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9280927 No.9280927 [Reply] [Original] [archived.moe]

This thread is for questions that don't deserve their own thread

Previous thread: >>9271297

>> No.9280930

Starting us off with an easy one.

How exactly do I show a sequence is Cauchy? Example: Prove [math]x_{n} = \frac{1}{n}[/math] is Cauchy

Writing the definition: [math]\forall \hspace{0.2cm}\epsilon > 0 \hspace{0.2cm} \exists \hspace{0.2cm} N \in \mathbb{N} \hspace{0.2cm} \text{such that} \hspace{0.2cm} \forall \hspace{0.2cm} m,n,\geq N, \hspace{0.2cm} |x_{n} - x_{m}| < \epsilon [/math]

One can write: [math]\left| \frac{1}{n} - \frac{1}{m}\right| <\epsilon [/math] but how do you generally proceed from here?

>> No.9280933

>>9280930
Is this correct?

Choose epsilon to be [math]\frac{\epsilon}{2}[/math] Since [math]m, n > N[/math],

[math] \left| \frac{1}{n} - \frac{1}{m} \leq \left| \frac{1}{n} \right| + \left| \frac{1}{m} \right| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon [/math]

>> No.9280935

>>9280933
[math] \left| \frac{1}{n} - \frac{1}{m} \right| \leq \left| \frac{1}{n} \right| + \left| \frac{1}{m} \right| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon [/math]

>> No.9280936
File: 16 KB, 652x96, reimannhypothesis.jpg [View same] [iqdb] [saucenao] [google] [report]
9280936

can someone help? I have absolutely no clue how to do it other than by solving the recurrence which I doubt we'd be asked to do for this given the wolfram alpha result.

>> No.9280938

>>9280930
but how do you generally proceed from here?
It depends on the situation.
On this one for example you can write [math] | \frac{1}{n} - \frac{1}{m} | < | \frac{1}{n} | + | \frac{1}{m} | = \frac{1}{n} + \frac{1}{m} < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon [/math] for some m and n greater than some number. [math] \frac{1}{n} < ε [/math] when n is greater than some number follows from the Archimedean property of the real numbers.

>> No.9280940

>>9280936
Write out the first few terms of the sequence to get an idea of what's going on. Can you show it's bounded and monotone? Then you will know if it is convergent or not.

>> No.9280945

>>9280927
Consider an insulating sphere with +q surface charge. Can a Gaussian surface within the sphere prove the field inside is zero for: (a) a uniform surface charge distribution; (b) a variable surface charge distribution?

>> No.9280946

>>9280933
You don't choose epsilon. You choose N as being larger than 2/epsilon.

>> No.9280948

>>9280936
[math] f(x)= \frac{1}{2} x + \frac{1}{x} [/math]
The sequence given a [math] x_0 [/math] is [math] x_0, f(x_0), f^2(x_0), \ldots [/math] .
Use Banach's fixed point theorem.
You can prove Lipschitz continuity using derivatives.

>> No.9280950

>>9280948
Oh and the fixed point is the limit.
You can find it solving this [math] x=\frac{1}{2} x + \frac{1}{x} [/math] .

>> No.9280952

>>9280930
Use the equivalence relation that says a sequence in [math]\mathbb{R}^n[/math] is convergent iff it is Cauchy. To prove that it is Cauchy it is enough to prove that it converges. In the case of [math]x_n=\frac{1}{n}[/math], it's clear that [math]0<x_n\leq 1[/math], meaning the sequence is bounded, and [math]x_m<x_n[/math] whenever [math]m>n[/math], meaning it's decreasing (i.e., it is monotone). The monotone sequence theorem tells you, then, that this sequence is convergent, and therefore it is Cauchy.

>> No.9280966

>>9280948
those things are new to me and it's supposed to be solveable with what we've already done in class >>9280940
seems right but don't I have to solve the recurrence to do that?

>> No.9280969 [DELETED] 

>>9280948
You need a compact set for Banach's fixed point theorem.
You can use Banach to prove that it converges if you start inside the set [math] A_N = [-N,\sqrt{2}] \cup [\sqrt{2},N] [/math] for [math]N > 2[/math].
Then use that fact to conclude it converges if you start in the set [math] A = (-\infty, \sqrt{2}] \cup [\sqrt{2},\infty] [/math] and only then prove the general case.

>> No.9280981
File: 53 KB, 978x307, Screenshot_20171107-142246.jpg [View same] [iqdb] [saucenao] [google] [report]
9280981

Reposting. Can anyone show how to get this solution?

>> No.9280990

>>9280966
Let's consider an easier example.

Only difference here is I have selected an initial condition.

Let [math]x_{1} = 1[/math] and [math]x_{n+1} = \sqrt{1+x_{n}} [/math].

First show the sequence is increasing by doing induction. Recall what it means for it to be increasing, [math]x_{n+1} \geq x_{n}[/math] for all [math]n[/math]. These problems are great for induction.

Next, the boundedness. It should be obvious that [math]x_{n} \geq x_{1}[/math] for all [math]n[/math]. The upper bound then is the challenge. Claim: [math]x_{n} \leq 2[/math] for all [math]n[/math]. This is another proof by induction.

Now you've shown it's monotone and bounded, therefore it converges, and this it's limit exists (and it's value is quite interesting).

>> No.9280994

>>9280936
Put x[n+1]=x[n]=x, to get
x=x/2+1/x
=> x^2=x^2/2+1
=> x^2/2-1=0
=> x^2=2
=> x=+/-sqrt(2)
If x[n] is +ve, x[n+1] must be positive. If x[n] is negative, x[n+1] must be negative.

Note that the recurrence is a specialisation of
x[n+1] = (x[n]+k/x[n])/2
for k=2. The generalised form is known as the "Babylonian method" or "Heron's method" for calculating the square root of k. If x=sqrt(k), then x=k/x=(x+k/x)/2

>> No.9280997

>>9280927
If [math]f : E \rightarrow F[/math] and [math]B[/math] is a subset of [math]F[/math], how does one show that [math]f^{-1}(F [/math]\[math] B) = E [/math]\[math]f^{-1}(B)[/math] ?

I guess I have to use double inclusion, but I don't really see where to go from there...

>> No.9280998

>>9280994
this works, thanks.

>> No.9281001

>>9280997
f^(-1) (F\B) = {x in E | f(x) in F\B}
= {x in E | f(x) in F} \ {x in E | f(x) in B}
= E \ f^(-1)(B)

>> No.9281003

>>9280969
Not really. You have to break it to two cases.
Case 1:
x0 is in the interval [math] [ \varepsilon , + \infty ) [/math] . xn converges to [math] \sqrt{2} [/math] . Contraction can be proven by the mean value theorem and the fact that the derivative is always less than 1/2.
Case 2:
x0 is in the interval [math] (- \infty, - \varepsilon] [/math] . xn converges to [math] - \sqrt{2} [/math] .

>> No.9281007

>>9280998
This "works" yeah, but you have to prove that it converges.
You really have to use >>9280948 , >>9281003 to actually prove it.

>> No.9281011
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9281011

>>9281003

>> No.9281013

>>9280981
Two forms solutions are [math]e^{\pm \sqrt(\epsilon)}[/math] and then using the boundary equations you get the correct constants, so you can write it into the form cosh

>> No.9281016

>>9281003
>the derivative is always less than 1/2
The derivative at x=0.1 is -99.5 for example.

>> No.9281017

>>9281013
I meant 1/squareroot

>> No.9281024 [DELETED] 

>>9281016
Oh fuck it needs to be |x|>1.
Well, in [-1,1]
|f(x)-f(y)| = 1/2 |x-y| |xy-2|/|xy| bla bla bla ... bound

>> No.9281048

Prove: Suppose [math](x_{n})[/math] is monotone and has a convergent subsequence. Prove that [math](x_{n})[/math] converges.

Suppose [math]x_{n+1}\geq x_{n}[/math] . We're also given, [math]\forall \epsilon > 0 \exists N \in \mathbb{N} [/math] such that [math] \forall k \geq N , |x_{n_{k}} - L | < \epsilon [/math]

[math](x_{n})[/math] converges, so [math]\forall \epsilon > 0 \exists N \in \mathbb{N} [/math] such that [math] \forall n \geq N , |x_{n} - L | < \epsilon [/math]

How do I relate these two?

>> No.9281050

>>9281048
It's a basic theorem.
Fucking google it.

>> No.9281052

>>9281048
Proof: Think

Because the sequence is monotone you can bound every term in the sequence between two of the terms of the convergent subsequence. And then as n goes to infinity, the distance between the terms of the convergent subsequence go to 0.

So you can proceed in either of two ways: Use this to prove the general sequence is Cauchy or even better: conjecture that the limit of the sequence is the limit of convergent subsequence and then do your epsilon magic.

>> No.9281053

>>9281048
Bolzano-Weierstrass theorem.

By the way, which book are you working out of?

>> No.9281055

>>9280945
Please help...

>> No.9281057

>>9281055
Proof: Think.

If the charge is distributed along the sphere, a Gaussian sphere with the same center but smaller radius will enclose exactly 0 charge. Apply Gauss law and then think.

>> No.9281062

>>9281057
*If the charge is distributed along the surface

>> No.9281090

>>9281057
Why do they make the distinction of considering a uniform and a non-uniform charge distribution as separate cases then?

>> No.9281131

>>9281090
To test if you are stupid

>> No.9281133
File: 16 KB, 653x231, 438b218a1e7cf4745b627d9b0b577aa5.png [View same] [iqdb] [saucenao] [google] [report]
9281133

>>9280870
The answer on my exam key is 2, so that is incorrect.

Pic related is the problem. I set my bounds as stated in my stack exchange post here: https://math.stackexchange.com/questions/2506874/triple-integral-bounded-by-a-cylinder-and-a-plane

Can anyone tell me why my bounds are wrong?

>> No.9281159
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9281159

The book says the answer to this problem is e^7 - 1
I get where the e^7 comes from but not the 1

>> No.9281167

>>9281159
It is a telescopic series.
Write out a partial sum and you'll see why. Everything except the first term e^7 and the last term -e^(n+1) cancel out.

>> No.9281169

>>9281167
>last term -e^(n+1)
-e^(7/(n+1))

>> No.9281198 [DELETED] 
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9281198

Thank you for the explaining the last problem.
The book also has a problem similar to this but n =0 making easier to find r if I am correct.
To find r you use
a_1r=a_2
But this series starts at n=1 making it more involved and I want to make sure im not missing something before I continue

>> No.9281201 [DELETED] 
File: 9 KB, 250x180, 1499464258015.jpg [View same] [iqdb] [saucenao] [google] [report]
9281201

>tfw you realize .05 seconds after posting

>> No.9281286
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9281286

Someone can explain to my why the phase spectrum of the sinc function is [math]\pm \pi[/math] when [math]\sinc(x) < 0[/math]?
Sinc is real so it shouldn't be all 0?

>> No.9281318

Can you expand conditional probability expressions like that ?

P(A) = P(c2) + P(c3) - P(c2) x P(c3)

P(c2|A) = [P(A|c2) x P(c2)]
P(A)

Can I then expand - P(A|c2) by replacing where c2 occurs in A with 1?

1+ P(c3) - P(c3)x1

Is there anything wrong with this ?

>> No.9281367

Simple question, why is [math]\phi_2=180°-\phi_0[/math] in the second quadrant of a unit circle? I understand if ex. you have 180°, and the first angle fills one portion, while the other fills the rest of the 180°. But doesn't make sense if both have the same starting point.

>> No.9281487

I'm new to general relativity and I am confused by notation

what's the difference between

[math] \Gamma^a_{\ bc} \quad \quad \Gamma_{ab}^{\ \ \ c} [/math]
?

>> No.9281494
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9281494

Am I doing this wrong, feel like I am doing this wrong.

>> No.9281520

>>9281494
They want me to show if I know how to get a trig identity. I am trash at them so I came here. The stuff on top is the original identity.

>> No.9281577

How does one derive the recurrence formula [math]T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)[/math] using the generating function [math]g(x, t) = \frac{1 - tx}{1 - 2tx + t^2} = \sum\limits_{n=0}^\infty T_n(x)t^n[/math] without using cosines in any way? The basic idea is clear: derivative wrt t and then playing with the coefficients, but I can't do this in practice for some reason, or maybe I'm just too blind to see how to do it.

>> No.9281595

>>9281494
It's literally multiplying both sides by the denominator. Then you see they are equal.
After that just work backwards: adding the stuff and then dividing it.

>> No.9281606

>>9281494
Anyway, you're doing it right.

>> No.9281717
File: 2 KB, 368x38, imabrainlet.png [View same] [iqdb] [saucenao] [google] [report]
9281717

?

>> No.9281733

>>9281606
Thx

>> No.9281761 [DELETED] 
File: 49 KB, 1781x277, Screenshot_20171107-163035_01.jpg [View same] [iqdb] [saucenao] [google] [report]
9281761

By what method can I evaluate this limit?

>> No.9281771

>>9281717
Did you try anything?

>> No.9281774 [DELETED] 

>>9281717
1/n^(1/3) < 1/n

>> No.9281779
File: 49 KB, 1781x277, limit.jpg [View same] [iqdb] [saucenao] [google] [report]
9281779

By what method can I evaluate this limit?

>> No.9281780

>>9281779
derivative

>> No.9281784

>>9281774
there was an attempt

>> No.9281798

1/n^(1/3)<ε <==> 1/n<ε^3 <==> n>1/ε^3
Ν:=Ceiling[1/ε^3]>=1/ε^3
n>N ==> n>1/ε^3 ==> 1/n<ε^3 ==> 1/n^(1/3)<ε

>> No.9281815

If two expressions are equal, is it a guarantee you can manipulate either one into the other form algebraically?

An example would be (n-1)*(n-3)*(n-5)...*(n-(n-1)) == n! / ((2/n)! * 2^(n/2)).

Is there a rigorous definition for equality between algebraic statements? Is the left hand side of that equality I posted even an algebraic statement if I have to include "..." to show what I mean?

>> No.9281830

>>9281815
Also: f(n) = (n/n) has a discontinuity at 0 but n/n = 1 and g(n) = 1 does not have a continuity.

If 1 = n/n then doesn't f(n) = g(n) but if they're equal why does f(0) not equal g(0)? Or does it?

Is this just a case of attributing more to the idea of equality than is there?

I would ask one of my profs this but I feel like this is something I should understand from elementary school....

>> No.9281835

>>9281780
Using l'hopital's rule?

>> No.9281840

>>9281835
no
definition of derivative, the limit

>> No.9281841

"In the card game of poker, each player is dealt 5 cards. Assume the game is played with all 52
cards. In how many ways can a player have a ‘four of a kind’ (that is, 4 of the 5 cards must be
from the same ‘denomination’ (e.g. the four 8s) - the fifth card can be any other card)?"

The answer is 624 but I don't know how this was reached.
Can someone do like a step-by-step?

>> No.9281849

>>9281771
no

>> No.9281856

>>9281841
There are 13 denominations in a 52 card deck so you can get 13 different sets of 4 cards of a kind.

For each of the 13 possibilities, you will have 52 - 4 = 48 cards remaining. So then there are 48 possible 5th cards for each 4 of a kind.

If there are 48 "versions" of each and 13 possible 4 of a kind hands, then you have 13 * 48 = 624 possible 5 card hands with 4 of a kind.

If you didn't follow that then you need some basic combinatorics knowledge: https://en.wikipedia.org/wiki/Rule_of_product

>> No.9281857

>>9281841
Judging by the answer, the order that the cards were dealt doesn't matter.
How many tetrads are there? 13
Pick one of them. 13 possible ways.
You are left with 52-4=48 cards.
Pick one of them. 48 ways.
13*48=624

>> No.9281878

>>9281487
There are two types of Christoffel symbols, one with only lower indexes (first kind) and one with one upper index (second kind). So long as you don't mix them, whichever way you write it is fine. I've seen it written both ways and also with the upper index in the middle.

>> No.9281960

>>9281878
so [math] \Gamma^a_{\ bc} = \Gamma_{bc}^{\ \ \ a} [/math] then?

>> No.9281967

why is every SQT just a bunch of math proofs questions that can be googled?

ask questions here that cant be googled and get ppls opinions

ill start: what causes cell apoptosis?

>> No.9281979

>>9281967
Sometimes having a dialogue with someone over a topic provides a deeper understanding of the concept, or allows understanding to take place at all.

>ill start: what causes cell apoptosis?
What do you mean by "causes" it? The mediating mechanism or the initiating one? Maybe neither, since both of these questions can be easily googled or found in the Wikipedia page on apoptosis and you said your question can't be.

>> No.9281981

13 dollar bet you go play again after big win

>> No.9281982

Can someone explain math/formulas? What are we actually doing here? Are we trying to find relations between some elements of a system and connect them through a formula?

I wouldn't know how to get a formula on my own, even though it may make sense.

>> No.9281997

>>9281960
Yes, but don't quote me on that.

>> No.9282003

>>9281997
>>9281960
An addendum: just look at the definition of the Christoffel symbol, it really doesn't care where you write the upper index. The only thing it cares about is the order of the lower indexes.

>> No.9282042

>>9281840
why not l hopital?

>> No.9282069

>>9281815
Equality is essentially defined as a relation where, if two things are equal, then you can substitute them for each other arbitrarily and everywhere:
[math](x=y)\Longrightarrow \left(P(x)\Leftrightarrow P(y)\right)[/math], no matter what proposition P represents.
In your case however, [math]f(n) = \frac{n}{n} = 1[/math] only when [math]n\not = 0[/math], and so [math]f(n)=g(n)[/math] doesn't hold when n is zero.

>> No.9282076
File: 25 KB, 549x97, problem 3.png [View same] [iqdb] [saucenao] [google] [report]
9282076

The generating function of a standard tetahedra die is: x^1 +x^2 +x^3 + x^4 and a standard cubic die is x^1 +x^2 +x^3 + x^4 +x^5+x^6.

So the sum would be: 2x^1+2x^2+2x^3+2x^4+x^5+x^6

How do I come up with the other dice that would generate this?

Am I even on the right track?

>> No.9282149

>>9282042
It can only be used in 0/0 etc type cases.

>> No.9282158

Should I study ODE with Boyce book or Tenenbaum book?

>> No.9282160

>>9282076
>Update
Well I figured out that the probability distribution of the sum of the two standard dice would be:
x^2+2x^3+3x^4+4x^5+4x^6+4x^7+3x^8+2x^9+x^10

Is there any way to find possibilities other than guessing and checking. Sorry if I am dumb.

>> No.9282182

>>9282158
Simmons if you're not a brainlet.

>> No.9282206
File: 1.04 MB, 3120x3120, IMG_20171107_174422.jpg [View same] [iqdb] [saucenao] [google] [report]
9282206

>>9281133
Your bonds are wrong. It tells you z is bound by the plane z = 0. That means your first set of bonds should be 0, sqrt(4-y^2). That gives you the right answer.

>> No.9282213

>>9281494
Write neater. Always. it doesn't matter if this is your rough work. That's illegible.

>> No.9282215

>>9282182
but I am a brainlet

>> No.9282219

>>9282149
I evaluated it using l'hopital's rule.
Given that F(gamma) = F0(cos(gamma*t) - cos(omega*t))
and G(gamma) = omega^2 - gamma^2
limit as gamma approaches omega for F(gam)/G(gam) evaluates to 0/0.

>> No.9282247

>>9281982
Entirely depends on what your talking about. 2+2 = 4 is simply simplifying the equation on the LHS into the RHS. This is often the case for equations including calculus.

An integral is solved from its integral form into an equation to make it understandable.

When we make a formula given elements of a system we are finding a relation between the elements and then expression that relation as a mathematical formula.

If for every 3 apples I have I can trade those apples for 4 pears a formula would be:

3a = 4p

>> No.9282300

>>9282160
Figured it out. There are four possibilities including the standard dice.

>> No.9282393

>>9282069
So then the statement "n/n = 1" is incomplete, or at least the "for n not equal to 0" is implicit. It seems like there is an analogy to the congruence relation in modular arithmetic, like a ≡ b (mod n), b ≡ c (mod n) then a ≡ c (mod n). Similarly f(n) = n/n = 1 = g(n), when we say f(n) = g(n) by the transitive property, we can't do so without maintaining the implicit restriction that they are equal WHEN n != 0, which arose when we set n/n = 1.

I've seen congruence modulo written with the modulus over the triple bar, so there is some sense in which the restrictions under which an equality holds are "attached" to the equality?

>>9281982
I think this question is hard to answer both well and concisely.

If you think a bunch about the rules of chess, you can come up with some theorems about how chess works. Maybe you can develop some new rules which follow logically from the "basic" rules which are set out in the definition of the game. Furthermore, maybe you can extend those idea outside of chess to tell you things about systems which are related to chess. Mathematics is about doing that, in general, for all kinds of different sets of rules.

You may find this channel interesting? https://www.youtube.com/watch?v=R7p-nPg8t_g

>> No.9282549
File: 14 KB, 554x59, problem 8.png [View same] [iqdb] [saucenao] [google] [report]
9282549

Any help on this would be appreciated. Obviously a rectangle (2n, 3m) or (3n, 2m) would work. What about other possibilities?

>> No.9282561

>>9282549
Both n and m have to be at least 2 and their product must be divisible by 6.

>> No.9282576

>>9282549
Key observation: A rectangle can be tiled by [math] 2 \times 3 [/math] if and only if that rectangle can be constructed by stitching together [math] 2\times 3 [/math] tiles. So lets instead investigate all the ways to stich together these tiles.

The easiest way would be by putting a [math] 2 \times 3 [/math] tile on the plane and calling that the main corner. And then stitching together some tiles next to it horizontally and some other tiles next to it vertically. You can then proof that the rest of the rectangle can be filled in properly.

If we let there be [math] x [/math] horizontal rectangles and [math] y [/math] vertical rectagles, the rectangle we created will have side lengths [math] m = 2x, n=3y [/math] so the set of pairs we are looking for is simply

[eqn] \{ (2x,3y), x,y \in \mathbb{Z} \} [/eqn]

>> No.9282598

>>9282576
You can very easily tile a [math]5 \times 6[/math] rectangle.

>> No.9282599

I found a story here some months ago about a hypothetical Russian rocket that used some insanely dangerous fuel/oxidizer combo because the CIA leaked that the US had found a way to make the combination work with use of a top secret stabilizer.
Anyone have a link to the story?

>> No.9282712

The probability density function

fx(x) = 1/((1+x)^2) for x>0, and 0 elsewhere.

How do I show that 1/x has the same distribution as x?

>> No.9282725

Brainlet incoming, shit deflectors activated.

What do you do when you're on the verge of failing (D or F) towards the end of the semester? I have three classes that are giving me a huge death wish. For one class I may get an instructor withdrawal because of circumstances, the other two have just been really difficult for me to balance with my stupid life and on top of that getting intuition has been difficult to me.

If anyone has any good resources that take things a bit slow for an engineering statistics and first course in linear algebra, please send. YouTube can be helpful sometimes, but there seems to be so many different approaches to these courses that not every video is directly relevant. LA has had a really bad exam average, but the hw is super long and I will probably have 2/3 of the total grade for that. Going to office hours for both classes tomorrow to be a little crybaby.

Anybody here fail out of uni before? If I actually pull my shit together I could be fine, but long term gf just left me so last week I did almost nothing (made my situation a lot worse).

>> No.9282730

>>9282712
[eqn] P(X \leq t) = \int_0^t \frac{1}{(1+x)^2}dx \\
= \int_{\frac{1}{t}}^\infty \frac{1}{(1+x)^2}dx \\
= P\left( X \geq \frac{1}{t} \right) \\
= P\left( \frac{1}{X} \leq t \right) [/eqn]

>> No.9282747

>>9282730

I see. Thank you...these are a little confusing for me.

How about..

If X has distribution R(0,1) and Y has distribution (-1/a)(lnX) with a>0,
how can I show that Y has the same distribution as the exponential distribution of a?

>> No.9282751

>>9282747
Stop asking other to do your homework.
Fucking do it yourself.
It's done the same way the other guy showed you.

>> No.9282758

>>9282751

I don't know how to.

>> No.9282763

>>9282758
You know the definition of a cummulative distribution function is.
Compute the cummulative distribution function of Y.
P(Y <= y) = ........

>> No.9282770

>>9282763

= int from 0 to y of (-1/a)(lnX)dx, no?

then what?

>> No.9282799

>>9282770
>then what?
Then solve the integral.

But you can also do this
y ranges from 0 to infinity.
P(Y<y) = P(-1/a lnX<y) = P(lnX>-ay) = P(X>e^(-ay)) = 1-P(X<e^(-ay)) = [math] 1- \int_{0}^{e^{-ay}}1dx = 1-e^{-ay} [/math]

>> No.9282818

Anyone can recommend me a book on power electronic? About step down/up converter and half/full bridge inverter and fly-back SMPS converter?

>> No.9282853
File: 2 KB, 149x18, MSP14952123e8772545b7b6d00000g52ah46dbba4eh2.gif [View same] [iqdb] [saucenao] [google] [report]
9282853

There is supposedly a trivial solution to this but I can't see how to do it without simplifying everything to the same base. Plz help

>> No.9282855

>>9282853
try [math]x = 0[/math]

>> No.9282869
File: 109 KB, 1667x348, Capture.png [View same] [iqdb] [saucenao] [google] [report]
9282869

>>9280927
How the heck do they draw that graph or whatever you call it? Half of those numbers seem like they were pulled from magic. Is there anyway to get good at drawing them? I can't find any tutorials for them.

>> No.9282884

>>9282869
law of sines

>> No.9282889

>>9282884
I get that part though where did they pull the 50 and 30 degrees from?

>> No.9282895
File: 93 KB, 1053x478, Capture.png [View same] [iqdb] [saucenao] [google] [report]
9282895

>>9282889

>> No.9282901
File: 92 KB, 878x295, Untitled.png [View same] [iqdb] [saucenao] [google] [report]
9282901

Pretty sure this can't be done as the two events are dependent, and you need them to be independent to do the first part,,,

Why am I wrong?

>> No.9282911

P(X=3,Y=2) = P(Y=2|X=3) P(X=3)
= 3 (0.6)^2 (0.4) (0.25)

>> No.9282922

>>9282901
Calculate binomial distributions where p=.6 and n=0,1,2 and 3. Multiply these by the p m f value for the corresponding X to get joint pm f. Rest is easy then

>> No.9282927

>>9282911
correct

>>9282901
Joint pmf:
[math] P(X=x,Y=y) = P(Y=y | X=x) P(X=x) = \binom{x}{y} {0.6}^y {0.4}^{x-y} p_X(x) [/math]
Marginal of Y:
[math] P(Y=y)= \sum\limits_{x=0}^{3} P(X=x,Y=y) = \sum\limits_{x=0}^{3} \binom{x}{y} {0.6}^y {0.4}^{x-y} p_X(x) [/math]

>> No.9282930

>>9282895
Thanks for the help anon. I see the error in my ways now.

>> No.9282967

Can someone explain to me how the raising operator [math]a^\dagger=\frac{1}{\sqrt{2}}\left(-\frac{d}{dq}+q\right)[/math] is the adjoint of the lowering operator [math]a=\frac{1}{\sqrt{2}}\left(\frac{d}{dq}+q\right)[/math]. This suggests that [math]\frac{d}{dq}[/math] is a pure imaginary operator, but I don't understand why this is the case.

>> No.9282968

Why is

[math] g^{bc}\nabla_b\nabla_cx^a = g^{bc}\left ( \partial_b \partial_c - \partial_d x^a \Gamma^d_{bc}\right ) [/math]

Does the metric kill all terms from [math]\nabla_b\nabla_c x^a[/math]
except for
[math] \partial_b \partial_c - \partial_d x^a \Gamma^d_{bc} [/math]

?

>> No.9282975

>>9282967

You can write d/dq as d/dq = ip where p is the momentum conjugate to q. q is a real observable, thus p is real as well. So d/dq is imaginary as you say.

You can also verify from stipulating that [a,a^\dag] = 1 (since you said they are raising operators). Then you can show [d/dq,q]=1 and since you know [x,p]=i, you can work out the rest.

>> No.9282993

>>9282968
Doesn't that just follow directly from the definition of the covariant derivative? You can see all the other terms cancel.

>> No.9283085

>>9282855
How does that help in any way?

>> No.9283087

Is the probability of getting HHT and HHH the same on a fair coin?
That doesn't sound right but if it' fair then they all = 0.5 so wouldn't it just be 0.5^3?

>> No.9283090

>>9283087
>Is the probability of getting HHT and HHH the same on a fair coin?
Yes.
>That doesn't sound right
It does, you are just confused. {HHT} is not {HHT, HTH, THH}.

>> No.9283230

>>9282993
I don't see how everything else cancels, maybe I'm doing something wrong

[math]
\nabla_b\nabla_cx^a = \nabla_b(\nabla_cx^a) = \partial_b( \nabla_cx^a)-\Gamma^d_{bc}(\nabla_dx^a)+\Gamma^a_{bc}(\nabla_dx^a)\\
=\partial_b(\partial_cx^a+\Gamma^a_{ce}x^e)-\Gamma^d_{bc}(\partial_d(x^a+\Gamma^a_{de}x^e)+\Gamma^a_{bd}(\partial_cx^d+\Gamma^d_{ce}x^e)\\
=\partial_b\partial_cx^a+\partial_b\Gamma^a_{ce}x^e+\Gamma^a_{ce}\partial_bx^e
-\Gamma^d_{bc}\partial_dx^a-\Gamma^d_{bc}\Gamma^a_{de}x^e
+\Gamma^a_{bd}\partial_cx^d+\Gamma^a_{bd}\Gamma^d_{ce}x^e
[/math]

Is it just some clever way of relabeling indices and most of this vanishes?

>> No.9283238

>>9283230
messed up a bit, rightmost term of uppermost equation should be
[math] \Gamma^a_{bd}(\nabla_cx^d) [/math]

>> No.9283245
File: 51 KB, 865x84, untitled0.png [View same] [iqdb] [saucenao] [google] [report]
9283245

What the hell?

Is this as obvious as I think? Can someone help me out?

>> No.9283250

>>9283230
What the fuck is this mess?

>> No.9283252

>>9283245
Yes, it is obvious. Are you supposed to proved it or what?

>> No.9283253

>>9283252
Time for you to go backto reddit

>> No.9283255

>>9283252

Yes and I think I did although I can't figure out LaTeX to share. So I come here to see if someone else will verify their proof with me

>> No.9283257

>>9283255
[math] f_{\vec{X},\vec{Y}}(\vec{x},\vec{y})=f_{\vec{X}}(\vec{x})f_{\vec{Y}}(\vec{y})>0 [/math]

>> No.9283264

>>9280927
Humanities brainlet struggling with optics here.
>At what distance from a converging lens with a focal distance of 40cm must an object be situated, in order to get an image 4 times larger than the object?

>The focal distance of a camera is f = 5cm. The image of a 6m tall house is 24mm large. From what distance was the house photographed?

>> No.9283279

How is the subgroup [math]nA[/math] in [math]A/nA[/math] defined exactly? Where [math]A[/math] is an abelian group.

>> No.9283297

i have a function from R^2 to R^1, how do i find its equivalent matrix operator?

>> No.9283301

>>9283279
nx = x+x+x...+x (n times)

>> No.9283305

>>9280927
I Have Never Deserved My Own Thread.

For I Gave You My Position in the Cosmos, for US.

I do mean 'all' of US.

>> No.9283311

>>9283230
Move to a geodetic coordinate system at x, all the non-differentiated Christoffel coefficients vanish.

>> No.9283314

>>9283279
nA is the identity element of A/nA. Therefore {nA} is the trivial subgroup of A/nA.

>> No.9283340

>>9283301
Yeah, but how is [math]nA[/math] itself defined? [math]nA := \{ na | a \in A \}[/math]? So all "multiples" of n?
>>9283314
Sorry if I wasn't clear, I meant something different.

>> No.9283432

>>9283085
[math]8^0 = 0.6*18^0+0.4*3^0[/math]
[math]1 = 0.6*1 + 0.4*1[/math]
[math]1 = 0.6 + 0.4[/math]

>> No.9283450

>>9283340
>Sorry if I wasn't clear, I meant something different.
Oh I see.
Well, if there's no other context, then it kinda Has to mean what you wrote.

>> No.9283466

How high in the atmosphere can a person live and not suffocate?

>> No.9283492

>>9280936
Separate the cases x_0 > 0 and x_0 < 0. Then, compute [math]x_{n+1}-x_n[/math] and deduce that (x_n) has a limit. Then, see what that limit must be.

>> No.9283493

>>9283466
mount everest is almost too high, they need oxygen masks to climb up there

>> No.9283508

>>9283492
>Then, compute xn+1−xnxn+1−xn and deduce that (x_n) has a limit.
that does not deduce that

>> No.9283626
File: 68 KB, 960x720, serveimage.jpg [View same] [iqdb] [saucenao] [google] [report]
9283626

for an NMOS, how come the drain current saturates after passing a certain drain voltage?

>> No.9283871

Thoughts on a non thesis M.S.? Will it help getting a reasonably good job? I have no plans to get a phd and i dont really want to stay in school for 2 extra years rather than 1

>> No.9283894

>>9283508
yes it does, but it requires a little something

>> No.9283902

>>9283626
The channel pinches off.

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

>> No.9283905

If I know the mean free path between particle collisions and between particle-photon collisions, in some part of the solar photosphere, and I know the average temperature difference between once such collision and the next, how do I tell if the system is in Local Thermodynamic Equilibrium? The temperature difference between particle-particle interactions is tiny, say 1e-6 K, but the temperature difference between photon-particle interactions is a few thousand K. Is this LTE or not?

>> No.9283907

>>9283466
A few miles, about 5 or 6.

>> No.9284105

>Every positive integer greater than one can be written uniquely as a product of primes, where the prime factors are written in nondecreasing order.

What’s important about the non decreasing bit. Why not omit it?

>> No.9284124
File: 27 KB, 628x290, d or a.png [View same] [iqdb] [saucenao] [google] [report]
9284124

>>9280927
What are two different methods to distinguish between a dissociative and associative pathway leading to the same product?
I don't think that labelling with deuterium would work here.

>> No.9284169
File: 25 KB, 1080x464, HALF-ADDER-FULL-ADDER-FIG-2-compressor[1].jpg [View same] [iqdb] [saucenao] [google] [report]
9284169

Here's my assignment: Design a combinational circuit that adds two 4 bits binary numbers together.

Steps given to design a combinational circuit:
1. determine # of inputs and outputs
2. derive truth table to get the relationship between input and output
3. obtain a simplified boolean function for each output
4. draw a logic diagram

So I'm guess there's going to be 9 inputs and 5 outputs. Right now I'm specifically wondering if I have to make a truth table for all the 9 inputs and outputs. And if not, can someone give me some suggestion on what to do? I'm completely lost.

>> No.9284172

Let's say that I have three vectors a,b,c that define the corners of a triangle. Now I want to show that the three medians of the triangle meet at the center of 1/3(a+b+c).
How do I show this? I don't know where to start at all.

>> No.9284190

Can I figure out if a linear transformation is orthogonal by running i-hat and j-hat through it and seeing if they retain distance? Or are there cases where this wouldn't work.

>> No.9284192

>>9284169
https://en.wikipedia.org/wiki/Carry-lookahead_adder

>> No.9284198

>>9284124

One of the methods is to determine rate dependance by with varying reactions. 1st order reaction versus 2nd order.

>> No.9284200

>>9284105
>uniquely

>> No.9284205

>>9284198
So, for 2nd order reactions, I should ask: "is the rate affected by the depletion of [B]"? And if it is, then it is indeed a 2nd order reaction. THank you anon.

>> No.9284212

>>9284190
wouldn't you just take the inner product of your input vector and the vector it maps to?

>> No.9284218

>>9284172

the median from a is the line from a to the midpoint of b and c
the midpoint of b and c is b + (c-b)/2 = (b+c)/2
the intersection is located on this line a+ ((b+c)/2 - a)*t for some t st 0<t<1

this must intersect the other 3 lines so

a+ ((b+c)/2 - a)*t = b+ ((a+c)/2 - b)*v
a+ ((b+c)/2 - a)*t = c+ ((a+b)/2 - c)*u
b+ ((a+c)/2 - b)*v = c+ ((a+b)/2 - c)*u

3 equations of three variables, solve for t,v,u.

>> No.9284236

>>9284212
Consider a linear transformation that simply scales i-hat by 5 and j-hat by 5. The inner product for the image and original will be zero, but distance between vectors is not conserved.

>> No.9284276

>>9284218
now I see. it actually isn't that hard. thanks.

>> No.9284370

>>9283894
>but it requires a little something
so... it doesn't

>> No.9284396
File: 79 KB, 1004x773, Axler.jpg [View same] [iqdb] [saucenao] [google] [report]
9284396

>>9284190
No, it would not work.
A map T is orthogonal iff ||Tv||=||v|| For All v in the vector space; not just the basis vectors.
For example consider the map which takes i to i=(1,0) and j to 1/sqrt(2)(1,1).
It's obviously not orthogonal.

But, if the images of i and j are orthonormal, i.e. have unit length like i and j do AND they are orthogonal, then the map is orthogonal.

>> No.9284397

>>9284169
The easiest solution is a ripple-carry adder: 4 full adders where the carry-out from one stage is fed to the carry-in of the next stage. Each full adder has 3 inputs, so that's only 2^3=8 rows in the truth table.

The disadvantage of ripple-carry adders is that the propagation delay from the least-significant input bit to the most-significant output bit is proportional to the number of bits.

More advanced designs avoid this at the expense of requiring more gates. The extreme is a minterm (disjunctive normal form, sum of products) solution which is only 2 gates deep but requires an exponential number of gates.

>> No.9284548

>>9284396
Much appreciated

>> No.9284554

>>9284190
Just take the determinant of the transformation.

>> No.9284561
File: 1.00 MB, 2404x751, 1486474084121.png [View same] [iqdb] [saucenao] [google] [report]
9284561

How do I calculate the arc length of this?
I got to a weirdish integral and I do not know if I'm on the right way.

>> No.9284563
File: 35 KB, 652x258, opera_2017-11-08_22-44-50.png [View same] [iqdb] [saucenao] [google] [report]
9284563

How do I pass my midterm on Friday?

>> No.9284567

>>9284554
Can you elaborate?

>> No.9284573
File: 2.00 MB, 3120x4160, 1499987398857.jpg [View same] [iqdb] [saucenao] [google] [report]
9284573

>>9284561
I have to calculate the arc length between x=0 and x=5 btw
This is how far I got manually. The integral is right but not sure if it's the proper way to do it "humanly". Wolframalpha got the same integral when solving it.

>> No.9284576

>>9284561
ez, Y=6*cosh(x) just solve it like a regular trig function then. Hyperbolic functions and regular trig functions have similar integrating rules.

>> No.9284583

>>9284576
Sorry but I'm the average calc II retard. Could you show me the trig identity you just used?

>> No.9284588
File: 3 KB, 415x170, opera_2017-11-08_23-07-37.png [View same] [iqdb] [saucenao] [google] [report]
9284588

>>9284583
I'm guessing you haven't covered hyperbolic functions yet. I suggest reading this link.

http://tutorial.math.lamar.edu/Classes/CalcI/DiffHyperTrigFcns.aspx

>> No.9284590

>>9284588
Yeah, we haven't. Hope I can learn it by myself pretty quick so I can get some hours of sleep before class.
Thanks for the help, anon.

>> No.9284591

>>9284590
No problem

>> No.9284611

>>9284591
What would the square root of 1 + 36sinh2(x) be?

>> No.9284631
File: 99 KB, 225x335, 1509527885766.png [View same] [iqdb] [saucenao] [google] [report]
9284631

Is there a single book that covers all of the following:
Banach algebras
Affine Hecke algebra
Beurling algebra
Duffin–Kemmer–Petiau algebra
Exterior algebra
Filtered algebra
Gerstenhaber algebra
Hall algebra
Koszul algebra
Poisson algebra
Rota–Baxter algebra
Weyl algebra

>> No.9284652

>>9284591
Reporting back. I tried as hard as I could and I failed. I got the length as 40.1 or 38.43.
Wolfram alpha said it's 439.56 from the start though.

>> No.9284673

>>9284652
seems like there is no simple closed form for this according to https://www.wolframalpha.com/input/?i=sqrt(36cosh%5E2(x)-35)

so the best you can do is approximate but if you really want the exact answer https://www.wolframalpha.com/input/?i=-i+EllipticE(i+5,+36)

but i may be wrong

>> No.9284680

>>9284673
I can't believe I just wasted ~5 hours on a teacher's typo.
No way this was intended in any way.
Can you spoonfeed me the steps up to the "exact answer"? I really don't know how to work my way there. I tried to use what I learned from the link you sent me while following what I learned about calculating curve length.

>> No.9284714

>>9284611
you can't simplify that. But it was probably meant to simplify in order to allow you to use 1 + sinh^2(x) = cosh^2(x)

>> No.9284796

>>9284680
It's fucked up. Mathematica took like 30 seconds to solve the indefinite integral and after like 2 minutes in, it still hasn't solved the definite one and I aborted it.
There's most certainly a typo.

>> No.9284936
File: 382 KB, 640x480, Asuka_smiling.png [View same] [iqdb] [saucenao] [google] [report]
9284936

What is the exact meaning of quotient spaces? i mean i know the definition, but what is the essence of it? take [math]\mathbb{R} / \mathbb{Z}[/math] for example.

>> No.9284943

>>9284936
It's just [0,1)

>> No.9284951

>>9284943
yeah i get it its homeomorphic to [math]\mathbb{S}^1[/math].... but what does it mean? something with translations...

>> No.9284966

>>9284936
You consider Z as zero.
..., -2+c, -1+c , 0+c , 1+c, 2+c, ... are considered as the same thing.

>> No.9284972

>>9284966
i see..thats why its the equivalence classes of [math]x\sim y :\Leftrightarrow x-y\in \mathbb{Z}[/math]

>> No.9284979

>>9284972
yep

>> No.9285024

> link jumping on wikipedia
> land on cadmium
> leading source of cadmium poisoning is tobacco/smoking
How the fuck does a toxic metal like cadmium end up in tobacco? I tried to find an answer but most hits seem to be around the percentages in various types of plant/cigarettes, but nothing on where it actually enters the chain? Non-smoker, but just curious.

>> No.9285070

I don't understand how the Cantor set is uncountable.
[math]\mathbb{Q}[/math] is countable, so if the Cantor set is uncountable that would mean it's not a subset of [math]\mathbb{Q}[/math], so my question is why is the Cantor set not a subset of [math]\mathbb{Q}[/math]

>> No.9285078
File: 358 KB, 552x543, 1510230564682.png [View same] [iqdb] [saucenao] [google] [report]
9285078

I have a real analysis exam on monday spanning from chapter 3 on baby rudin till chapter 5 and I've barely started studying, will I make it?

>> No.9285122
File: 91 KB, 1420x599, 6543654584.jpg [View same] [iqdb] [saucenao] [google] [report]
9285122

How do I go about solving this? I thought about breaking up the nonparallel plates into infinitesimal parallel strips, but I got nowhere. Please help.

>> No.9285146
File: 366 KB, 583x608, 1508341370410.png [View same] [iqdb] [saucenao] [google] [report]
9285146

Here's a step in the proof of [math]f(x) = x^{2}[/math] being continuous at [math]x=2[/math]:

Let [math]\delta \leq 1[/math] Then [math]|x-2|<1 \implies |x-2|\cdot |x+2| < \epsilon [/math]. Note [math] |x+2| \leq |x-2| + 4 < 5[/math], so [math]|x-2|\cdot |x+2| < 5\delta [/math]

What's the reasoning behind: [math]|x-2|\cdot |x+2| < 5\delta [/math]?

>> No.9285148

>>9285146
wtf is this nonsense

>> No.9285153

>>9285146

If [math]|x - 2| < 1[/math] then [math]|x+2| < 5[/math]. In this case, if [math] |x - 2| < \delta[/math] we have that [math]|x-2| |x + 2| < 5 \delta[/math].

The second sentence is wrong; [math]|x-2||x+2| < \epsilon[/math] is something you want to prove, rather than an implication. You can do this by putting delta less than 1 and also less than epsilon / 5.

>> No.9285155

>>9285078
Try these problems:
https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-fall-2010/exams/

Also try the problems from the UCLA "Basic Qualifying Exam" or "Berkeley Problems in Mathematics" on the relevant topics (the Berkeley Problems in Mathematics book is arranged by topic).

>> No.9285156

>>9285122

https://physics.stackexchange.com/questions/148283/capacitance-of-two-non-parallel-plates

>> No.9285157

>>9285156
Do you think I don't know how to Google, you dipshit? The problem isn't asking for capacitance, it's asking for potential. Kys.

>> No.9285174
File: 457 KB, 772x804, 1508341079554.png [View same] [iqdb] [saucenao] [google] [report]
9285174

>>9285153
I see. Thanks.

>> No.9285177

what is
>t. me
?
It looks like a citation or something but I've never heard of it being done that way. Is it a new thing?

>> No.9285181

If the instructor doesn't include a grading scale on the syllabus does that mean that usually mean they curve the entire class's final grades?

>> No.9285182

Given the pipe diameter, the volumetric flow rate of the water and the viscosity of the water, how do I calcualte the reynolds number?
Re=(D) (U) / V
d=diameter, u = fluid velocity and v=viscosity.
I just need to find the value for U but the only information I have is the volumetric flow rate, so I have to convert it to linear fluid velocity. How do i do this? What is 0.2L/min in m/s?

>> No.9285183

Why is it when two elements are placed next to each other with no operator, it is assumed to mean multiply?
So 2a is 2*a. What's so special about multiplication?

>> No.9285193

>>9285024
Cadmium, and others heavy metals, accumulates in the tobacco plant, but given the shit they put in cigarettes, I wouldn't be surprised if it also comes from other sources.
Also, tobacco is pretty good at accumulating heavy metals.
https://www.google.it/url?sa=t&source=web&rct=j&url=http://www.globalsciencebooks.info/Online/GSBOnline/images/0712/TAET_1(1%262)/TAET_1(2)46-53o.pdf&ved=0ahUKEwjCrdfR2rHXAhXJmBoKHeFMDGEQFgg8MAU&usg=AOvVaw09uZFhf0nG1KowgNpt62_X

Apparently plants can be used to clean contaminated soils. (keywords: bioremediation, phytoremediation)

>> No.9285205

>>9285177
It's a meme from /int/ that comes from the way Finnish people sign messages such as letters, e-mails, etc.

>>9285183
It doesn't necessarily mean multiplication. I'd argue that it's more notation for addition as 2a means that you have a quantity of 2 of whatever a is. For example, [math]4\lambda[/math] could be interpreted as 4 [math]\lambda[/math]'s -- i.e., [math]\lambda+\lambda+\lambda+\lambda[/math] -- or [math]\lambda[/math] 4's -- [math]\sum_0^{[\lambda]}4+([\lambda]-\lambda)4[/math] where [math][\lambda][/math] is the greatest integer in [math]\lambda[/math].

>> No.9285207

>>9285205
I meant [math](\lambda - [\lambda])[/math] not [math]([\lambda] - \lambda)[/math].

>> No.9285321

>>9285183
Nothing is special about it. You just do it to write less.

>> No.9285405

How difficult can I expect a course in projective geometry (undergrad) to be? The course description says it will cover: projective space, the principle of duality, mappings in projective space, conics and quadrics. I did okay in linear algebra.

>> No.9285429

>>9282149
Not him but it really does look like that limit is creeping up on 0/0. That also looks like trig identities night be helpful.
t. Guy who models oscillations

>> No.9285452

www2.lawrence.edu/fast/GREGGJ/Math150/032Work.pdf

can anyone tell me why they don't multiply gravity in finding work here?

they eventually get 1.38*10^12 ft-lbs and I understand everything they went through to get it except I thought work would have to be Ms*g*y (where Ms is mass of slice of pyramid) making the integral be ∫(0,h)Ms*g*y (and since Ms=p*x^2dy and x=(b/h)(h-y) the integral is ∫(0,h)p*((b/h)(h-y))^2 *g*y dy = p*g*∫(0,h)((b/h)(h-y))^2 *y dy) but they have the integral as just ∫(0,h)Ms*y (or p∫(0,h)((b/h)(h-y))^2 *y dy), literally the same thing just without being multiplied by g.

it took me like 2 hours to go through all this since i suck
but the point is that they have W=1.38*10^12 ft-lbs and I think it should be 32.2(1.38*10^12) ft-lbs. (since 32.2 ft/s^2 = 9.8 m/s^2 probably obviously)
W=Fd so what gives? where is the F in what they did?

>> No.9285458

>>9285452
learn latex you mong

>> No.9285466

>>9285458
sorry
But I kind of think all the shit is besides the point Almost, or maybe it isn't idk. But they're just calculating the work done to get a slice of a pyramid up to a point y. And they don't multiply gravity by that y to get W. I don't know why they don't

>> No.9285471
File: 13 KB, 225x225, 1503247350492.jpg [View same] [iqdb] [saucenao] [google] [report]
9285471

how do I show [math]x_{n} = \frac{1}{2^{n}}[/math] is cauchy using the definition:

[math]\forall \hspace{0.2cm} \epsilon > 0\hspace{0.2cm} \exists\hspace{0.2cm} N \in \mathbb{N}\hspace{0.2cm} \text{such that}\hspace{0.2cm} \forall\hspace{0.2cm} m,n \geq N, |x_{n}-x_{m}| < \epsilon [/math]

I get as far as [math]\big| \frac{1}{2^{n}} - \frac{1}{2^{m}} \big| = \big| \frac{2^{m}+2^{n}}{2^{n+m}} \big| [/math]

>> No.9285475

>>9285471
* [math] \big| \frac{2^{m}-2^{n}}{2^{n+m}} \big| [/math]

>> No.9285611
File: 160 KB, 629x645, Capture.jpg [View same] [iqdb] [saucenao] [google] [report]
9285611

any german speaking anons here who can help with anything here, literally any help would do since I'm really lost

>> No.9285620

>>9285611
Really would help you but I'm too busy. Just run it through
http://imtranslator.net/compare/
and check individual words with https://www.dict.cc
all the best

>> No.9285637

>>9285620
thanks german speaking anon, but could you help with the examples that are in question, since I'm not only looking for help with the translation but some help with solving these fuckers

>> No.9285643

>>9285637
bin ein Hirnlein, also kann ich nur übersetzen

>> No.9285645

>>9285471
wlog let n<m

1/2^n - 1/2^m < 1/2^n

>> No.9285655
File: 291 KB, 750x750, 15092396555577.png [View same] [iqdb] [saucenao] [google] [report]
9285655

>>9285645
t-thanks

>> No.9285663

brainlet here, I have trouble multiplying matrices together

>> No.9285672

>>9285663
think of it as a dot product of the rows of A with the columns of B

>> No.9285673

>>9285663
https://www.intmath.com/matrices-determinants/matrix-multiplication-examples.php

>> No.9285683

>>9285672
this is a very good advice

>> No.9285707

>>9285663
The n-th column of a matrix is where it sends the n-th dimension.

A*B

Take the columns of B, break them up into the sum of their dimensions, see where A sends each dimension individually, sum them together to find the net effect, and you have the columns of the result AB.

>> No.9285800

>>9285452
>>9285466
Okay I asked a friend
It was an issue of units
Pounds are not mass, but are basically newtons. So that's why they don't need to multiply by g, they already have their dimensions correct

>> No.9285854

aerospace engineering senior brainlet here. How hard is it going to be getting an internship with sub 3.0 GPA?

>> No.9285936

I'm in EE, and I'm thinking of going into grad school to specialize in microprocessor systems, or semiconductor physics. Do either of these require any proficiency in digital systems, or combinatorial circuits?

>> No.9285953

What's a good book on signal processing?

>> No.9286010

>>9285953
The Analytical Theory of Heat

>> No.9286073

Is it possible for the joint probability distribution of two purely random variables to have a greater variance or standard deviation than either one of the distributions its composed of?

>> No.9286078

How hard would it be to get a master's in Mech E if I have a bachelors in EE?

>> No.9286132

>>9285936
I'm just a second year EE, but shouldn't microprocessor systems? That sounds like it would be a lot of digital/combinational/sequential circuits.

>> No.9286139

>>9285611
hey girl
I have a phd in spurtopologie

>> No.9286140

>>9286132
I'm more interested in the physics of things, but you're probably right.

>> No.9286149
File: 67 KB, 736x736, 1447542508836.jpg [View same] [iqdb] [saucenao] [google] [report]
9286149

>>9280927
One of the exercises in this textbook has the solution listed as (-∞,3) U (3,∞), while my answer was (-∞,3) U [4,∞)
Would a professor mark that wrong? I obviously see why the former is the superior choice, but they both describe the same set, right? It's like in programming (int i = 0;i<=10;i++) vs (int i = 0;i<11;i++), right?

>> No.9286174

Was there any follow up to that /sci/ anon who made the hypothesis of some steroid being able to cure cancer?

>> No.9286184

Please help a brainlet.
The volume of toothpaste has an average of 3.4 ounces with a standard deviation of 0.1. What is the probability that arandom sample of 9tubes is within 0.05 of the stated average?

>> No.9286186

>>9286139
sup my man, then these examples should be no-brainers for you tbqh, help out

>> No.9286220
File: 92 KB, 959x617, math +.png [View same] [iqdb] [saucenao] [google] [report]
9286220

>>9286149
Think about 3.5 and where it is.

>> No.9286223

>>9285953
http://4chan-science.wikia.com/wiki/Electrical_and_Electronics_Engineering#Signals_.26_Systems
http://4chan-science.wikia.com/wiki/Electrical_and_Electronics_Engineering#Digital_Signal_Processing

>> No.9286230

>>9286010
>implying anyone will get the start with the Greeks reference

>> No.9286234

>>9286220
tfw im a brainlet

>> No.9286251

TeX question:

What happened in this post:
>>9280933

and how did he fix it so quickly:
>>9280935

?

>> No.9286252

>>9282818
Check the sci wiki for books, I think they have a few recommended texts for power electronics

>> No.9286261

Need to pick my specialty for MS in EE soon, torn between photonics and optics or RF/mixed signal. Which might be better for the long term? I plan to get a PhD after my MS if that matters

>> No.9286263

>>9286251
\left| \frac{1}{n} - \frac{1}{m} \leq \left| \frac{1}{n} \right| + \left| \frac{1}{m} \right| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon
\left| \frac{1}{n} - \frac{1}{m} \right| \leq \left| \frac{1}{n} \right| + \left| \frac{1}{m} \right| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon

he forgot "\right|"

>> No.9286272

>>9286230
??

>> No.9286299

>>9286272
It's what started Fourier Analysis (which is used heavily in signal analysis):
https://en.wikipedia.org/wiki/Joseph_Fourier#The_Analytic_Theory_of_Heat

>> No.9286307

>>9286299
What is "Start with the greeks"?

>> No.9286335
File: 71 KB, 600x493, 1433662924954.jpg [View same] [iqdb] [saucenao] [google] [report]
9286335

>>9286307
The beginning (and end) of Philosophy.

>> No.9286337

>>9286335
I dunno, I could make a really good case for Blaise Pascal...

>> No.9286359
File: 66 KB, 662x580, 1509496616332.jpg [View same] [iqdb] [saucenao] [google] [report]
9286359

Would increasing the oxygen content in our atmosphere by 1% have any noticeable effect?

>> No.9286371

>>9286359
That picture is depressing pls delet

>> No.9286376

>>9280927
I've been trying to get a grip as to why I shy away from math even though I can understand the big picture type of concepts.

I have pinned it down to my mental processing speed in regards to reducing fractions like 135/168 to their lowest whole number form.

I know the steps to perform in my head, but I easily get lost.

Should I just cave and use a calculator for this, so that I could progress quickly through my math learning? Or should I try to memorize some type of fraction table?

>> No.9286495

What bands does Ammonia (dissolved and nitrated) absorb? Trying to build a coloriometer to monitor aquarium ammonia levels. Maybe there is a better solution? I already have a pH probe.

>> No.9286521

In epsilon delta proofs, why do we always start with the epsilon expression and then go <= ... <= ... until we arrive at the delta expression? Why does that work?

>> No.9286528

>>9286521
Because less than is transitive and therefore a partial order

>> No.9286539

>>9286528
huh

>> No.9286581

>>9285471
>this fucking retard again
here you brainlet, here: >>9280935 >>9280938

>> No.9286667

bro how the FUCK do i get LaTeX to work? and DONT just say read the wiki, i dont get it. there is just a guide on how to type with no links to whatever extension/plugin i need.

>> No.9286713

Can a diverse population of a single organism go for several generations of still having the exact same allele frequencies? If so, how?

>> No.9286752

>>9286667
open your .tex document and compile it to a pdf.

A VERY basic document looks like this:

\documentclass{article}

\begin{document}
Your text goes here.
\end{document}


Then use a compiler such as pdflatex to compile it to pdf.

If you are on windows you should probably download texworks/texstudio/texmaker. They should allow you to easily install a tex distribution.
On linux (or others) you should just run pdflatex mydocument.tex and look at the created pdf, of course you can also use an editor with built in pdf viewer.

>> No.9286916

>>9285663
Take the first row of your A matrix and multiply it by the first column of the B matrix, the sum of the products will give you your first entry for AB. Then multiply the first row from A into the second column of B and continue until you've run out of columns. This forms your first row for AB. To get your second row, take the second row from A ..... and I think you get the pattern.

>> No.9286926

>>9286667
Download Miktex
Download an editor like texstudio
Read the books:
http://4chan-science.wikia.com/wiki/Universal_Material#LaTeX

>> No.9286929

>>9285663
It's simple as the other guys told you.
To understand why it is defined that way see here:
https://www.youtube.com/watch?v=XkY2DOUCWMU

>> No.9287163

>>9286376
a-anyone?

>> No.9287191

>>9285193
Thanks man, will read when I sober up

>> No.9287312
File: 394 KB, 649x524, 5Ha0Ypd.png [View same] [iqdb] [saucenao] [google] [report]
9287312

i have to prove that this is true for my homework. M and N are sets that contain other sets.
[math](\cap M) \cup (\cap N) \subseteq \cap(M \cap N)[/math]
but to me this makes no sense at all so i've been thinking that my lecturer might have made a mistake when he created the assignment. does anyone else think this isn't true or am i just an idiot?

>> No.9287322

>>9287312
Not once in my life have I seen this notation.
What does ∩N mean?

>> No.9287323

>>9287312
Yeah it is wrong.
Consider M={A} and N={B} with A not empty or B not empty.
You get non-empty set subset of empty set.

>> No.9287326

>>9287323
oh and A cup B = empty

>> No.9287330

>>9287312
>sets that contain other sets
do you mean an intersection of an indexed family of sets?

>> No.9287333

>>9287322
it is an intersection of every set that is an object of N.
For example if N={A,B}, then ∩N is A∩B.

>> No.9287347

>>9287333
Thank you. Never saw it before.

>> No.9287384

>came from one of the worst school districts for math in my state
>never cared about studying in high school
>played video games from 4 pm to 1030 pm every day
>never took precalculus in high school

>now a junior in college
>C in precalculus
>C in calc 1
>terrified to take calc 2

Have been held back by math my entire life. I fucking suck at algebra mates. That’s all there is too it. In calc 1 I was phenomenal at integrating and deriving equations but I couldn’t get correct answers because I couldn’t simplify complex algebraic expressions. The only test I did well in for precalculus was logarithms. How can I truly fucking finally get good at algebra and beat this monster that has been on my back since 7th grade?

>> No.9287386

>>9287384
It's really nothing to learn precalculus.
Just take a book and read it for 1-2 weeks and you'll be done with it.

>> No.9287404

>>9287386
Any book you recommend?

>> No.9287418

>>9287404
Dunno.
check math.stackexchange.com

>> No.9287427
File: 206 KB, 1600x1201, joker-prison-close-there-s-a-secret-hidden-behind-heath-ledger-s-joker.jpg [View same] [iqdb] [saucenao] [google] [report]
9287427

Why do mathematicians assume infinity even exists. Literally nothing we know of is infinite in the real world. Also geometry is bullshit too. Why the hell are we using points and lines to build more complex geometric shapes. Points and lines don't exist in the real world either.

Someone please convince me math isn't a bunch of dumb rules some fag from ancient greece made up.

>> No.9287459

I get the idea of prime implicants in boolean algebra, but I don't quite get which terms are the essential prime implicants.

>> No.9287599

>>9287384
unironically make an account on khan academy

>> No.9287606

>>9287427

maybe the only reason you believe in a real world is that you left out the infinite your real world can't see?

>> No.9287700

>>9287323
kinda late but yeah, that's what i've been thinking

>> No.9287703
File: 56 KB, 514x432, 1498992183284.jpg [View same] [iqdb] [saucenao] [google] [report]
9287703

Which grad school rankings are the best? Looking for top schools in EE, specifically photonics/optics related

>> No.9287725

>>9286713
yes, if those allele combinations that aggrandize are regularly thinned out by environmental factors or even by the colony itself

>> No.9287729

>>9286359
yes

>> No.9287738

>>9285611
If I'm really gonna translate THAT for you, I'm afraid I'm going ask you to pay for that.

>> No.9287789

With the word "Deleted", what is the probability that the two Ds in a permutation of the word will be separate?
I got 300/420. Is that right?

>> No.9287824

>>9287789
Yes.
[math] 1-\frac{6!}{3!1!1!1!}/\frac{7!}{3!2!1!1!} = 5/7 [/math]

>> No.9287842
File: 55 KB, 619x247, Untitled.png [View same] [iqdb] [saucenao] [google] [report]
9287842

Hello I am struggling with this classic mechanics question, The first part I am fine with.

ai) a=F/m so a=4/3ms^(-2)
ii)W=1/2mv^2 so v=sqrt(2W/m)=sqrt(2*96/18)=4sqrt(2/3)ms^-1
b) P_initial=F*v=24*0=0W
p_final=24*4sqrt(2/3)=96sqrt(2/3)W

Now for part c I am getting different results of time pulling due to the velocity changing, how do I go about calculating this? or have I completely fucked up as I go along now?
Thanks in advance.

>> No.9287895

>>9287842
[math] P = F \cdot \frac{S}{T},\space 96 = 24 \cdot \frac{4}{T}, \space T = 1 \space s [/math]

I didn't check your numbers, though. But unless you got a multiplication or a square root wrong, everything seems fine.

>> No.9287912

>>9287895
I missed the square root there, it's actually 96 sqrt(2/3). So the answer is 1,225 s.

>> No.9287959

plz, how do you prove that lim cos(n) ( or sin(n) ) when n goes to infinity is 0 ( in the distrubitions sens).

>> No.9287980
File: 23 KB, 600x484, helper.jpg [View same] [iqdb] [saucenao] [google] [report]
9287980

>>9287959
>( in the distrubitions sens)

>> No.9288007

>>9287912
But why would the velocity/acceleration time be different? I am getting root 6 for that as a result.

>> No.9288012
File: 45 KB, 821x869, 1510158230500.jpg [View same] [iqdb] [saucenao] [google] [report]
9288012

>>9287980
>distrubitions
no bully plz.

>> No.9288128
File: 2 KB, 134x23, heyho.png [View same] [iqdb] [saucenao] [google] [report]
9288128

Help this brainlet whos getting into ODEs out. Tried using integrating factors and now stuck at e^(1/2(-x^2))

>> No.9288233

>>9288128
looks like you picked one hell of an ugly ode

>> No.9288236
File: 15 KB, 642x194, Unbenannt.png [View same] [iqdb] [saucenao] [google] [report]
9288236

>>9288233
forgot pic

>> No.9288294

>>9288233
>>9288236
wow had no idea it was going to get this complicated, thanks for the help though

>> No.9288342
File: 95 KB, 564x401, equilibrium.png [View same] [iqdb] [saucenao] [google] [report]
9288342

What would be the approach to this question? I tried substituting the value of mew given to obtain the respective partial pressures but my Kp value is 6.649 instead of 6.049

>> No.9288364

>>9281815
Depends on what you consider an "algebraic manipulation". Consider f(n)=(-1)^n, and g(n)=1 if n is even, and -1 if n is odd. It also depends on what you consider to be an "expression". Consider that x^2-x=0 for all [math] x\in\mathbb{Z}/2\mathbb{Z}[/math].

>> No.9288451

What's the proper latex symbol that represents the relationship between [math]SO(3,1)^+[/math] and [math]x^2 + y^2 + z^2 - t^2[/math]

>> No.9288512

I belatedly realized, if one can do such a thing and revert to basic knowledge, that 'range' and 'domain' are essentially synonyms intentionally chosen as specific separate terms

>> No.9288519

How are you supposed to insert latex into a word doc? (i'm using libreoffice btw)

>> No.9288546

What is the "acceleration of acceleration"?
If acceleration is the change in velocity over time, then what's the term and the symbol for the change in acceleration over time?

>> No.9288576

>>9288546
That's jerk, which is acceleration applied over time. Acceleration is the instantaneous tangent (derivative) of jerk. Yank is jerk times mass.

>> No.9288588

>>9288576
These terms sound suspicious.
Are you yanking my chain, anon?
Please don't jerk me around like this.

>> No.9288592

>>9288576
>>9288546
>do my due diligence and look them up
>yank and jerk are real professional terms even though they sound like a three-year-old coined them
The world gets weirder every day, and I get slightly less stupid. Ever so very slightly.

>> No.9288598

>>9288576
>Acceleration is the instantaneous tangent (derivative) of jerk
other way around, also a derivative is the slope of the tangent, not the actual tangent itself

>> No.9288607

>>9288592
look up snap crackle and pop

>> No.9288655
File: 18 KB, 232x254, i guess.jpg [View same] [iqdb] [saucenao] [google] [report]
9288655

>>9288607
>look it up
>still not being fucked with
Science has gone too far.

>> No.9288731

If air pressure lowers with wind speed, then how come box lids only fly off if wind gets under them?
Surely having stationary air below and moving air above should lift the lids?

>> No.9288741
File: 272 KB, 600x600, __doremy_sweet_and_yakumo_yukari_touhou_drawn_by_kuroba_rapid__d1957fc2c6003338387fb4a418dde6c9.png [View same] [iqdb] [saucenao] [google] [report]
9288741

>>9284631
>single book
No.
>Banach algebras
Any functional analysis.
>Affine Hecke algebra
Any mathematical text on conformal field theory.
>Exterior algebra
Any differential topology book.
>Koszul algebra
Any homological algebra text.
>Poisson algebra
Any classical mechanics text.
>Weyl algebra
Any mathematical text on quantum field theory.
>>9284936
Let [math]X[/math] be a [math]G[/math]-space and denote the action of [math]G[/math] on [math]X[/math] by [math]g \mapsto (x \mapsto g(x))[/math], then the [math]G[/math]-quotient is the space of orbits [math]\mathcal{O}_x = \{y \in X\mid y = g(x) \}[/math].
>>9287959
Let [math]f\in\mathcal{S}_c[/math] be a test function and suppose [math]\phi[/math] satisfies the equation [math]\phi'' + \phi = 0[/math], then [eqn]\int_K dxf(x)\phi(x) = -\int_K dxf(x)\phi''(x) \rightarrow_{x \rightarrow nx} -\frac{1}{n} \int_K d(nx) f(nx)\phi''(nx) = -\frac{1}{n}\int_K dx f(x)\phi''(x) = -\frac{1}{n}\int_K dx f(x)\phi(x) \rightarrow 0[/eqn]
Remember that scalings of compact sets are still compact so [math]f_n(x)\equiv f(nx) \in \mathcal{S}_c[/math].

>> No.9288792

>>9288741
See global rule 13.

>> No.9288880

>>9287895
>>9287912
>>9288007
Ive gone back through and had a look and I still cant see why the time calculated through power and work is different to my acceleration velocity time? can anyone clarify why this it the case or at least check if I have fucked up in prior calculations?

>> No.9288946

>>9280930
Starting us off with an easy one.

How sequence a Cauchy sequence exactly ? Example : Prove [math]x_n=\frac{1}{n}[/math] is Cauchy

Definitely Write :
Writing the definition: [math]\forall \hspace{0.2cm}\epsilon > 0 \hspace{0.2cm} \exists \hspace{0.2cm} N \in \mathbb{N} \hspace{0.2cm} \text{such that} \hspace{0.2cm} \forall \hspace{0.2cm} m,n,\geq N, \hspace{0.2cm} |x_{n} - x_{m}| < \epsilon[/math]

You can write : [math]\left| \frac{1}{n} - \frac{1}{m}\right| <\epsilon[/math] ; But how do you generally proceed from there?

Sequence Complete.

>> No.9288951
File: 2 KB, 252x55, unknown.png [View same] [iqdb] [saucenao] [google] [report]
9288951

>>9280927
I can't solve this, I think I'm not cut out for math. Help.

>> No.9288962

>>9288951
solve what?

>> No.9288967

>>9288962
finding the minimum or maximum

>> No.9288971

>>9288967
wait the max.

>> No.9288981

>>9288967
>>9288971
How the fuck can you not find that? The problem is as simple as it gets.
Open your textbook and read the relevant chapter.

>> No.9289030

>>9288880
Post your calculations. It's hard to guess what you did.

>> No.9289121

Are there legal restrictions to making your own medications (in the US primarily but also generally)? Like, I imagine with an understanding of fundamental chemistry it's easy enough to make something that's already been made, tested, mass produced and sold.
There's obviously the risk of getting no-knock raided as a suspected illigal substance lab but people have gotten raided for less.
I ask because I take medication daily and it would be comforting to know that, should I run out of refills in a wierd situation (come to think of it, that's exactly what I'm dealing with right now...) or there's a disaster of some sort preventing pharmacy access, that I could just make a batch.
I'm not talking about controlled substances like testosterone or opiates, things like antidepressants and dermatoligy meds that require a perscription but aren't "monitored" or restricted.

>> No.9289194

>>9287427
>Why do mathematicians assume infinity even exists
Because math has nothing to do with reality.
You have to ask the physicists why they use mathematics which relies on infinities to describe the real world.

>Why the hell are we using points and lines to build more complex geometric shapes.
Physicists use them because they are a good approximation, if you are talking about the distance from NYC to LA and you are not using a point somewhere in the middle, then you had to describe the distance as m +-100km or something, which is absolutely retarded.

>Someone please convince me math isn't a bunch of dumb rules some fag from ancient greece made up.
That is easy, the "dumb rules" were made up by a German and a German/Isreali jew.

>> No.9289391

>>9289030
okay

ai) [math]F=m\cdot a[\math] so [math]a=\frac{F}{m}[\math] which ends up giving me an acceleration of [math]\frac{4}{3}ms^{-2}[\math]
Then [math]W=F\cdot s[\math] which gives work as 96.0J
ii) so assuming that W=E here its going to be [math]v=\sqrt{\frac{2W}{m}}[\math] which gives a speed of [math]4\sqrt{\frac{2}{3}}ms^{-1}[\math]

b) the power applied at the start is nothing right due to the initial velocity being 0 seeming it starts from rest so the ending power will be calculated as [math]P=F\cdot v[\math] which I get as [math]96\sqrt{\frac{2}{3}}W[\math]

c) This is where things get fucked up and the prior power calculation gives me the same value as you and my acceleration and velocity time gives me root 6, so where am I going wrong. Also sorry if my latex typing is shit, fingers crossed and I typed it correct, if not you will have to puzzle it out.

>> No.9289398

>>9289391
Cool I did backslashes like a retard, oh well.

>> No.9289516
File: 10 KB, 1331x80, Capture.png [View same] [iqdb] [saucenao] [google] [report]
9289516

I'm getting confused with the algebra of this problem.

I am able to break the hypothesis to:

(1-(-7)^(k+1)/4) + (2(-7)^(k+1)

at that point, I find a gcd with 4 and multiply to get

(1-(-7)^(k+1)/4) + (4*2(-7)^(k+1)/4)

I don't actually get how I would simplify that to equal

(1-(-7)^(k+2)/4)

>> No.9289570

Could someone explain to me what a force actually is? Is it just to describe a transfer of energy?

>> No.9289707

>>9289516
[math]
2\sum_{k = 0}^{n+1}(-7)^k =
[/math]
[math]
2(-7)^{n+1}+2\sum_{k = 0}^{n}(-7)^k =
[/math]
[math]
2(-7)^{n+1}+\frac{1-(-7)^{n+1}}{4} =
[/math]
[math]
\frac{8(-7)^{n+1}+1-(-7)^{n+1}}{4} =
[/math]
[math]
\frac{1+7(-7)^{n+1}}{4} = \frac{1-(-7)(-7)^{n+1}}{4}= \frac{1-(-7)^{n+2}}{4}
[/math]

>> No.9289713

>>9289707
beautiful. Thanks anon.

Thinking from a simplification standpoint and not an induction standpoint, I was unable to see the 8 turning into 1+7.

>> No.9290056

Is it trivial that [math][a,b)\cup=[a,b][/math]? [math][/math] being the singular point [math]b[/math]

>> No.9290067

How do you solve

1 / (2 + x) = 1/4

I get it, it's the most basic algebra.

>> No.9290094

>>9290056
pretty much. If you want to do it riugorously you have
[math] [a,b) \cup \{ b \} = \{x\in \mathbb{R}~|~a\leq x <b \}\cup \{ b \} = \\
\{x\in \mathbb{R}~|~((x\geq a) \wedge (x<b))\vee (x=b) \} = \\
\{x\in \mathbb{R}~|~((x\geq a) \vee (x=b)) \wedge ((x<b)\vee (x=b)) \} = \\
\{x\in \mathbb{R}~|~(x\geq a) \wedge(x\leq b) \} = [a,b] [/math]

>>9290067
multiply by (2+x) and 4 and you get 4 = 2+x, subtract 2 and you get x = 2

>> No.9290122

>>9290056
yeah
[math][a,b)\cup\{b\}=\{x\in R:a\leq x<b\}\cup\{x\in R:x=b\}=\{x\in R :a\leq x<b\vee x=b\}=[a,b][/math]

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