Questions that don't deserve their own thread

> capture light from 360°
is it going to be floating in the air ?

>>7568695
I was thinking a slender pillar, but now that I think of it, floating isn't entirely unreasonable.

Is it math or maths?

>>7568732
"math" for North America, otherwise it's "maths".

Why aren't there less even numbers than integers?

>>7568741
I heard there was in fact less even numbers than integers.

>>7568742
Can you explain why?

>>7568746
I think it was in a lecture transcript, but I forget the logic.

>>7568741
In any arbitrary bit fixed interval there are far fewer even numbers than integers, but if you take the entire real line, then you can match each even number to an integer and never run out of either.
1-2
2-4
3-6
4-8
Etc.
Infinit sets do weird things, and this is one of them.

>>7568692
An anon last thread replied to this >>7565879 but I wasn't too sure, so repost...

Is there any good way to derive (A U B)=B U (A ∩ B')?

Clearly, I can see the truth of it (use the distributive laws), and Venn diagrams point to it quite clearly, but I'm looking for a more traditional derivation (I'm tempted to say algebraic, but I'm not too sure if this qualifies as algebra).

How do I do math proofs properly? I started spivak hoping to learn to learn/approach "pure math" rather than "cookbook math" but when i check an answer to a problem set it's like his proof arbitrarily combines/divides steps.

When i learned to do proofs in symbolic logic we'd number the lines and list the operation or rule applied, one per line. So:
1.p v q
2. ~p
3. q 1,2, DS

Wtf am i missing with math?

>>7568692

>>>7568044

Anyone use a science dictionary plugin for spellcheck?

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>>7568692
Can someone tell how to simplify this integrand?

>>7568980
Sorry. I'm tired, this is easy as fuck. Just separate the shit into two integrals and then do substitution.

My professor told me that my proof is wrong but did not tell me why, here it is:

Theory: If A1 and A2 are two sets of real numbers that Sam likes, then Sam also likes A1 or A2.

Proof: Since Sam likes A1 we know that for all x in A1, "a" is less than or equal to x.(*)

Since Sam likes A2 we know that for all x in A2, "a" is less than or equal to x.(**)

I need to prove that for all x in A1 or A2, "a" is less than or equal to x.

Let x be in A1 or A2. Either x is in A1 which in this case "a" is less than or equal to x by (*).

Or x is in A2 so "a" is less than or equal to x by (**). In both situations "a" is less than x so I am done.

How is this wrong?

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On the off chance there are any NLP people here, what are some simple projects to help me learn and start actually doing NLP?Thanks.

>>7569046
Literally what the fuck? Are you mentally retarded? This is like an incoherent mix of middle-school grade logic questions and Dr. Seuss' Cat in the Hat

>>7569057
>not realizing that its a discrete math question
>too new

>>7569046
Ok, I'll try to help Anon.

First of all, you forgot the "Assume" portion. You need to declare that you're assuming the hypothesis is true to begin with. Let me see if I can figure the rest out.

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>>7569046
I'll try to help, right after I'm done with this question.

>>7569064
>>7569067
Thanks anons, I just started learning proofs so this is all new to me.

>>7568980
\int{\frac{x+3}{x^2+4}}dx = \frac{1}{2} \int{\frac{2x}{x^2+4}}dx + \frac{3}{2}\int\frac{d(x/2)}{(x/2)^2+1} = \frac{1}{2} ln\Big(x^2+4\Big) + \frac{3}{2}arctan\Big(\frac{x}{2}\Big) + C

>>7569067
\int{\frac{x+5}{x+6}}dx = \int{\frac{(x+6) - 1}{x+6}}dx = \int{\Big(1 - \frac{1}{x+6}}\Big)dx = \int{1}dx - \int{\frac{dx}{x+6}} = x - ln(x+6) + C

Dude these are considered very easy integrals. If you ask for too much help with these, instead of trying to figure them out on your own, you're not going to learn how to do them yourself. Then you will fail the exams. So no more help unless you're truly stumped.

>>7569091

\int{\frac{x+3}{x^2+4}}dx = \frac{1}{2} \int{\frac{2x}{x^2+4}}dx + \frac{3}{2}\int\frac{d(x/2)}{(x/2)^ 2+1} = \frac{1}{2} ln\Big(x^2+4\Big) + \frac{3}{2}arctan\Big(\frac{x}{2}\Big) + C

fucking latex

>>7569091
>>7569096

fuuuuuuuu

\int{\frac{x+3}{x^2+4}}dx = \frac{1}{2} \int{\frac{2x}{x^2+4}}dx + \frac{3}{2}\int\frac{d(x/2)}{(x/2)^ 2+1} = \frac{1}{2} ln\Big(x^2+4\Big) + \frac{3}{2}arctan\Big(\frac{x}{2} ) + C

last try

Honestly, whats the best way to learn math. For me its always been a memory thing. Remembering formulas and such. I really want to master this shit already. College intermediate alg here.

How would I "formally" find the inverse of a function? I know the middle school procedure of swapping y with x and then solving it, but I don't believe it's enough.

>>7568963
Use LaTeX instead m8.

>>7568796
Start from the right

B\cup (A\cap B')

By the distributive property

=(B\cup A)\cap (B\cup B')

The union of a set and its complement it's the Universal Set, and the intersection of any set with the Universal Set is the same set, so

=B\cup A

>>7569105
I mean that's what you'd like to do. The formalities come in when you'd like to discuss if the function is appropriate for an inverse--i.e., the trig functions must be restricted to a smaller domain to consider their inverses. This is what trips people up in intro physics courses, you can only get -pi/2 to +pi/2 if you use inverse tangent, you need to see if that's actually your value or if you need to shift your domain by pi. Similarly, if your function is only one-to-one in a specific region, you need to restrict yourself to that region to form an inverse function with the set of its image.

>>7568692
I'm not actually sure that's geometrically possible with optics alone.

>>7565253 old thread
>>7566978 old thread
>>7569046

>My professor told me that my proof is wrong but did not tell me why, here it is:
>...

Your homework is not due until October 5-6! You are currently taking Math 137Y at the University of Toronto with Alfonso Gracia-Saz, and you were asked to sign this cover page

http://www.math.toronto.edu/~alfonso/137/137_1516_HW1_CoverPage.pdf
http://www.math.toronto.edu/~alfonso/137/137_1516_HW1.pdf

Nevertheless, here you are on /sci/ soliciting for help from non-students under false pretenses.

Now, I could report you to the OSAI to have them investigate the campus network logs and get you expelled from U of T. But I'm not going to do that, anon. However, you should realize that you're playing with fire here regarding your future and career.

I suggest you carefully review the legal document you executed and the University Code of Behaviour on Academic Matters. Then seriously consider your actions and behavior.

>The University of Toronto takes academic integrity very seriously.
>We are obligated to report all suspected instances of misconduct to OSAI.
>Please do not force us to do so. Every year, multiple students in this
>course are disciplined. For more information, see
>http://www.artsci.utoronto.ca/newstudents/transition/academic/plagiarism.

>http://www.governingcouncil.utoronto.ca/policies/behaveac.htm

how 2 use latex in /sci/?

>>7569198

[*math]\sum_{\int{dx}}^{\frac{A\cup B}{\inf}}cos(\theta)[/*math]

remove the two * from above to get this

\sum_{\int{dx}}^{\frac{A\cup B}{\inf}}cos(\theta)

>>7569180
>>7569046
>my proof
>plagiarism

shouldnt claim its YOUR proof when its your profs, even if its incorrect.

>>7568948
>arbitrarily combines/divides steps
This is very common. Just as steps involving nothing more than associativity of real numbers under addition will not be explicitly announced in the meat of the book.

Dividing steps can help as a pedagogical tool if a certain leap might prove too confusing all at once (check out baby rudin for an author who refuses to do this, and tends to confuse the novice as a result.)

How can I prove that \frac{a}{x}-\frac{b}{x^2} is bijective? The constants are throwing me off a bit.

>>7569248
Ugh, that looks horrible. Let's try again

\frac{a}{x}-\frac{b}{x^2}

>>7569105

That's fine for deriving a candidate g for the inverse function. To prove it's truly an inverse ( g=f^{-1} ) you have to show that g \circ f = id_D is the identity map on the entire domain D of f. It unambiguously brings you back to the original point for every point in the domain.

http://en.wikipedia.org/wiki/Inverse_function

>>7569278
Ah, I understand now. So I can just propose a function g, and then if I prove that g \circ f = id_D, then I'd have proven g is the inverse of f?

>>7569248
You can't because it's not.

>>7569293
Alright, how can I disprove it?

>>7569248
>>7569250
bijective between what? It's certainly not the real numbers, since it doesn't map 0 to anything. From (0, inf) to itself? Either way, the method is the same. Show that it's one-to-one and onto; i.e. show that every value it maps is mapped uniquely and that it maps to all of the points in the set.

>>7569281
yeah that's what i said. you can find a function any way you want, but you have to prove it has the correct properties to be an inverse according to the definition.

>>7568948
A good book is Velleman's 'How to Prove It.' If you're familiar with intro. logic, the first two or so chapters should be review, but DO MOST OF THE PROBLEMS/EXERCISES. Follow the examples, apply those methods to the problems. If I recall correctly, up to chapter 5 or 6 is essential--any and ALL rigorous mathematics texts (i.e. those that would have upper-level undergraduate or graduate material in them) will assume this much. After that I think it goes into a set-theoretic construction of the real numbers via Dedekind cuts and an exposition of useful theorems like Bolzano-Weierstrauss. The book makes a really good intro/refresher of the stuff needed for a first class/text in analysis.
You should pick up the kinds of tricks/tools needed for proofs after analysis and I'd recommend going on to abstract algebra to expand upon these methods and how you conceptualize mathematics.. After that you can really do anything you want. The standard track is multivariable analysis/abstract linear algebra into complex analysis into manifold theory/differential geometry and beyond.

>>7569295

not the other anon, but just evaluate your expression at x = \frac{2b}{a \pm 2 b c} where c \ne \mp \frac{a}{2 b} is any other real number that doesn't kill the denominator.

You'll see that your expression is two-to-one.

>>7569198
[Catalog]
>> /sci/guide
>> https://sites.google.com/site/scienceandmathguide/
>> /sci/ LaTeX Tutorial

I'm shit at maths, really really bad.
how do I stop being shit?

>>7569308
neat, i'll check that out, thanks.

>>7569180
You're right anon. I"ll never do it again., I guess I should have more faith in myself.

>>7569104
identify a topic that interests you, find some ``survey articles" on that topic, determine the ``easiest" theorem in the survey, download the paper that proves the ``easiest" theorem, spend the rest of your life trying to understand this theorem

If the universe does end, would it eventually go through another Big Bang? Or what would make it impossible for it to ever happen again?

>>7569421
What have you tried?

>>7569679
I'm fairly certain that with our current understanding of the Universe no one can give you answer on that. In fact I think it might be centuries early before anyone can.

Also since time itself is a concept valid only on inside the Universe - next big bang happening could pretty much be the same one all over again.

If you need the answer now, I suggest you pick some dogmatic religion. They will give you one.

>>7569119
>By the distributive property
...I really need to learn that shit, like how do I 'expand' set theory? I understand simple cases, but what's the general method?

>>7570169

https://en.wikipedia.org/wiki/Algebra_of_sets

https://www.youtube.com/watch?v=1TKSfAkWWN0

In this video I get confused when he's talking about length contraction.
According to him the length contraction occurs from the cat's frame of reference as the protons are seemingly moving.
But why does it not occur for his frame of reference? There should be a field there because the electrons are moving which causes the change in charge density.

How do i write that if B then A but if A then B or not B
Without a contradiction
In orher words B needs A but A doesn't need B

(A \rightarrow B) \and B
That you mean?

>>7570434
I meant
(A\textrightarrowB)\wedgeB

>>7570443
I'm just starting with these codes
(A \rightarrow B) \wedge B

>>7570220
That doesn't answer my question, it only gives results - not methods.

For example, the page gives A U (B ∩ C) = (A U B) ∩ (A U C)

But what is the method for expanding more complicated cases?

For example, (A U B) ∩ C = (A ∩ C) U (B ∩ C)
Now, I fully admit that I can see that this is very much like A ∩ (B U C) = (A ∩ B) U (A ∩ C). Indeed, it's just the same fucking thing with the symbols moved around a little, so it that all I'm meant to do? Just recognise the similarities to the rule and substitute as appropriate?

If so, what if I try A ∩ (B U C U D)?
Is it just (A ∩ B) U (A ∩ C) U (A ∩ D)?

>>7569062
you're retarded moron
without the conquest he provided here >>7566978 and in the language here >>7565253 it's just plain nonsense

the answer is obvious (a in A1 and A2 isn't necessarily the same number) but based only on >>7569046 this is just retarded

I have a graduate school application with a section that says

work experience (Include teaching and research experience at the undergraduate or the graduate level)

I'm wondering if they mean all work experience like working as a delivery driver and irrelevant stuff like that or ONLY to include teaching/research experience (which I have none of)

any ideas

>>7569360
Thanks m8, it's just what I needed. However, how did you manage to obtain that expression?

Could someone explain the induction proof of the binomial theorem? I'm using this page https://proofwiki.org/wiki/Binomial_Theorem/Integral_Index and all is fine and dandy until the

\displaystyle {n+1 \choose 0} x^{n+1} + {n+1 \choose n+1} y^{n+1} + \sum_{k \mathop = 1}^n {n \choose k} x^{n+1-k} y^k + \sum_{k \mathop = 1}^n {n \choose k-1} x^{n + 1 - k} y^k

part. I'm having trouble understanding why they changed the binomial coefficient and the exponents like that after the second sigma.

>>7570524
stop thinking about "methods" and "rules" - the only rules are the definitions of the operations, anything else is constructed by applying them logically.

if you can prove it, you can use it.

>>7570782

y = \frac{a}{x}-\frac{b}{x^2} = -b \left(\frac{1}{x} - \frac{a}{2b}\right)^2 + \frac{a^2}{4 b}

If the goal is to find two different x-values in the domain that are mapped over to the same y-value in the range, you can guarantee this by taking

\frac{1}{x} - \frac{a}{2b} = \pm c.

for the quantity squared. Now solve this for x.

>>7570920
Much appreciated anon. One more question: how did you come up with this equality? \frac{a}{x}-\frac{b}{x^2} = -b \left(\frac{1}{x} - \frac{a}{2b}\right)^2 + \frac{a^2}{4 b}.

Last ice age, NA was pretty much covered in ice from a certain point up. The ice retreated/melted whatever as it heated up, leaving lakes behind. How did fish get in all those lakes that were landlocked by miles and miles?

Checkmate someone.

>>7568692
>be OP
>come back to thread
>fairly simple question still not answered
Either there's an SEP field over that question, or I'm wrong about it being simple.

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>>7571144
>SEP field

>>7570838

Those fiendish bastards silently changed variables on you, anon, but then they underhandedly kept using the same dummy index, k.

Their wicked aim is to combine the two binomial n-sums that result from distribution over (x+y) \cdot \sum ... into a single, monstrous binomial (n+1)-sum. The bulk of their heinous "proof" is all about making that despicable unification happen; but it CAN'T happen, unless they FORCE all the sums into the same debased scheme. This is at the rotten core of all their strategems.

You can plainly see how they conspire to do it, too. They won't tell this to the uninitiated, but they secretly define a veiled variable j=k+1. Now everybody thinks that k runs from 0,...,n-1. But this j, in all its vanity and supposed superiority, tries to do one better: it runs from 1,...,n.

All their diabolical incantations still call it k, but that's just to camouflage their true vile intentions from the rest of us. I tell you, anon... if you peer deeply into their cryptic arcanum... it's secretly j!

In their raving delusions they believe we can all be plied and manipulated, just like their "dummy variables". But now I will EXPOSE their black arts! And their FOLLY will be plain for ALL to SEE!! WITNESS AND BEHOLD THaT Their MACHINATIONS HAveN'T ChAnGED ANyTHING!!!!!!!!!

\sum_{k \mathop = 0}^{n-1} {n \choose k} x^{n - k} y^{k+1} = \sum_{j \mathop = 1}^n {n \choose j-1} x^{n + 1 - j} y^j

>>7571166
Goddammit anon, I can't focus on the proof because my sides hurt.

>>7570969

Transportation by natives/stocked by DNR

how do you guys approach problems where you have to construct an algorithm to do something and you're given a running time you have to get it under? i'm interested in hearing different thought processes

>>7571206
I like to think of how I can most efficiently sacrifice accuracy for speed, then try and figure out how to bring up the accuracy again.

Okay, so my university uses Stewart for its Calc I, II, and III sequence.

As you would expect, all of this stuff is extremely boring and rote and doesn't test for any depth at all. All that is asked for is the ability to bang out the standard old techniques and strategies.

After making the mistake of getting into Calc II classes because I didn't know I already knew the material, I'm very, very reluctant to spend a whole semester on Calc III.

What's the fastest and most efficient way to bang out Calc III so I can try pass a proficiency exam for it? I need to find some sets of practice problems somewhere. Paul's works for an outline but I don't think they have enough problems.

Has anyone else been in this situation? What did you guys do?

>>7571213
oh, that's interesting. what do you mean by sacrificing accuracy? I imagine that the logic is either wrong or right, not a spectrum of truth.

with typing code I can see that, by ignoring edge cases perhaps. but would you use the same technique when writing proofs?

>>7571206
>>7571213
>>7571217
he probably means using probabilistic solutions. a solution which is only right 99.9% of the time can be much, much faster than one that is always right

Can someone give me an example of a function where x is a set of all rationals but not all reals? Does such a thing exist?

>>7571206
think about the bounds of the problem, think what that means for you (you need O(n^2)? can you do something by going through all pairs of numbers for instance?)

think on a brute force algorithm and how to improve it (you can think of an O(n^3) algorithm which can be improved to O(n^2 *lg(n)) with clever data structures), try small cases, bruteforce small inputs to see patterns,

consider the structure of the problem, are you calculating something over and over again? can you build the solution from smaller problems? (dynamic programming)

think about classical algorithms and problem reductions: can you reduce the problem to something you've seen before? does a part of the problem look oddly familiar? would it be easier if it had another form, and can you transform it into that form with a clever algorithm?

think about what's stopping your solution from working: can you isolate a more specific problem and solve your problem by solving it instead?

etc

>>7571226
>>7571226
you have a fundamental misunderstanding of functions. a function is just a domain plus a correspondence rule. so for example x^2 in Q is a function. every function from R to R can be restricted to a function from Q to R, and any function from R to Q to one from Q to Q

>>7571232
fantastic, thank you anon!

>>7571226
does your question ask you to give a rational function where x isn't the set of all reals? if not, ignore me

>>7571238
if you're interested in the discipline of solving tasks using classical algorithms in limited time, look up competitive programming

http://www.codeforces.com
http://uhunt.felix-halim.net/

>>7571238
>>7571235

Like for -x^2 + 4x - 5 < 0, it is true for all real numbers. Is there any sort of function where any rational number you plug in would be true, but an irrational wouldn't, without explicitly stating on the side "Let x be any rational number"

Any book recommendations for the finite element method? I'm doing a self study project required for my degree and I have no prior knowledge of the method.

So when particles appear and don't appear at the same time, isn't that basically magic?

>>7571244
I find it really fun, but it's not the area I'm most passionate about. I'll check those out though. Thanks!

>>7571251
this is more interesting. any such equation would be equivalent to "x is in Q", or "x is the quotient of two integers". anything you build will just be a complicated way of saying that. there's no simple way and certainly none using elementary functions

>>7570965
That just comes from a very important and ancient technique, which is now called "completing the square" and is usually considered trivial/elementary. But it took humanity over 2 millennia to fully develop it: the Babylonian mathematicians, Greeks (Euclid, see Elements Book II, Prop 4-6), Indians (Brahmagupta), Persians (al-Khwarizmi, Omar Khayyám), and Europeans all contributed. Today you can google it. Well worth knowing "in your bones" so to speak.

why do my feels approach infinity as my number of gfs approachs zero over time

Why is the sound of rain relaxing?

Hard mode: none of this shit
https://m.reddit.com/r/NoStupidQuestions/comments/25a0wb/why_is_rain_so_relaxingenjoyable_to_listen_to/

>>7571339
That's mean that feels = c + k/gf

Is there a good tutorial on active analog filter design somewhere? I know the basics of making a simple lowpass / high pass etc but I'm having trouble combining several filters into a 3 band EQ thing. I have a passive EQ that I could use but I'd like the ability to independently boost the respective frequency bands.

Where can I find a collection of non-SMBC or xkcd funny maths related pictures? I've been sending a friend of mine a few of them for his birthday every year for the past few years and I'm starting to run out of good stuff, the usual /sci/ humour thread is often just shitty reposts.

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Saw this on another board but couldn't figure it out.

Wolfram Alpha gives the exact solution but how to arrive at the solution?

>>7571639
By integrating.

>>7571639
Substitute u = sqrt(x)

\int_0^\infty \frac{\sqrt{x}}{1+x^2} dx = 2 \int_0^\infty \frac{u^2}{1+u^4} du = \int_{-\infty}^\infty \frac{u^2}{1+u^4} du

Use (complex) partial fraction decomposition, view the integrals as curve integrals around the upper half-plane and apply the residue theorem.

>>7571666
Thanks. This is way more complicated than I expected.

>>7571671
It isn't that complicated.

1 + u^4 = (u - \alpha_1)(u - \alpha_2)(u - \alpha_3)(u - \alpha_4)
with
\alpha_1 = \frac{1}{\sqrt{2}}(1+i)
\alpha_2 = \frac{1}{\sqrt{2}}(-1+i)
\alpha_3 = \frac{1}{\sqrt{2}}(-1-i)
\alpha_4 = \frac{1}{\sqrt{2}}(-1-i)
so

\int_{-\infty}^\infty \frac{u^2}{1+u^4} du = 2 \pi i \left( \frac{\alpha_1^2}{(\alpha_1 - \alpha_2)(\alpha_1 - \alpha_3)(\alpha_1 - \alpha_4)} + \frac{\alpha_2^2}{(\alpha_2 - \alpha_1)(\alpha_2 - \alpha_3)(\alpha_2 - \alpha_4)} \right)

Just plug the definitions of the \alpha_i in and multiply it out.

With a maths degree, what job would you go for?

I'm hoping to get into research, but finance or engineering seem more likely, and for reasons I can't quite put my finger on, engineering looks shit (well, thinking about it, why would anyone employ a maths grad for engineering when an appropriate engineering exists?).

What is entropy?
I'm too illiterate to understand the google definition

does black body radiation start from the lowest frequencies? like we all know anything warm but not hot enough to glow visibly is still throwing off a few infra-red photons, but am I a source of radio waves as well as IR? Is static on a TV or radio the disorganised radio waves coming from the sun and everything else bouncing around?

>>7571639
>>7571666
>>7571671
>>7571702

sometimes quick n dirty is best way

\int_0^\infty \frac{\sqrt{x}}{1+x^2} dx = \int_0^\infty \bigg( \frac{(\frac{i}{2\sqrt{x}})}{1 + ix} + \frac{(\frac{-i}{2\sqrt{x}})}{1 - ix} \bigg) dx

= \int_0^\infty \sqrt{i} \cdot \frac{(\frac{\sqrt{i}}{2\sqrt{x}})}{1 + (\sqrt{ix})^2} dx + \mathrm{c.c.}

= \sqrt{i} \cdot \tan^{-1}(\sqrt{ix}) \vert_0^\infty + \mathrm{c.c.}

(Choose either branch of \sqrt{...}, tan() is even, but conjugate has to be consistent.)

= 2 \cdot \mathrm{Re} \Big( \frac{1+i}{\sqrt{2}} \cdot \frac{\pi}{2} \Big)

= \frac{\pi}{\sqrt{2}}


>>7572530

Yes you radiate across the whole spectrum, anon, but...

For hf << kT the intensity behavior is only quadratic instead of cubic.
For hf >> kT there is an exponential supression.

so most of the radiation output really is near the peak given by Wien's displacement law.


>>7572517

OK lets cut through all the bullshit, anon.

Let's say you have a closed physical system. Could be anything. You know some things about it. Maybe it has a total amount of energy in it or a known number of particles. Maybe it is confined to some known volume. Maybe it has a mean pressure that you measured, or a mean magnetic field coming off of it. Whatever; it's different for every system. Lets call all this gross "external" knowledge you have the "macrostate" of the system.

>>7572517
>>7573319

Now, what don't you know? Presumably the actual constituents of the system are in some particular detailed arrangement at some time. This could be the positions and momenta of all the atoms in the system, the composition of each molecule, the spin state of each nuclei. Whatever; it's different for every system. Let's call all this detailed "internal" knowledge you don't have the "microstate" of the system.

You can think of the entropy of the system like this. Given your knowledge of the macrostate, the entropy is the number of bits (ones or zeros) of additional information you would need to be able to completely specify the exact microstate of the system, times a stupid constant (k_B \cdot ln(2)). So the entropy is just a measure of the information lacking about the COMPLETE state of the system.

Can anyone hint me on how to prove that for all x in an ordered field F\{0}, x E (1, inf) is equivalent to 1/x E (0, 1)?

>>7574133
question is not clear but say x is in F\{0}.

Suppose x \in (1,inf). Then x>1>0. If 1/x <= 0, then mult by x>0 would imply 1 <= 0 which is false. So 1/x>0 as well. Then since x>1 we also have 1 > 1/x. Therefore 1/x \in (0,1).

Next suppose 1/x \in (0,1). Then 1/x > 0. If x <= 0, mult by 1/x > 0 would imply 1 <= 0 which is false. So x>0 as well. Then 1/x < 1 implies 1 < x. So x \in (1,inf).

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As the picture says, how the fuck do I prove that without just drawing a venn diagram of it and going "Yeah it's there, I can see it"

>>7575539
Just prove that you can rewrite your left hand side argument into the right hand side argument, there. Proved.

>>7568692
do preferential trade agreements help or hinder multilateral tariff reductions?

>>7575539
This is one of the fundamental laws of set algebra called the Distributive Law. See https://en.wikipedia.org/wiki/Algebra_of_sets

If you are uncomfortable using it in homework, because you haven't covered that material in class yet, then the general method for proving any two sets C and D are equal is this:

First prove that C ⊆ D (every element of C is also in D). Next prove D ⊆ C (every element in D is also in C). Then you can conclude C = D because the two sets contain exactly the same members.

To do (A∩B)∪R ⊆ (A∪R)∩(B∪R) in excruciating detail:

Suppose x∈(A∩B)∪R is an arbitrary element.
Then x∈(A∩B) or x∈R or possibly both.
If x∉R, then x∈(A∩B); so x∈A and x∈B; so x∈(A∪R) and x∈(B∪R). Therefore x∈(A∪R)∩(B∪R).
On the other hand, if x∈R, then both x∈(A∪R) and x∈(B∪R) are true. So x∈(A∪R)∩(B∪R) as well.
In either case we have x∈(A∪R)∩(B∪R).
Since x was arbitrary, we may conclude that every element of (A∩B)∪R is also in (A∪R)∩(B∪R).
Therefore (A∩B)∪R ⊆ (A∪R)∩(B∪R).

I will leave the other direction as an exercise for the reader.

>>7575539
Its true via the distributive property. You could go through the trouble of a direct proof by cases, but fuck man that's harder.

You should read all the identities/theorems in your book, it should be in there.

>>7568748
\mathbb{Z}\cap E \neq \mathbb{Z}\newline
\mathbb{Z}\cap E= E\newline
\mathrm{Thus, \left |\mathbb{Z} \right |> \left | E \right |}

>>7576315
>>7576319
committing sudoku

\mathbb{Z}\cap E \neq \mathbb{Z}
\mathbb{Z}\cap E= E
\mathrm{Thus, \left |\mathbb{Z} \right |> \left | E \right |}[m/ath]

>>7575539
Reference the distributive law if you've covered it in class, otherwise draw a truth table to show the two are logically equivalent.

>>7576320
cool story bro


(a finite mind stares off into the infinite having not understood it)

>>7576773
let me have my laughs friend.

everyone knows \left | \mathbb{Z} \right | = \left | E \right |
Since the function f(x)=2x will match any x \in \mathbb{Z} with an element in E.

not sure if i should post this here or on /ohm/.
I need a way to detect used cigarettes butts and only used cigarettes butts for a project of mine.
can't use size cause each cigarette is different once smoked, can't use color because its different depending on the brand.
I was thinking about using an electronic nose or whatever it's called to detect smoke/cigarette smell but that would mean i'd detect everything that smell like cig , not just cigarette butts. it's driving me crazy.

>>7577337
Hire a team of unemployed midgets to act as used cigarettes butt detectors.

What the hell is /ohm/ ?

Why is it that when I do this integral:

int ( x/(x^2-1) dx )

using u-sub, I get (1/2) ln|x^2-1| + C

but when I use the trig substitution x = sec(θ)

I get ln|x^2-1| + C.

>>7577916
I get (1/2) ln|x^2-1| + C both ways anon.

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I feel like the answer should be some oscillating sequence (i.e. trig but those don't seem to work)......Maybe something recursive?

>>7578693

Harmonic series.

Seriously. get your shit together.

I have my 2nd calc 1 exam next week. Its about getting derivatives and using all the rules, quotient rule, chain rule, etc.

Does anyone have some advice for this stuff? Any sources of good learning?

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I'd try an extension cord and a lamp

2 questions
What the fuck would I even do with a math degree other than academia? It's interesting to me but I want something practical that lands me a job.

Second, has free will been scientifically disproven?

Assuming realistic propulsion and no outside influence:

Would an object's maximal average speed for any journey be half c?

>Half time accelerating to C, half time decelerating.

>>7579035
Q1:
Go into accounting/ banking. You'll blow your brains out from boredom but still paid better than academia.

Q2:
Nope. But it can probably be shown philosophically (logic).

>>7578720
normally I would advise you to study the topic until you have grasped everything, but you probably just want to pass the exam.
just solve a shit ton of practice questions of varying difficulty. if you have serious problems, ask one of your peers that knows his stuff and get a study buddy.
at this level you don't need much abstraction, since you just use mathematical tools, which you probably don't even understand properly.

BTW in my country, basic calculus is taught in the 11th grade of high school and college/university level curses start right away with real analysis.

if you already have a hard time dealing with derivatives and simple calculations, don't even bother to take more 'advanced' math lectures, because real math is nothing like the stuff you have experienced so far.

>>7579101
i know I come off as condescending, but many of my friends who went though similar struggles with e.g. math in engineering programs, had to realize the hard way, that it is of no avail to continue their studies, if you already fail multiple times taking the easiest math-related course.
most of them then went on to become either geologists or biologists

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Any idea how to approach pic related?

>>7578714
but i need a sequences, not a series.

>>7577456
/ohm/ is the electronical stuff general on /g/ but they don't have a thread open ATM.

bump for >>7577337

>>7579393
it's on /diy/ not /g/

>>7578720
What exactly is giving you trouble?

>>7578341
I found my mistake. I mislabeled one side of the right triangle as x^2 - 1 instead of sqrt(x^2 - 1).

>>7579081
Your total distance travelled plays into this. And your method of braking. if you slow down at a rate that is faster or slower than your initial acceleration, just that alone would throw the whole thing off.

>>7570344
https://en.wikipedia.org/wiki/Truth_table#Logical_implication

>>7568948
Ilya, you're a fucking idiot

>>7578693
>>7579360

No anon, the other dude >>7578714 is saying to define your sequenece as a partial sum of the harmonic series, like this

a_n = \sum_{k=1}^{n} \frac{1}{k} .

It's well known how this series diverges logarithmically.

Even so,

a_{n} - a_{n+1} = \sum_{k=1}^{n} \frac{1}{k} - \sum_{k=1}^{n+1} \frac{1}{k} = -\frac{1}{n+1}

clearly converges to 0.

But you don't want to do it this way, anon. This is the sissyfag way. And you DON'T want to be a SISSYFAG, do YOU?!?!

So be a real man, and define

a_n = \ln(n)

Obviously this diverges, but

a_{n} - a_{n+1} = \ln\Big(\frac{n}{n+1}\Big)

tends to \ln(1) = 0 as n \rightarrow \infty .


But if I were you, I'd say fuck all that pansy shit and aim up to becoming a truly rugged individual, and define

a_n = \sqrt{n}

Again, divergence is obvious, but

a_{n} - a_{n+1} = \sqrt{n} - \sqrt{n+1} = \frac{1 - \sqrt{1 + 1/n}}{n^{-1/2}}

and L'Hospital's Rule on the RHS says it converges to 0 as n \rightarrow \infty .

FUCK YEAH!

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>>7579142

Don't ask me why it's the wrong orientation

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>>7579874

Beyond what I've done I have feeling you'd be best served to use a computer to solve the system of linear equations that results from comparing the coefficients of powers of x, but it might be possible to make some assumptions about what is and isn't zero to come up with some solution. Seems a little strange to me that there should be a completely unique solution, at least if you consider a_5 = 0 a solution.

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lines 1 and 2 are given, is line 3 necessary? line 10 is justified because if you repeat the operations of lines 9 and 6 they overlap, did I show this correctly?

Pls help /sci/. I started spivak and I don't have access to a teacher or other math resources.

>>7580114
Line 3 is "necessary" if you are being very formal. The more words you use, the less stuff like line 3 you have to write down.

Also, "6.9,pattern" is not a very good justification, but if you say "repeated application of line 6 to ___ yields (line 10)" or something like that, it's easier to follow.

There aren't great resources on learning how to write proofs. Read and write a lot of them. Spivak is a great start. Notice how he writes his proofs.

>>7580114
Also, https://en.wikipedia.org/wiki/Mathematical_induction

>>7579142

I got

f(x) = 4096 x^13 - 13312 x^11+ 16640 x^9 - 9984 x^7 + 2912 x^5 - 364 x^3 + 13 x

So the answer is 9,1,2.

>>7580379
latex sux

f(x) = 4096 x^{13} - 13312 x^{11}+ 16640 x^9 - 9984 x^7 + 2912 x^5 - 364 x^3 + 13 x

>>7579630
Holy shit brilliant, you need to be my math tutor haha :) Thank you!

What makes me so stupid compared to even smarty pants people on /sci/?

What In my brain makes me such a dunce when everyone else seems to be able to learn and understand stuff with the greatest ease?

Is there anyway to start being smart?