit much more from it than from titles dealing with Riemann geometry or topology
.

Moreover, I'm pretty sure it's up to him to decide whether chasing girls is a worthy life purpose.

Indeed .But why not to do both in a reasonable weighted proportions?

 

[complexPHILOSOPHY]

Don't get an impression I discourage young Tom from doing math.
Just on the contrary:I encourage him to do it but in a different way.
The way he is doing it now seems to me completely unnecessary and premature no matter how advanced he may be.
If he wants to excell in math and be a successful & productive mathematician there is much better route to take than to study ton of the books and doing every single exercise from them.
I bet there is still a lot stuff from areas of so called elementary math he needs to work on before he starts to study Riemann geometry.
For example : https://www.physicsforums.com/showthread.php?t=145089" is a sort of problem I would expect from a talented 14 year old to deal with.Problem in a common Euclidean plane.Nothing less and nothing more.Not trivial though.
Nobody will stop young Tom to learn about Riemann geometry .However,as they say the science isn't a rabbit ,won't flee from you..
Except advanced course in Linear algebra and maybe first course /introduction to group theory I would rather recommend him title like:
Arthur Engel :"Problem Solving Strategies"
I think he may benefit much more from it than from titles dealing with Riemann geometry or topology
.

Indeed .But why not to do both in a reasonable weighted proportions?

Nah FER SURE! You just didn't explicitly imply those intentions initially, my friend. (say that 10x fast).

I have to work through lots and lots of problems on my own so that I can subjectively reason through all of them correctly, before I feel confident enough to proceed to a higher level of mathematics. So when I hear that young kids are doing this level of mathematics, it makes me feel stupid.

I get stuck on a field of maths for a while because of this and I don't know how to transcend that boundary. Even if I get an 'A' in the course, there are still lots of problems in my book that I can't work through the first time around so I obsess over that and continue to work on the course even after I have 'finished' it. It's not a problem right now since I am still in lower-division courses, however, I am afraid it will eventually make an impact.

 

[verty]

I'll just add that if someone has a passion for exploring, they should explore where they want to explore rather than following a recipe they don't agree upon or understand. Of course it probably would be fortuitous to investigate recipes but certainly one should always follow one's own recipe.

 

[kdinser]

As if I didn't feel dumb enough around math people already, this thread has wrecked me for life.

Great read, even though I only understood every 5th word.

 

[Tom1992]

these arguments are all due to cantor. you might enjoy reading his own work, contributions to the founding of the theory of transfinite numbers.


the cardinality of the set of all maps from S to T is #(T)^[#(S)].

thus the cardinality of the maps from a set say Z to {0,1} is 2^alephnull.

the basic argument shows that this is always larger than the cardinality of S, if #T > 1. I guess.

Equivalently, since a subset of a set S is equivalent to a map from S to {0,1}, the set of subsets of S always has greater cardinality than does S.

It was knowing these arguments that got me into honors calc as a freshman in college, since it showed my interest in math. I read them in high school.

thanks for your insight. i went ahead and tried to prove the #(T)^[#(S)] formula, which was trivial in the finite case, though I'm not sure if my proof is valid in the infinite case. i then had to ask myself, how many injections are there from S to T (assuming #T > #S) and and how many surjections are there from S to T (assuming #T < #S)? again, i obtained the formulas in the finite case (using only high school math interestingly). as to how to do it in the infinite case i have no idea. it's interesting how the more exercises you do from a textbook the more new questions you wonder about.

 

[TQFTsRcool]

Linear algebra is just the representation theory of a field, and that is a trivial subset of far more interesting subjects.

What a great line! I wonder whether Matt considers differentiation to be just the projection of a curve on a manifold onto the manifold's tangent space, and therefore a trivial subset of a far more interesting subject, too.

 

[Caramon]

I thought I was ahead of the game when I took AP Calculus in Grade 11 (at age 15) and finished Multivariable Calculus when I was 16... since when is it normal for people to know calculus when they are 8-11 years old? D:

 

[Cider]

This thread is over four years old.

 

[Ryker]

This thread is over four years old.

Your point being?

 

[Cider]

My point being that there is no reason to post here anymore, as the purpose of the topic does not merit any more posts since it is no longer relevant to the original poster. That and to inform the person who revived it, clearly someone who is new, that they posted in a thread that is four years old, otherwise they might go reviving other threads that need not be revived.

 

[Ryker]

If it bothers you other people find interesting stuff in old threads and actually want to comment on something that struck their attention, surely it's not that hard avoiding such threads, is it? A glance at the date of the post just before the new ones takes a second, after all.

 

[hjaramil]

Tom I am trying to solve problem 4 of page 193 of Munkres "Analysis of Manifolds".
It confuses me that he suggest the use of SIX (6) points to find tha area of a triangle.
Why six points?
Any hint?

thanks.

 

[hjaramil]

I think I solved it.

The idea is to apply the mean value theorem six times: For each of the three components in each of the two directions. Since the mean value theorem is for functions from R^n -> R, I could have a different point for each of the (3) components on each of the (2) directions. That is why I need six points. This was killing me...

Thanks anyway.

H.