It looks more like "is an element of," as in, P2 is a space (or subspace) of polynomials, and p(t) is an element (vector) thereof. It's a little hard to tell for sure what symbol you're typing, but if I'm interpreting it right, that's what it means.

...!

:O

Oh my...I think my brain just slapped the back of my own skull for daring to open this thread at all...

I can't tell what's worse--that horrifying equation or semck83's probably-smart-and-correct-but-totally-over-my-head-by-like-a-skyscraper's-height answer...

You math folks should lobby for that to count as a second-language credits-wise in high schools and colleges...yikes! xD

@semck Yeah that was it, thanks! I was having trouble finding the symbol on word so I could type it out and that was the closest thing I could find. X_X

@Obiwan Not to worry! With a little sweat and tears (mainly tears), you can figure this stuff out too. =P

@obi, Haha. It does certainly have a lot of its own words. I sometimes stop and reflect how many.

Still, I strongly believe that at least a good deal of math is actually quite accessible if well explained, and I dislike talking over people's heads. To that end, I'd like to try to briefly explain what is going on above. Of course, you didn't ask, and it's hard to get notation down well on a message board, so feel free to ignore. I just don't like people feeling alienated by my subject!

So first, a word on notation in a message board. I use the symbol ^ to mean "to the power of." So for example, 3^2 = 9 ("three squared equals 9"). I use * to mean times.

Now. Three concepts to explain.

First: sets. I'm not going to use this in much detail, so suffice it to say that a set is a bunch of things collected together. For example, I might say the set S is {Putin,Obi,Thucy}. It would then be true that "Obi is an element of S," or Obi Є S. (The symbol Є just means "is in" in the context of sets. That's all it means. So if N is the set of positive whole numbers, then 3 Є N).

Second: vector spaces.

Think of a plane (like, the infinite kind), and mark one point as zero. We can think of all the other points as in relation to this one. For example, if the plane is a (flattened) US and downtown Topeka is the zero, then Chicago would be "570 miles Northeast" (approximately, at some specific angle, not just "Northeast," obviously).

Now, this allows us to do a couple things. First, we can "add" two points on the plane. We can think of each point on the plane as having an arrow going to it from the zero. We call the arrows "vectors," and we tend to get a little fuzzy and not distinguish between a point and its arrow/vector. For example, we might think of Chicago as a point, or as an arrow 570 miles long going northeast from Topeka.

If we have two points A and B, the way we add them is we pick up the arrow that goes to B, and stick its tip at the point of the arrow that goes to A. The "sum" of the arrows is then the new end of B. For example, Denver might be the arrow 530 miles long pointing west (roughly). If A is Chicago and B is Denver, then A + B would be wherever you get by first going to Chicago, and then going 530 miles west. (I don't know where that is. I don't want to know).

OK, so much for adding "vectors" (just points!). We can also multiply them by numbers. This just scales their length by the number, and if the number is negative, it also makes them point in the opposite direction. So for example, if A is Chicago, then -0.5A is the arrow going southwest from Topeka for 285 miles. We can do this with any arrows.

Now, here is the key point, and it's somewhat confusing, but it's hopefully not too bad if you take a moment to take it in. It turns out that these two properties -- having a bunch of things that you can add together, or multiply by a number -- show up again and again in very useful ways in mathematics, and moreover, there are a LOT of useful things (theorems) that you can derive knowing only those abstract facts, and knowing nothing about the things that you're actually adding. So for this reason, mathematicians define an "abstract vector space" (or just "vector space") to be any set S of items such that you can add two of them and get another element of the set, or multiply one of them by a number and get another element of the set. (There are a few requirements -- for example, we require that A + B = B + A, as you'd expect, and so on. Nothing too surprising).

So, a plane is a vector space. It turns out that normal three-dimensional space is too (the arrows just have one more dimension to move in), etc., etc.

Third topic: polynomials. I'll stick to third-degree polynomials just for ease of notation on a message board.

Consider a rule, or function, that takes one number to another in a prescribed way, say by squaring it. You give me a number, and I square it. The common notation for this, as you may know, is f(x) = x^2. I can then plug in specific values -- f(3) = 9, f(4) = 16, etc.

A monomial is a function, like this example, where the rule consists of raising the input number to some power, and then maybe multiplying it by a constant. For example, if f(x) = 3x^4, then that is a monomial. I would have f(1) = 3, f(2) = 48, f(3) = 243, etc.

A polynomial is a function whose rule is just a sum of several monomials. For example, f(x) = x^2 + 2x. x^2 and 2x are each monomials, and f is their sum, so it is a polynomial. I have f(1) = 3 in this case, f(2) = 4, and so on.

A "third-degree polynomial" is a polynomial whose highest power is 3. That means nothing gets raised to a higher degree than 3. So f(x) = -2x^3 + x + 1 is a third-degree, while f(x) = x^4 - 1 is not.

Now, here's the interesting thing. Consider the set of all polynomials third-degree or less. That's going to be things of the form ax^3 + bx^2 + cx + d, where a, b, c, and d are fixed numbers that don't depend on x. (A few examples: x^3 + 55x^2 - 2; x^3 + 1; x^3 + x^2 - x - 1; x^2; and so on). If you add two of these, you get another third-degree (or less) polynomial, and if you multiply any of them by a number, yet again you get a third-degree (or less) polynomial: if f(x) = x^3 + x^2 - 1, then 3f(x) = 3x^3 + 3x^2 - 3.

All that means that S is actually a vector space.

Which brings us to tboin's problem. In the example, P2 is a vector space of polynomials (at least if I'm right!), and p(t) is a polynomial, and the notation

p(t) Є P2

just means that the polynomial p(t) (whatever that may be) is in the space.

For example, consider our space S of third-or-less degree polynomials, and let p(t) = t^3 + t^2 - 3t + 4, and q(t) = t^4 + 1. Then

p(t) Є S

is true, as p is third-degree; but

q(t) Є S

is false, as q is fourth-degree.

Well, I hope this was at all interesting or helpful. Sorry if not. I'm admittedly not the best teacher, and as I say, you didn't ask anyway. I just read (or perhaps misread!) some amount of math intimidation in your remarks, and I try to discourage this, as I say.

Wow... are you majoring in math or a math professor? I bet you could explain this stuff better than my professor. X_X But if you wouldn't mind, could you check if this is right? I believe the question was something like...

If W1 is defined by p(t) is an element of P2 and p(1)=0, then give a basis for W1. I put t^2-1, and I also thought t-1 could be an answer as well?

tboin, I do study math, yes.

I'm not sure what P2 is, which I think would be important to know for answering your question. Is it the space of second-degree polynomials?

Anyway, if that is the question, then you are correct that a basis is {t - 1, t^2 - 1}. Those two are both obviously in the space, and they are linearly independent. Moreover, P2 is only three-dimensional, so if we found one more, that would span all of P2. But not everything in P2 is in W1. So therefore, there can't be anymore.

Yep that makes sense. Now I know what the heck I'm doing lol. Thanks a lot! I should go to the forums more often...

+1 to semck on that excellent post.

I hope obi takes the time to read it; I sometimes feels he prides himself in being ignorant of math.

@semck83:

LOL...yes, mathematics and I are not on good terms...

But you did a good job explaining all of that--I at least followed to the point with Denver and chicago and the vector points...after that, less so, but getting that far and understanding was an accomplishment...

I agree with tboin4, you seem talented making this stuff intelligible, you should be a professor! :)

(As for me, though, I just want to take my last, LAST EVER MATH CLASS--Stats, Math 115!--and be done with mathematics forever...I'm not very good with them, and just don't like them...I like words and texts because THERE I can manipulate them around and interpret different things my way to prove my point, and so long as I can find or make the material to back up my claims in a Lit. or Poly Sci class, they can stick...

But no amount of Shakespearean referencing or existential reasoning or New Critical or Deconstructionist reading can change mathematical facts...I guess that's the beauty of them, they're set facts and certainties, but for me it always just translates to my needing to know exactly what to do and recall each step, or I get it wrong...I know abgemacht's said before there are multiple ways to solve the same math problem, and there probably is...I'm just hardly able to remember one, let alone many different theorems and formulas and just choose whichever seems convenient for the problem...

And then in Lit. or Poly Sci., concepts and answers, again, can be manipulated and the best reasoned argument wins...but no amount of argument changes the single-answer system of math I learned...no matter how well you can craft an argument, 2+2=4...

Only O'Brien torturing Winston Smith on end for days or weeks in Room 101 can make 2+2=5.

And Math 101 was already torture enough!

But I can at least see the whole sets-thing making sense...I sort of try and do the same thing for Lit. classes, I suppose...you can group Dickens, George Eliot, and the Bronte Sisters as all being Victorian Novelists, and as they abide by the same tropes and are reacting to a similar environment, you can expect similar things in their works, so if you're studying one and haven't read them, New Criticism would teach that you should be able to connect them to a similar author stylistically or periodically, so if you haven't read Charlotte Bronte but have read her sisters Emily or Anne, or Eliot or Dickens, you have a period connection to interpret in her works, and if you've read Virginia Woolf and Jane Austen, you can interpret on a "Feminism-Set" basis--albeit with the stipulation Woolf called for androgynous writing and called Bronte out for being too "melodramatic" with her brand...but whatever, I've rambled long enough, and it's still not changing the fact that 2+2=4, is it?) ;)

@abgemacht:

I read the post...

And I don't pride myself on being poor with math...

But I feel that if I'm going to be arrogant enough to insist that I am somewhat skilled in the Literature/Philosophy/Poly Sci field--I'll actually qualify for my AA in English as of next week's finals being completed barring anything drastic--I should fully own up to my shortcomings.

So, in order for me to fairly say "I'm intellectually-skilled to an extent with Literature, and I'm going to act like it," I think it's fair to also point out "But I'm not claiming that I Am The Very Model Of A Modern Intellectual...when it comes to mathematics, I'm a relative n00b, and you all can pwn me!"

;)

@obi

"still not changing the fact that 2+2=4"

And you can debate Shakespeare all you want, but it doesn't change the fact that we spell "the" t-h-e. It's really the same thing.

There are plenty of areas of math where the answers aren't written in stone, such as Numerical Analysis, where sometimes one way is better at some things and worse at others. It's up to you and the situation to decide what's best. It terms of the freedom you have, it's much closer to Shakespeare than you may suspect. Just like in math, there are boundaries in literature, too. I'd probably be hard pressed to convince you that Macbeth was actually a criticism of the slaughter of the Aztecs, no? : )

"And I don't pride myself on being poor with math..."

I just like giving you a hard time.

I'll freely admit that I find Shakespeare rather tedious; feel free to harass me all you like : )

LOL...

Interestingly enough, according to some literary theories, you CAN read-in extra-textual meanings based on something that would generally be considered apocryphal IF you can make that reinterpretation mesh with the unity of the text that you either presuppose or create yourself...

It'd be essentially saying "Macbeth" is a text, and that text "exists" both as its own entity and connected with other texts, so while you could say Shakespeare never could have written "Macbeth" as an intended criticism of slaughtering indigenous people, if you wanted to take an--albeit extrreme--New Criticical or New Historicist approach to its extreme, you COULD potentially argue for the text supporting the upholding of the righteousness of a monarchal position, and that the destruction of that position only leads to chaos and death, and, so, in the same way Macbeth taking the throne may be viewed as a negative invasion and usurpation to an extent, stealing the rightful land of the Aztecs and taking their seat of power away from them may be seen "reflected" in a text like "Macbeth"...even though this is entirely apocryphal from a historical point of view (and would actually work a LOT better with "Julius Caesar," since that particular play WAS written in part for a political message, ie, "Don't rebel against the Queen...look at what happened in Rome when Caesar was treacherously murdered..."

So yeah.

You CAN pull of really insane things like Macbeth-with-Aztecs in literary theory...

But that's why I love it--it sounds insane, but since everything's connected to something else, if you know how to pull the strings and know enough authors, theories, and people and political movements, or psychology or philosophy or astronomy or whatever else you can read in, the chances are increased you'll be able to manipulate meaning...

And THAT makes reading and writing = CREATING the text anew in a way...

There's probably a way to do that with Math, too--

But I can just remember authors and movements a lot easier than theorems and formulas...so I'm a poor math-manipulator, I need to use my words, words, words!

(And LOL--for some reason this has been the semester for everyone, friends included, to tell me why they all seem to find Shakes tedious or overrated or one-dimensional...I never heard too much about him one way or the other, and suddenly, this semester, EVERYONE seems to suddenly hate him!) XD

And that's why I can't stand literature. It just seems like you can make anything up you want and you're good to go.

eh, Shakespeare. I just feel that people told the same stories better before and after him. Also, reading plays suck. I do enjoy watching Shakespeare being performed.

Well, you can't just make up ANYTHING...you can't say Hamlet lives happily ever after, or that Winston Smith succeeds in beating the Party in "1984"...

And it should bear mentioning that what you can and can't do with a text depends greatly on what you subscribe to...

My friend and I get in a lot of debates--he's firmly of the belief that if you have something to say you should be concise about it if you can, and that texts can be completely understood by themselves...whereas I agree with those who say that texts are all interconnected, and so the more connections you bring in, the greater the picture you'll get, much like shedding more light on a darkened spot from several different flashlights to create one really, truly illuminated spot.

And again, even if you want to go all T.S. Eliot--like I try to--and be as Inter-textual as possible, you have to respect the unity of the text...a lot of it is based on the idea that certain words/characters/texts can code for different meanings, so you would have to find a "synonymous reinterpretation," otherwise you reach an impasse because you've selected antonyms or non-combinant literary ideas or characters or texts and the unity is wrecked...sort of the same way, I guess, that putting in one of semck83's Four-power numbers wouldn't work if the set requires on Three-power numbers:

You can argue for Hamlet = Eugene Onegin, or Hesitance, or Existentialism...

But if you stick an interpretation in that doesn't fit with those common concepts, ie, Hamlet = Communist Hero, you've effectively confirmed you ARE talking out of your ass...about the most you could do there would be to argue that since the King is corrupt and Hamlet wants him dead, it's a case of the Hegemony being opposed--but as Hamlet's also of the upper class and WOULD, in fact, have been King had not Claudius stolen his throne, AND the fact that Hamlet debates killing Claudius for just about every reason BUT "The working people are being oppressed, I must liberate them!" you're not getting far with this...Communist Hero =/= the rest of that "Hamlet Set" of "Hesitance, Existentialism, Eugene Onegin, etc.," so even New Historicism would frown on that sort of stretching, generally.

There's a limit to how much you can bend, just like at some point you run out of canvas-space for your painting.

(And yeah, Shakespeare, as with most playwrights, needs to be performed...about the only exceptions to that rule I can think of are "The Glass Menagerie" by Tennessee Williams--he gives so much set description and stage direction it could very well have been a novel if he'd wanted to tweak it just a bit, so you can read it that way--and "Man and Superman" by George Bernard Shaw, as that's basically a Socratic dialogue with the play as an excuse for it to happen.

+1 semck83

MS Word actually features huge amount of symbols in regard to math, including the one we looked for - ∈ - and it's set cousin ⊊, assuming P2 is not a poor constrained fella containing only p(t). One will still probably use specialized software like MathType for larger works but its equation editor is a decent tool nevertheless.

Of course this forum doesn't allow inserting stuff that would really give obi a headache ;)

+1 smeck83

Great explanation, in all my undergrad and gradutate electrical engineering and math courses, I never had a good explanation of set theory, so I work out that understanding on my own.

I couldn't have explained it that well, and I am using set theory and bases daily in my thesis work.

Great thread. On the subject of math word processing, do any of you guys use LaTeX?

I do.

Yes, I use latex. Before that, I used OpenOffice for all my equation writing.

Shakespeare needs to be performed?! How do you perfomr a sonnet? hehehe

I love the whole lit/language versus math/science argument. Especially when yu consider that language is required for the understanding of mathematics at the conceptual level. Of course, I love both (philosophy and interpretting history beyond the facts are my weaknesses).

Oh, and Obi, the many ways to solve a problem dilema... I can solve 19*19 two different ways. One is tradiitonal long multiplacation where you get 81+90+90+100 (9*9 + 10*9 + 9*10 + 10*10) andthe other is quick reduction where you get 400-20-19.(20*20 less 20 becomes 20*19 less 19 is 19*19).

It's a simpistic but very demonstrative example.

@Draug/Obi

I was going to ask Obi this, but it fits so well into what Draug just said.

I think a lot of the humanities are very important. For instance, I have a great respect for History. I also think English is important. English is a skill-set that needs to be learned and practiced. English is a tool used for communication and is very important to understand, if not master.

However, I cannot justify the study of literature. I think literature itself (plays, books, poems) are great and should be read by everyone. But the study of people's work? What does it accomplish? If I need a B.S. in Lit to understand Catcher in the Rye, then didn't Salinger fail as an author? I just don't see the point.

For instance, I had to take a lit course in college. We spent a week doing "close readings", where you take a passage out of context from the rest of the work and the rest of the world. Just study the words. How stupid is that? What is the point of studying something out of context? In the end, what was supposed to be 50 minutes of close reading turned into 50 minutes of me demanding the professor justify the practice of close reading. The end result was the same: we accomplished nothing.

Thoughts? : )

@abgemacht:

Alright, I'll take a crack AT IT...when I get back from the doctors. TBC! ;)

(Because I've actually thought about that and DO have a reason I think you might buy...)

I would be very interested in hearing your response. I've asked this to a number of people and have never gotten a good answer.

Looking forward to reading from you.

I'm looking forward to this reply because I've also never understood the point of Literature classes. My AP Literature class in high school consisted of my playing copious amounts of euchre and passing the test at the end of the year with ease. Much the same with my required college literature classes, though the professors tend to get more annoyed with euchre playing now.

@abge or pepijn: do you guys know of any good free resources with which to learn LaTeX?

@Frank: Try http://www.latex-project.org/guides/ ... It's not how I learnt the ins and outs, that was the LaTeX Companion, though that's not free as far as I know, but maybe it's a start.