/sci/ - Science & Math » Thread #12244696

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integral

Anonymous Sun 18 Oct 07:28:35 2020 No.12244696 View Reply Original Report

Quoted By: >>12244707 >>12244723 >>12244757 >>12244797 >>12244919

alright /sci/, give me your best proof/explanation of this result

Anonymous

Anonymous Sun 18 Oct 2020 07:30:35 No.12244699 Report

Quoted By: >>12244789

rationals are measure zero on the real line

Anonymous

Anonymous Sun 18 Oct 2020 07:30:40 No.12244700 Report

Quoted By:

https://m.youtube.com/watch?v=cyW5z-M2yzw
Use the idea in this and think.

Anonymous

Anonymous Sun 18 Oct 2020 07:36:44 No.12244707 Report

Quoted By:

>>12244696
Irrationals are the closure of the rationals.

Anonymous

Anonymous Sun 18 Oct 2020 07:48:23 No.12244723 Report

Quoted By:

>>12244696
infinitely more irrationals than rationals

Anonymous

Anonymous Sun 18 Oct 2020 08:08:40 No.12244757 Report

Quoted By: >>12244770 >>12244774 >>12244789 >>12244793

>>12244696
Firstly it's obvious that
$\int_0^1 f(x) dx \leq \int_0^1 1 dx = 1$
The rationals are countable so let $q_0, q_1 , \ldots$ be an enumaration of the rationals and let $\varepsilon > 0$ then
$\int_0^1 f(x) dx = 1 - \int_0^1 (1-f(x)) dx \geq 1 - \sum_{k=0}^\infty \int_{q_k - \frac{\varepsilon}{2^{k+2}}}^{q_k + \frac{\varepsilon}{2^{k+2}}} 1 dx = 1 - \sum_{k=0}^\infty \frac{\varepsilon}{2^{k+1}} = 1 - \varepsilon$

So we have for all $\varepsilon > 0$ that
$1 - \varepsilon \leq \int_0^1 f(x) dx \leq 1$
This is only possible if
$\int_0^1 f(x) dx = 1$

Anonymous

Anonymous Sun 18 Oct 2020 08:22:43 No.12244770 Report

Quoted By: >>12244776

>>12244757
is this meaningful?

Anonymous

Anonymous Sun 18 Oct 2020 08:23:43 No.12244774 Report

Quoted By:

>>12244757
damn im impresed, thanks anon

Anonymous

Anonymous Sun 18 Oct 2020 08:24:33 No.12244776 Report

Quoted By: >>12244789 >>12244795

>>12244770
It's a very elegant proof.

Anonymous

Anonymous Sun 18 Oct 2020 08:31:39 No.12244789 Report

Quoted By: >>12244793

>>12244757
>>12244776
Why can't you just state >>12244699?

Anonymous

Anonymous Sun 18 Oct 2020 08:33:18 No.12244793 Report

Quoted By: >>12244805 >>12244833

>>12244789
Because that's very convincing at all, is it? What the fuck does "measurable" mean. I'm a fucking stupid engineer, and I need something like >>12244757 to demonstrate the proof to me.

Anonymous

Anonymous Sun 18 Oct 2020 08:35:42 No.12244795 Report

Quoted By: >>12244797

>>12244776
Of what?

Anonymous

Anonymous Sun 18 Oct 2020 08:37:24 No.12244797 Report

Quoted By: >>12244804

>>12244795
Of >>12244696 you mongoid

Anonymous

Anonymous Sun 18 Oct 2020 08:40:50 No.12244804 Report

Quoted By: >>12244827

>>12244797
is this meaningful?

Anonymous

Anonymous Sun 18 Oct 2020 08:41:03 No.12244805 Report

Quoted By: >>12244814

>>12244793
Sorry, I thought I was on /sci/ for some reason.

Anonymous

Anonymous Sun 18 Oct 2020 08:44:52 No.12244814 Report

Quoted By: >>12244833

>>12244805
????????

Anonymous

Anonymous Sun 18 Oct 2020 08:49:40 No.12244827 Report

Quoted By: >>12244839

>>12244804
oh my god what happened to /sci/? it's full of nutbars now. anyway guys thanks for the proof

Anonymous

Anonymous Sun 18 Oct 2020 08:52:10 No.12244833 Report

Quoted By: >>12244834

>>12244814
Holy shit you're dumb. The implication was that a student of science and math would not be so retarded as >>12244793

Anonymous

Anonymous Sun 18 Oct 2020 08:53:28 No.12244834 Report

Quoted By: >>12244840

>>12244833
That's me. What did I say that's retarded?

Anonymous

Anonymous Sun 18 Oct 2020 08:55:28 No.12244839 Report

Quoted By:

>>12244827
>can't answer it
yikes bro.
maybe talk to someone about getting schizophrenia medication.

Anonymous

Anonymous Sun 18 Oct 2020 08:56:48 No.12244840 Report

Quoted By: >>12244843

>>12244834
>Because that's very convincing at all, is it?
>What the fuck does "measurable" mean.
>I'm a fucking stupid engineer
None of it is retarded inherently. It is just that you are retarded.

Anonymous

Anonymous Sun 18 Oct 2020 08:58:12 No.12244843 Report

Quoted By: >>12244845

>>12244840
I'm actually smarter for being above shit as gay as analysis, I just thought that proof was slick.

Anonymous

Anonymous Sun 18 Oct 2020 08:59:41 No.12244845 Report

Quoted By: >>12244849

>>12244843
How is 8 lines of pure tedium more slick than a single line?

Anonymous

Anonymous Sun 18 Oct 2020 09:01:04 No.12244849 Report

Quoted By: >>12244857 >>12244861

>>12244845
Because I don't need to have previous knowledge of gay words like """"measurable""""" to be very confident in the result.

Anonymous

Anonymous Sun 18 Oct 2020 09:03:59 No.12244857 Report

Quoted By: >>12244860

>>12244849
Yeah, no point in developing new ideas and theories to make life easier.

Anonymous

Anonymous Sun 18 Oct 2020 09:05:18 No.12244860 Report

Quoted By: >>12244882

>>12244857
I'm not trying to shit on you too hard, anon, I'm just being extremely honest. I have legitimately no fucking idea what "measure zero" means. However, I was able to comprehend the longer proof.

Anonymous

Anonymous Sun 18 Oct 2020 09:06:10 No.12244861 Report

Quoted By: >>12244870

>>12244849
if you accept the sup axiom and the uncountability of the reals, then it's not a stretch to accept the idea of measure and sets of measure zero

Anonymous

Anonymous Sun 18 Oct 2020 09:10:21 No.12244870 Report

Quoted By: >>12244882 >>12244900

>>12244861
No, it isn't. But in order to accept those I had to go to wikipedia to find out what those words mean. Math is really boring desu.

Anonymous

Anonymous Sun 18 Oct 2020 09:15:16 No.12244882 Report

Quoted By:

>>12244860
same here, I'm only now learning measure theory, hence why I made the thread. obviously in a couple years I'll be right alongside him using the one-liner proof but until then I like the riemann intergral proof

>>12244870
>Math is really boring desu
well if you're that same engineer from the earlier post, then I understand why you don't like abstractions, but I think it's important since using only engineering logic you can make a ton of mistakes (convergence, swapping order of summation and integration, etc.)

Anonymous

Anonymous Sun 18 Oct 2020 09:24:40 No.12244900 Report

Quoted By:

>>12244870
I'm an EE and I went into math mid undergrad because I found it to be compelling.
A class into analysis or algebra is thousands upon thousands of times more interesting and fulfilling than any class in EE. Start with something simple like Abbott's understanding analysis and see for yourself.

El Arcón

El Arcón Sun 18 Oct 2020 09:35:37 No.12244919 Report

Quoted By:

>>12244696
>alright /sci/, give me your best proof/explanation of this result
If f(x)=1 over the entire integration region, then the integral is equal to one. Now remove the area associated with x=0.5. This has zero area so the integral is still equal to one for
$f(x)=\begin{cases} 0&x=0.5\\1&\text{otherwise} \end{cases} .$
The rationals are countably infinite so infinity countable zero areas removed from the integral of f(x)=1 will result in A = 1-0 =1. This is the OP case
$f(x)=\begin{cases} 0&x \text{ rational}\\1&\text{otherwise} \end{cases} .$

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