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Wednesday, October 25, 2006

Whatz prooved & Whats not :: A DEAD END SUPPLIMENT !

Hey all !

Hmm ...
Another Dead End ? Nope I'm saturated of those now. But this one's interesting enough to be brought to notice of the blog.

A seemingly remarkable proof indeed for prooving that all numbers are interesting !
But Can a subjectivity of the problem be discussed mathematically ? The answer seems to be yes !?!

Now lemme introduce you 2 a couple of DEAD ENDS :

(1)

To Proove : Every natural number N can be described within 15 words.
Assumption : Let there be a set S of number which cannot be expressed within 15 words.
Ordering the set S & pickig out the first element, say P we get P as :
P is the "the smallest natural number that cannot be unambiguously described in fifteen words or less".

:. P can be described within 15 words.
By Reductio Ad Absurdum ;
We herey proove the Statement that :
EVERY NUMBER CAN BE REPRESENTED WITHIN 15 WORDS.

(2)

I begin this one, relating to the THEOREM OF MATHEMATICAL INDUCTION as quoted by Arun in a comment to the previous post.

"

The principle of mathematical induction basically is a logical proof of extention. If you can show that:
1) The statement is true for the base case, i.e. 0 generally,
2) That if it is true for n, n-1, n-2, it IMPLIES truth for n+1.

"


Okay. Let there be a statement S(n) depending on n which can be either TRUE or FALSE.

I DEFINE S(n) as :
S(n): "In ANY group of 'n' blog-members, all have equal ages."
OR
"In a set of blog-members-ages from ANY particular blog, all values are equal."

Now proceeding on the lines of the above discussed theorem, lets discuss S(1).
: " In ANY group with 1 blog-member, All blog-members have equal ages ( Takin one member at a time since n = 1 )."
Clearly S(1) is TRUE.

Assume that S(k) is TRUE which implies that :
: " In ANY group of 'k' blog-members, all have equal ages."

Now to proove S(n) for all N belongs to (1,infinity), We must proove S(k+1).

So now we have a set of (k+1) blog-members. Of those, we can choose different combinations of 'k' blog members. In each of these combinations, all blog-members would have equal ages. Therefore, All blog-members in the (k+1) blog-member set are equal in age.
:. S(k+1) is TRUE if S(k) is TRUE.


S(1) is true. S(k+1) is TRUE When S(k) is TRUE.

:. by the THEOREM OF MATHEMATICAL INDUCTION ...

S(n) is TRUE.

Implying this for the blog KVPY2005 : Fused Bulbs... I hereby proove that all memebers in this blog are equal in age, which is conincidentally true within the limits of error. lol.



Ill leave the implications to be discussed by the blog members.

WAITING FOR COMMENTS

(desperately hehehe)

Twish

22 comments:

Arun Chaganty said...

Um, Twister, are you ill or something?
So now we have a set of (k+1) blog-members. Of those, we can choose different combinations of 'k' blog members. In each of these combinations, all blog-members would have equal ages. Therefore, All blog-members in the (k+1) blog-member set are equal in age.
:. S(k+1) is TRUE if S(k) is TRUE.

Your statement said that a particular 'k' members of the blog are of equal age, but not that every k blog member set have the same age. When you add the new member, you create a problem. That member may or may not be the same age.
Oh, and I think you should make the point that as there are 'k' people in this group, they have the same age, statement (though it is false) more explicit, so that it'd confuse more people.

TwIsTeR said...

Sry Arun But its the proof's we've been discussing that are ill.

Newayz.

So u forgot the statement ?
That in ANY set of blog-members...
I can choose many sets of 'k' members from the set of 'k+1' members.

No fault over here.
Check with a thermometer.

Arun Chaganty said...

My apologies, your statement was right. Ouch, I remember seeing one such thing which was related to all straight lines are concurrent, but it broke down somewhere. Hmm, I'll have to look into this more carefully.

I think the best I can reason is that for any comparison, you should have atleast 2 people in your set. Hence the first thing to prove is not S(1) but S(2). It doesn't make sense for S(1). But your statement that all ages of any k-membered set makes me think twice about it.

Just as a interval, I hope all of you realise that this statement is a stupid (no offense) casting of a much larger set of proofs. All of mathematics would break down if this were true. For example, we could say, and k-membered set has all elements the SAME. This argument has just been put in a more colorful guise.

TwIsTeR said...

Agreed that its not true. Of course we realize that Arun, for I know not all of us are the same age. Thatz enuff a reason !

Though wats wrong ?

& wat about the first proof.

Arun Chaganty said...

@twishter: Ya I know you'd have thought about it, but I just was kinda stating it out loud. That we're the same age is a small thing compared to saying all sets are the same. That means if you have a set of dogs, cats and humans, we'd be the same in all respects.
As for the first statement, one doesn't know how to define the interesting-ness of a number, so you can't say anything about it.

At this point, I've come to a realisation of our proofs, which should have obviously come sooner. Suppose you have a particular number, whose interesting-ness you are discussing. Now if you say it isn't interesting, then it becomes the first uninteresting number, hence it becomes interesting. But then it no longer satisfies the only definition that does make it interesting, i.e. that it is the first uninteresting number. Hence, if you look at it, you can't say anything about it. It can neither be proved interesting nor uninteresting.
How's that take on things?

TwIsTeR said...

In my opinion,
You should define interesting in the first place & then proceed.

Though wat about the EXPRRESSOING NUMBERS WITHIN 15 WORDS proof ??

Arun Chaganty said...

Halt the presses, I finally clarified what's wrong with your set proof. It's a very small little rat. The problem is in the way your proving the S(k+1). What you've done is decompose (fancy for make subsets) your set into subsets of k each. And by equating the terms in all of these subsets, you go on to say that it one subset where a_k (say) is absent, there's another where it is present, There will be common terms in these two sets, hence we will be able to equate this term a_k with the rest of the elements in set A and prove S.
But that assumes that there will be common terms in the two sets, which is only possible for cases after n = 2. Thus it must be proved for n = 2, and there it fails.

StaN said...

first november 2006
HAPPY FAMILY OF MY SCHOOL
out on PDB
praavandablog.blogspot.com

the biggest rap
ever written by STAN

p a y a v i s i t

Anonymous said...

Im in class 11 and will be giving the kvpy exam on 12th nov.what kind of questions should i expect and are there any specific topics or areas i should focus on?

Arun Chaganty said...

I think we've already answered such a Q once. Take the exam lightly. You'll mainly be tested on 10th class stuff, and some general knowledge (scientific stuff). Also you'll be writing all Maths, Phy. Chem. and Bio. Maths will have some slightly tougher problems but you should be able to do it. If your genuinely interested in science, and have been for a long time, you won't have much trouble.

Anonymous said...

that doesnt exactly help much.could you be more specific?

Arun Chaganty said...

i can't actually. Just study whatever you had in 10th class, and study it well, that should be enough. Maths may be a little tougher than the rest, and be more logical and more applicative. You may want to look at some Maths Olympiad questions just to get an idea.

Ankit said...

Hey "listen" (->class 11 student)
In fact read.
Dont worry for your exam.
As arun puts it -"if your genuinely interested in science, and have been for a long time, you won't have much trouble."(i like what you said arun except of course the 'long time')
The Kvpy paper last year had 2 sections per subject.one with single choice correct, other with multiple choices.They had 4:1(ratio)(i think it was +2 -0.5) marks so intelligent guessing is always a good idea in single choice.(simple maths figure it out)
Just for your confidence, i attempted all questions(might seem strange but this is how i generally attempt a paper).
The questions were non standard. Importance was given to maths with non routine questions (almost small puzzles).I dont think u have given a lot of "thinking" 3 hrs paper so the secret lies in keeping ur cool and go for it even when u find the first few questions un-do-able.I think that should solve ur problem.

Gr81 said...

I agree with u guys - I never studied for any such tests - got thru VMC's test, NTSE, KVPY (So I can't call all of em a fluke ;))

Well, I got a mail from another student (Anmol) who found my id thru the blog (Pretty cool :D)

Here's my reply to his ques: (Mayb it'll help others too!)

PS: ITS A BITTTOOOO LONG.......
- as for preparation, shud i revise the X class science-math course or the XI course or even further, i.e. XII

First of all, KVPY is a scholarship given to students who have the potential to take up research in future. But it doesn't really mean u should be a very geek and bookworm sort of a person. They need students with good base, not just those who memorize facts, problems etc. (A bit of this is my personal opinion, but I hope u agree with me...)

Anyways, they say they'll ask stuff from class Xth course - so do it thoroughly. Some (or many) topics in XIth are linked to Xth (Quad Eq, Trigo, Circles etc.) and u should revise them too. A bit of idea about Complex might help too.

I heard they have changed the pattern of exams a bit this time, so I'm not sure what they'll give. But it is not as easy as NTSE (incase u know what it is) or like CBSE (which is just memorizing and nothing else..)

- do i need to be good at calculus


Calculus is something that is hyped a lot in XIth. It is very simple (really!!) U must be having idea about Diff. and Integral Calculus a bit. I dont remember getting a ques on that, but still just see the basic idea.

- is science G.K. type or manipulative type


manipulative type??? Couldn't understand that! But yeah if u meant taking out the answers by looking at the option - well that isn't a good idea. Still I found the ques from general chem and phys that we studied in Xth (and a bit in XIth). Maybe u should do a bit more of chem (I'm pretty weak at it ;)

- what kind of preparation did u do for kvpy , not many give this from my school thus i am seeking guidance from u


Frankly speaking, I never prepare for such exam like I do even for my school test. These tests hardly test your memory, they actually test your knowledge. (And incase u think I just got thru by fluke... well then mayb it worked in NTSE, KVPY and EROSE too... So mayb I'm too lucky!!)

Still, as I have said above, your base should be strong, the concepts is what is most important.

- am also appearing for olympiads, is the preparations interlinked ??


I have always thought of olympiads for people with a really scientific mind - the ques there are few and really challenging. In my paper, the ques weren't (or mayb didn't look) that tough as they are in oympiads, but if u get good ranks in the olympiads, then u can easily do the KVPY paper. (It is the interview stage that will really test you !!)

PPPSSS: Gr8 if u read till here - And yeah u dont need to post anonymous comments... u can leave your name atleast :D

Arun Chaganty said...

Suddenly everyone wakes up.
@ankit: I mentioned for a long time because the experience and love and knowledge that you gather can't be adiabatic. It's gotta be done nice and slowly. And you pretty much scared me off when you said:

The questions were non standard. Importance was given to maths with non routine questions (almost small puzzles).I dont think u have given a lot of "thinking" 3 hrs paper so the secret lies in keeping ur cool and go for it even when u find the first few questions un-do-able.I think that should solve ur problem.

@Rash: I haven't a clue what EROSE and VMC are, but I'll assume their tough tests that no one does well. But here's an inherent contradiction: It isn't memory-based, but requires knowledge.
As far as I can tell, knowledge is very much memory. It's understanding that it requires. I actually didn't do maths very well (I never come up against those type of problems before, but I still took a good shot at them), but I definately felt that the physics chem and bio were lets be frank easy. The topics weren't a bit of XI and XII.
Calculus rocks, and its quite easy too, as Rash said. Don't ever be intimidated by calculus. (Have you ever had that urge that everything you see isn't so nice and clean and simple and continuous. Actually calculus is a deep rich topic, whose deeper workings are very complex. Life's boring these days, because I haven't seen an anomaly in a long time (not since the Chaos Game)).

Oh, a girl in my jr. college (call it school if you like though) has been hell-bent on this exam (thats a good thing). She asked me if she needed to learn Quantum Theory and Quantum Mechanics! I told her to lay off the heat. The exam was cinch, and for the interview, she could learn it if she genuinely was interested in it. (I kinda explained stuff about the Theory of Relativity in my interview). I think we should post a little 'sticky' of sorts on our page to pacify such people.

Gr81 said...

Yeah Arun u r correct. Waise by Knowledge I meant Knowledge not Information, where Knowledge involves more of thinking than mere cramming of facts and figures (Information acoording to me is more of facts than understanding)

I wish we could have other papers similar to the KVPY one (So that even if u cram less, u still get good marks if ur basics are good - kinda IIT style)

And yeah, Erose is a National Scholarship Test (pretty ok kind) and VMC: Vidya Mandir Classes are considered the best coaching classes in Delhi (PS: I dont think so, but there is some kinda hype about it)

Anyways, best of luck to those giving the paper!

Arun Chaganty said...

Aha. I guess every place has there own exam for stuff. In HYD there's Ramaiah (which I consider equivalent to hell), which has the SAT (not THE SAT). That paper is about as tough to crack as my skull (its do-able, but not by me). Actually thats just the maths exam. The chem and phy aren't so much. I generally feel that that exam did require some experience before, or more time (*much* more time).

Ankit said...

We have managed to put in 17+1 comments(I wonder how many are actually about the dead end).
You guys talking about Vmc??
Well That is the place i go to.Infact we have a test this Sunday.And mind you my coaching institute does not give tough questions.
They have a simple stratergy- jee level.What that means is 75% for the topper and down to 60% for the top 10(~jee top 100).
Anyways, Why should we be discussing that?
@arun- What is adiabatic knowledge? i saw the two words together for the first time.
@all bloggers- Why do i see almost a drought of people on the site(maybe down to some table-chair studying)?
And why dont we get posts that can be read and understood within 10 mins(of course i wrote the shortest one)?Ah! I dont remember the last time i read one(maybe the size scares me off)
Lets not put the comment into a similar kind of category.
Enjoy!!

Arun Chaganty said...

@ankit:
Sadly, this isn't going to end until someone posts something new. But I may try this weekend, but I'm supposed to be hitting the books really hard.
By adiabatic knowledge, I mean stuff that you learn so quickly that you don't have time to take stock, and fully grasp the knowledge. Similar to an adiabatic process where the process occurs so fast that it doesn't have enough time for the temperature to come to equillibrium with the surrounding. It was just a random play of words, but looking back adiabatic knowledge sounds rather nice...

Jonh Neo said...

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i found many information here.
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Gr81 said...

@John: Didn't find anything there...
So was this (stupid) advertising?

Arun Chaganty said...

drat! I thought he was talking about this site!