/sci/ - Science & Math

[Anonymous No.6710371]

/sci/, could understanding functions be this fucking easy?

I'm talking to my friend about hardcore math (Fibonacci sequence, Mandelbrot set, Julia sets, etc.). I'm not even a college grad. However, I'm trying to comprehend Mandelbrot Set [https://en.wikipedia.org/wiki/Mandelbrot_set] when I got stuck on "infinity when a particular mathematical operation is iterated on it"

More precisely, the link to "iterated".

This led me to the wikipedia "iterated function" page here: https://en.wikipedia.org/wiki/Iterated_function

Then I realized, shit, I don't even know what "function" means in math.

My buddy sends me the wiki link to "Function (in mathematics): https://en.wikipedia.org/wiki/Function_%28mathematics%29

See attached pic.

My question to you all is this.... so it could be anyfuckingthing in the middle as the process??
If function means: input > process > output
then wouldn't function be like 3 > (3+2) > 5
amirite?

Because im starting to think
if its just
input > process > output
thats fucking easy as shit
like
that would mean
input > fibonacci equation > output
or
input > whatever equation > output
so we just need confirmation of this. From /sci/

...because then I'll join mensa, shit. How is that even hard?
(sorry for the strange text arrangement above. Literally posted that from our gchat.)

[mtp !EKFQOBEsKo No.6710386]

yes but for implying that mensa has anything to do with mathematics heres an obligatory GET THE FUCK OUT

[Anonymous No.6710387]

4/10 wrote out bewildered response before deleting it

[Anonymous No.6710393]

>>6710386
but seriously, is the term FUNCTION in math summed up as "input > process > output" every time? Show your intellect.

[Anonymous No.6710399]

Remember, simplicity is one of the three virtues of math.

A function takes an element from a set A and maps it to another element in another set B. If A and B are the collections of blue and yellow points respectively, then f is basically the collection of all arrows between them. The only strict requirement is that each point in A is attached to one arrow.

[Anonymous No.6710404]

>>6710399
winrar

[Anonymous No.6710407]

>>6710371
>hardcore math
>functions
>mensa
This kills the /sci/

[Anonymous No.6710408]

>>6710399
OP here, so basically what you're consequently saying is that all math acts this way because all numbers are fixed points of values across the number line. Thus, each "function" always will have a corresponding precise output. Yes/no?

[Anonymous No.6710411]

>>6710399
How do they come up with this terminology?

[Anonymous No.6710413]

>>6710408
No, not all math can be described through functions. And a function doesn't even have to necessarily manipulate numbers.

[Anonymous No.6710417]

>>6710408
And no, the domain of the function does not have to include all numbers. Some inputs will have no defined output.

[Anonymous No.6710422]

>>6710408
No, a function is just a special type of relation (the one element of the domain maps to only one element of the image).

And Functions don't need to operate on numbers, they can operate on anything, including other functions, so you will have functions that map functions to functions for example.

[Anonymous No.6710429]

>>6710413
Ok, if not all math can be described through functions, give at least 1 example to back this statement up then please.

[Anonymous No.6710430]

>>6710411
What terminology?

[Anonymous No.6710435]

>>6710417
>the domain of the function does not have to include all numbers. Some inputs will have no defined output.

What do you mean by "domain"?

What is one example where an input has no defined output?

[Anonymous No.6710439]

>>6710422

What? Please give us an example of where a function maps to functions to functions.

[Anonymous No.6710441]

>>6710429
Simple, any relationship in which an element of the domain goes to multiple elements in the range is not a function.

[Anonymous No.6710443]

>>6710429
Graph theory

[Anonymous No.6710445]

>>6710435
The domain is the group of inputs. The range is the group of outputs. The function is the collection of relationships that go from one input to one output.

[Anonymous No.6710448]

>>6710439
Galois groups

[Anonymous No.6710459]

>>6710435
>What is one example where an input has no defined output?
x=0 is not in the domain of the function f(x) = 1/x

[Anonymous No.6710461]

>>6710441
and by "Range" I'm assuming you mean group B here?? >>6710399

[Anonymous No.6710471]

I gotta pee but you fuckers better keep talking about this by the time I get back.

[Anonymous No.6710474]

>>6710461
Almost. The range is the set of points in group B that have an arrow going to it. The domain only includes inputs that have a defined output. The range only includes values that have a defined input or inputs.

Example: f(x)=2 is not in the range of sin(x)

[Anonymous No.6710486]

>>6710443
Graph theory can be described as mapping subsets of a set of elements to either "not connected" or "connected", or something else for directed/labeled graphs, etc.

Checkmate atheists

[Anonymous No.6710492]

Looks like it's gonna be a long ride until OP gets to double integrals.

[Anonymous No.6710493]

>>6710474
Note however that f(x)=2 is in the codomain of sin(x). Range is a subset of the codomain.

[Anonymous No.6710496]

>>6710411
inject = throw in
surject = throw over
biject = throw both

[Anonymous No.6710500]

>>6710493
Depends, some people use range as codomain, I was using it as image.

[Anonymous No.6710515]

>>6710407

[Anonymous No.6710517]

>>6710486
There is a one-line definition of all possible graphs:
Firstly, take the system consisting of two objects E and V and two arrows d, c from E to V.
Now map this system onto two sets FE, FB (the sets can have any cardinality) and two functions Ff, Fg.
If you view FE as the names of graph edges and FV as the names of graph vertices, then you can read the functions Fd and Fc as specifying the domain and co-domain of each edge. This covers all graphs in existence.

It is the third example here:
http://en.wikipedia.org/wiki/Functor_category#Examples

[Anonymous No.6710520]

(the one line definition being "it's the presheaf category from the cat with only two parallel arrows")

[Anonymous No.6710526]

>>6710371
yeah man, you were supposed to learn that in algebra 1. thats the whole point of that class.

idk why youre having trouble with the mandlebrot set either. its basically just the same thing as when you call a funciton like

mandlebrot(int c){
mandlebrot(c^2 + c);
}

but you are allowed to see the results inside the call

[Anonymous No.6710531]

>>6710526
since when is algebra about functions?

[Anonymous No.6710554]

>>6710531
http://en.wikipedia.org/wiki/Elementary_algebra

crtl+f function

you cannot have algebra, you cannot have relational math, you cannot have tables, without functions.

i'm posting this in case you're seious, but i choose to believe this is a joke.

[Anonymous No.6710557]

>>6710554
beyond the operations, ive not seen algebra to involve functions. how is group theory about functions?

[Anonymous No.6710562]

>>6710557
>>6710531

Hey anon, which math have you taken up to this point?

You really sound like you know what you are talking about.

[Anonymous No.6710573]

>>6710492
>double integrals
OP here, back from taking a piss and walking to grab a hot dog from 7-11. Ok WHAT are double integrals???

[Anonymous No.6710579]

>>6710531
https://www.youtube.com/watch?v=shEk8sz1oOw

fundamental theorem of algebra:

all odd-powered polynomials have a least 1 real root.

all even funcitons may not have any real roots, but if it has one real root, then it is either repeated or a second will be found.

this is shown by the fact that odd funcitons have the opposite ends' behavior (going +/-) and even functions have same end behavior (going +/-)

if that isn't a proof that algebra is about funcitons then i don't know what is.

[Anonymous No.6710585]

>>6710573
First integral: find the area under the cross section

Second integral: find the volume under the surface

Third Integral: find the Hyperspace under the volume

it goes on like that in this pattern

Integral n:
find the n+1 Dimensional space under the n Dimensional space

[Anonymous No.6710595]

>>6710579
yeah, the fundamental theorem of algebra is not a theorem in algebra. its called so for historic reasons.

[Anonymous No.6710605]

>>6710595
I definitely learned that theorem in my Aglebra two class years ago.

What was it then?

[Anonymous No.6710606]

>>6710585
>the cross section
What cross section?

>find the volume under the surface
Why under and not over or around?

>Hyperspace
huh?

[Anonymous No.6710607]

>>6710605
http://matheducators.stackexchange.com/a/1590/1895

[Anonymous No.6710613]

>>6710557

Given a set G and a function f such that
f:GxG -> G

Then (G, f) is a group if the following hold for all g in G:

f(g1,g2) is in G
f(f(g1,g2),g3) = f(g1,f(g2,g3)
f(e,g1) = f(g1,e) = g1
f(g1,g-1) = e

Or if you don't want a facetious answer, you study particular functions in group theory, ones that take one group into another such that it preserves group structure.

[Anonymous No.6710619]

>>6710607
ok so what you're saying is that I learned the theorem and what it says many years before i could read the proof for it?

i find that pretty plausible and realistic. regardless, i refuse to accept that algebra is not about functions when so much of what you do in that class is literally functions and identifying their characteristics.

>>6710606
read a textbook before posting seriously

[Anonymous No.6710620]

>>6710606
integrals go from lower dimension to higher dimensions, and what he wrote is both the intuition behind the first integrals and the visualization usually used to introduce them. when you integrate a one-dimensional real function, you get the size of the area between the line described by the function and the coordinate line. when you integrate over both dimensions of a two-dimensional real function (i mean a function that takes two inputs), you get the space between the area over which you integrated and the coordinate plane. when you integrate over all dimensions of a three-dimensional real function, you get the hyperspace (meaning 4-dimensional space) enclosed by the space over which you integrated and the coordinate space. and so on for higher dimensions.


>>6710613
>Or if you don't want a facetious answer
i already mentioned the operators, so thats obvious. also, everyone know that homomorphisms are important in group theory. but you dont study the functions, you study the groups via the functions. in analysis, you use sequences and series to figure out functions.

[Anonymous No.6710621]

>>6710517
please draw this for us in MS paint. Can't really visualize this

[Anonymous No.6710624]

>>6710620
>i already mentioned the operators

You expected me to read your post?

[Anonymous No.6710626]

>>6710606
By cross section, he mines the line from a to b. An integral is the area bounded by the function, the x-axis, and the lines x=a and x=b.

Double, triple, quadruple, etc. integrals generalize this to higher dimensions.

[Anonymous No.6710627]

>>6710619
what text book? Why can't you explain? The way you explained it makes it sound like someone worked on the cross section of a banana and then for it the dentist the shit is bananas.

[Anonymous No.6710628]

>>6710624
considering its 16 words, seemed like the reasonable thing to expect.

>>6710619
>regardless, i refuse to accept that algebra is not about functions when so much of what you do in that class is literally functions and identifying their characteristics
what class is that? high school sophomore year?

[Anonymous No.6710630]

>>6710628
>what class is that? high school sophomore year?

In defense of this dude, he posted a link to the Elementary Algebra, which is high school algebra

[Anonymous No.6710633]

>>6710626
THANK YOU!!!!!

Woulda never understood this if that prior poster was the teacher.

[Anonymous No.6710636]

>>6710626
Here's a double integral picture for comparison. You'll have to use your imagination for anything beyond that.

Just as the single integral represents a two dimensional area, the double integral represents a three dimensional volume.

[Anonymous No.6710637]

>>6710628
yes. that was the year i took algebra 2. that is what we did. we also went over imaginary numbers, series, matricies, and discontinuities. after that i took trig into precalc and then calc.

headed into calc 2 because i good ap scores. i thought we were talking about the high school class. if we were talking about the college class algebra 2, which is something else entirely, then i admit i havent come that far yet.

[Anonymous No.6710643]

>>6710637
seeing how im not american, i really have no idea what algebra 1, 2, calculus 1 2 3 and whatever numbered thing those classes are called in the US cover. but the field algebra is not about solving equations, if thats what you mean. it enables you to, but thats not what its about.

[Anonymous No.6710644]

>>6710461
Are you some sort of retard? Domain and range are calc 1 day 1. You're not OP right?

[Anonymous No.6710647]

>>6710399
>Remember, simplicity is one of the three virtues of math.
what are the other two, abstraction and brevity?

[Anonymous No.6710649]

>>6710647
yes, I was wondering same. Please tell

[Anonymous No.6710650]

>>6710643
oh ok. yeah my calc teacher last year was russian, he always talked about the way we number our math classes being weird. cool teacher, he always taught through generalized proofs.

he said that he would have taught all the classes in a different order and would have added/taken out parts of each curriculum.

>>6710644
i think it was earlier than that...

[Anonymous No.6710655]

>>6710647
>>6710649
Clarity and generality

[Anonymous No.6710657]

>>6710371
Avoid terms like "fucking easy" and "hardcore" when trying to learn something in order to avoid looking like a fool.

[Anonymous No.6710658]

>>6710655
werent those popularized as virtues of programming? obviously they apply to math as well.

[Anonymous No.6710662]

>>6710657
OP here. But I am a fool when it comes to math. However, getting laid I got a PHD in that. Come over to my dimension and see how you do.

[Anonymous No.6710671]

>>6710662
I may or may not be able to go into you dimension depending on the function between us. But I get the feeling it's something like the identity matrix.

[Anonymous No.6710673]

>>6710662
>posting on a math/science forum
>to brag about getting laid
>thinking that impresses anyone
>believing scientists don't get laid


you're a hot streat buddy, I gotta say.

[Anonymous No.6710677]

>>6710671
oooooooh

sick burn bro
(i got it and actually laughed)

[Anonymous No.6710681]

>>6710671
no idea what that was supposed to mean. "go into your dimension"?

[Anonymous No.6710688]

>>6710681

OP's dimension.

[Anonymous No.6710691]

>>6710681
Think vector space.

[Anonymous No.6710694]

>>6710691
...no, really? hes talking about matrices and im supposed to think of a vector space? still dont get it.

[Anonymous No.6710698]

>>6710691
>alternate mathematical description where its easy to get laid frequently

sounds pretty theoretical

[Anonymous No.6710707]

>>6710694
I am that guy. Here's some hints:
>vector spaces have dimensions
>linear functions have matrix representations
>functions can take in single values, vectors and, like your mother, a large sum of dicks

[Anonymous No.6710709]

>>6710707
Yep, confirmed neckbeard.

[Anonymous No.6710713]

>>6710709
You'll take that class your first year at uni, kid.

[Anonymous No.6710716]

>>6710713
I retract all my prior negative statements. Much respect to those who are comprehending these complex math concepts.

[Anonymous No.6710724]

>6710707
yeah but a joke isnt just throwing semi-relevant words together, they gotta make sense as well.

[Anonymous No.6710729]

>>6710707
yeah but a joke isnt just throwing semi-relevant words together, they gotta make sense as well

[Anonymous No.6710753]

>>6710729
But it does make sense. I'll explain it to you.

First of all, the intent of the joke was to be related to what he just said, as well as the purpose of the thread, while simultaneously establishing myself as one who is well-versed in the language of snu-snu. Moving on.

His dimension is one where he gets a lot of poontang. In linear algebra, a dimension is related to vector spaces. Depending on the function/matrix transforming vectors from one vector space to another, we can input a 10-dimensional vector and output a 5-dimensional vector. The first statement I make in the >>6710671 post was a play on words, namely "dimension", and it is from there that the humor arises.

By saying that it was "something like the identity" between us, I implied that I am already in his dimension, but perhaps a different location. The words in my post are not "semi-relevant"; they form a coherent joke that I expected to elicit at least a chuckle from someone (which it did). If you mean that my post and his are "semi-relevant", well, that is the nature of the joke.

[Anonymous No.6710774]

>>6710753
yeah thats entirely unfunny.

[Anonymous No.6710778]

>>6710774
Well I certainly hope you didn't find a long-winded explanation for a joke you didn't get "funny".

[Anonymous No.6710791]

>>6710439
>what is a functor???
>what is a linear map between operator algebras???
>what is a differential operator

[Anonymous No.6710794]

>>6710557
Algebra is all about morphisms. Specifically, group theory is the study of group homomorphisms.

[Anonymous No.6710796]

>>6710694
Matrices are nothing but linear operators on finite-dimensional vector spaces.

[Anonymous No.6710800]

>>6710794
in my experience, all you do is assume you have a morphism and then study what you can tell about the groups. the functions are of very limited interest and are only a means to an end, as opposed to analysis where you study the functions themselves.

>>6710796
my post was a rather long "duh".

[Anonymous No.6710812]

post my legitimate freshman algebra question on /sci/

>thread deleted

Raving mensa madman posts high school math question on /sci/

>endless answers and people helping him

where did ii go wrong? is it his enthusiasm?

[Anonymous No.6710813]

>>6710800
>in my experience, all you do is assume you have a morphism and then study what you can tell about the groups
>in my experience
Your experience is very limited then. After babby's first college abstract algebra class, algebra becomes very heavily categorical. Normally I'd give a few examples but you really should try to expand your view of algebra on your own.

[Anonymous No.6710814]

>>6710812
I'm sorry man. Post it here, and we'll see if we can help you. If it means anything, I've been mocking the OP this whole time.

[Anonymous No.6710819]

Input. Output. Transition is called a function. Close thread.

[Anonymous No.6710822]

>>6710819
#include <iostream>
using namespace std;

int main() {
cout << "Input. Output.Transition is called a function.";
return 0;
}

How's it feel to be a simple program?

[Anonymous No.6710824]

>>6710822
Feels better than writing a simple program.

[Anonymous No.6710984]

>>6710636
Except you can use double integrals for finding areas of 2D regions and triple integrals for finding volumes of 3D shapes.

[Anonymous No.6711048]

>>6710984
over nonconstant functions?

[Anonymous No.6711061]

>>6711048
Yes.

[Anonymous No.6711080]

>>6711061
got a link or something? have not come across something like that yet

[Anonymous No.6711089]

>>6711080
Let <span class="math">C: x^2 + y^2 \leq r^2[/spoiler] be a circle.
Say you want the area of this circle, solving the equation for y yields: <span class="math"> y = \pm \sqrt{r^2 - x^2}[/spoiler]
Then you end up with the region: <span class="math">R = \{ (x,y) \in \mathbb{R}^2 : -r \leq x \leq r,\; -\sqrt{r^2 - x^2} \leq y \leq \sqrt{r^2 - x^2}\}[/spoiler]
Which simply means your radius varies form -r to r, and y varies between those functions.
<span class="math">A = \int\int_R dydx = \pi r^2[/spoiler]

or in polar coordinates:
<span class="math">R = \{ 0 \leq R \leq r, 0 \leq \theta \leq 2 \pi\} \rightarrow A = \int_0^{2\pi}\int_{0}^{r} rdrd\theta = \pi r^2[/spoiler]

[Anonymous No.6711110]

>>6711080
Do you not remember in Calc 1 (possibly 2, I took them at my highschool and it was never distinguished when we transitioned) when you found the arc length of curves with a single integral?

<div class="math">
\displaystyle
A = \int^a_b\! \sqrt{1 + \frac{\mathrm{d}y}{\mathrm{d}x}\,\mathrm{d}x
</div>
I believe was the Arclength formula.

[Anonymous No.6711112]

>>6711110
<div class="math">
\displaystyle A = \int^a_b\! \sqrt{1 + \frac{\mathrm{d}y}{\mathrm{d}x}\,\mathrm{d}x
</div>
fuck

[Anonymous No.6711116]

>>6711110
<div class="math">
\displaystyle A = \int^a_b\! \sqrt{1 + \frac{\mathrm{d}y}{\mathrm{d}x}}\,\mathrm{d}x
</div>

[Anonymous No.6711180]

>>6711089
Or for a sphere:
<span class="math">S: x^2 + y^2 + z^2 \leq r^2[/spoiler]
Variation on z: <span class="math">\{z \in \mathbb{R}: -\sqrt{r^2 - y^2 - x^2} \leq z \leq \sqrt{r^2 - y^2 - x^2} \}[/spoiler]
on y: <span class="math">\{y \in \mathbb{R}: -\sqrt{r^2 - x^2} \leq y \leq \sqrt{r^2 - x^2}\}[/spoiler]
and on x: <span class="math">\{x \in \mathbb{R}: -r \leq x \leq r\}[/spoiler]
<span class="math">V = \int\int\int_R dzdydx = \frac{4}{3}\pi r^3[/spoiler]

And on spherical coordinates:
<span class="math">R = \{0 \leq \rho \leq r,\; 0 \leq \phi \leq \pi,\; 0 \leq \theta \leq 2 \pi\},\; |\,J| = \rho^2 \sin(\phi) \rightarrow V = \int\int\int_R \rho^2 \sin(\phi) dr d\phi d\theta = \frac{4}{3}\pi r^3[/spoiler]

Or for a 4D sphere:
<span class="math">S: x^2 + y^2 + z^2 + t^2 \leq r^2[/spoiler] with <span class="math">(x,y,z,t) \in \mathbb{R}^4[/spoiler]
It's the same thing as for the sphere above, except you add the variation for t:
<span class="math">-\sqrt{r^2 - z^2 - y^2 - x^2} \leq t \leq \sqrt{r^2 - z^2 - y^2 - x^2}[/spoiler]

And <span class="math">V = \int\int\int\int_R dtdzdydx = \frac{\pi^2 r^4}{2} [/spoiler]

Or in spherical coordinates:
<span class="math">R = \left\{0 \leq \rho \leq r,\; 0 \leq \phi \leq \pi,\; 0 \leq \varphi \leq \pi,\; 0 \leq \theta \leq 2 \pi \right\},\; |\,J| = \rho^3 \sin^2(\varphi) \sin(\phi) \rightarrow V = \int\int\int\int_R \rho^3 \sin^2(\varphi) \sin(\phi) dr d\varphi d\phi d\theta = \frac{\pi^2 r^4}{2}[/spoiler]

[Anonymous No.6711186]

>>6711180
how do i latex here, lemme try $f(x) > 0 \rightarrow \varphi \circ \theta$

[Anonymous No.6711188]

Have you read your SICP today?

https://www.youtube.com/watch?v=2Op3QLzMgSY

http://deptinfo.unice.fr/~roy/sicp.pdf

http://www.call-cc.org/

[Anonymous No.6711192]

>>6711186

[Anonymous No.6711200]

>>6710393

Only with one stipulation:

Given a relation f, f can be a function if and only if for all x over which f is defined, if f(x)=a and f(x)=b then a=b.

Essentially, each input leads to exactly one output.

[Anonymous No.6711202]

>>6711192
cheers

[Anonymous No.6711205]

>>6711200
personally, i dont like the understanding that functions "process" something. that suggests that something happens, some sort of computation, but thats not what functions do. they actually dont do anything, theyre just there, static and unmoving

[Anonymous No.6711217]

>>6711202
I'd recommend grabbing one of the latex preview scripts for /sci/ on userscripts.org.

It's priceless to make sure you're not fucking up.

[Anonymous No.6711223]

>>6710371
>I'm talking to my friend about hardcore math
>Then I realized, shit, I don't even know what "function" means in math

And this is why CS majors DO NOT BELONG HERE

>so we just need confirmation of this. From /sci/

No, a function is a collection of (input, output) with the understanding that each input only appears once.

[Anonymous No.6711228]

>>6711223
shut the fuck up already

[Anonymous No.6711239]

>>6711223
>CS majors DO NOT BELONG HERE

Speak for yourself. I'm math and CS double major, fuck off

[Anonymous No.6711253]

>>6710371
Given two sets X and Y, the Cartesian product <span class="math">X \times Y[/spoiler] is the set of all ordered pairs (x, y) with <span class="math">x \in X[/spoiler] (i.e., x is an element of X) and <span class="math">y \in Y[/spoiler].

A function f from X to Y, denoted <span class="math">f: X \to Y[/spoiler], is a subset <span class="math">f \subseteq X \times Y[/spoiler] such that, for each <span class="math">x \in X[/spoiler], there is exactly one <span class="math">y \in Y[/spoiler] such that <span class="math">(x, y) \in f[/spoiler]. We write <span class="math">f(x)[/spoiler] for the unique element of Y such that <span class="math">(x, y) \in f[/spoiler]. The set X is called the "domain", and Y is called the "codomain".

That's what a function is. You can think of it as being like a "process" if you want, but that's just intuition; formally, it's just a set of ordered pairs with the above property. A function doesn't have to be defined by an equation or an algorithm; in fact, some functions (such as "busy beaver" functions, which arise in computer science) *cannot* be computed by any algorithm.

Given a function <span class="math">f: X \to X[/spoiler] whose domain and codomain are the same, we can "iterate" f by applying it repeatedly: for example, <span class="math">f^2: X \to X[/spoiler] is the function defined by <span class="math">f^2(x) = f(f(x))[/spoiler] for all <span class="math">x \in X[/spoiler]. Likewise, for any positive integer n, <span class="math">f^{n+1}: X \to X[/spoiler] is defined by <span class="math">f^{n+1}(x) = f^n(f(x))[/spoiler] for all <span class="math">x \in X[/spoiler]. Repeatedly applying a function to an element <span class="math">x \in X[/spoiler] gives a sequence of points <span class="math">x, f(x), f(f(x)), f(f(f(x))), ...[/spoiler], called the "orbit" of x.

For example, let <span class="math">\mathbf{C}[/spoiler] be the set of complex numbers, and for <span class="math">c \in \mathbf{C}[/spoiler], let <span class="math">f_c: \mathbf{C} \to \mathbf{C}[/spoiler] be given by <span class="math">f_c(z) = z^2 + c[/spoiler] for all z. We can look at the orbit of 0, which is the sequence <span class="math">0, f_c(0) = c, f_c(c) = c^2 + c, f_c(f_c(c)), ...[/spoiler]. The Mandelbrot set is the set of all <span class="math">c \in \mathbf{C}[/spoiler] for which this orbit is bounded in the complex plane.

[Anonymous No.6711258]

>>6710371
>hardcore math (Fibonacci sequence, Mandelbrot set, Julia sets, etc
please fucking go.

[Anonymous No.6711290]

>>6710557
Historically, group theory came about from studying permutation groups and they obviously play a huge role in anything group theoretic that you happen to do.

[Anonymous No.6711293]

> Haskell Curry

[Anonymous No.6711415]

>>6710399
I feel like a bunch of definitions would really be cleared up if we started with maps first and then total functions as a special case. Same types of diagrams and everything.

I mean look at how many people don't know the difference between the range/image and the co-domain. Worse the number of people who don't know the difference between the pre-image and the domain.

[Anonymous No.6711425]

>>6711253
Actually, your definition is for a total function. There are two types of functions, total and partial. A total function is defined for every element in the Domain, a partial function is not (the pre-image is a proper subset of the domain). Both are functions.

A map, <span class="math">M[/spoiler], from <span class="math">X[/spoiler] to <span class="math">Y[/spoiler] is just a set <span class="math">M[/spoiler] such that M is a subset of <span class="math">X \times Y[/spoiler]. Then a function F is a special type of map with the added requirement that if <span class="math">(x_1, y) \in F[/spoiler] and <span class="math">(x_2, y) \in F[/spoiler] then <span class="math">x_1 = x_2[/spoiler].

The definition of injective is analogous to the definition of function
and the definition of surjective is analogous to the definition of total map.
Similarly total function is analogous to bijective.

I'm really phoning in this explanation but I hope it's understandable.

[Anonymous No.6711426]

>>6710621
If you're still there and interested, I might do it in the break. Just because I was surprised how it works out the first time I learned it.
If you don't know other functor categories, then it will not be so interesting and I'd rather spare my time.

>>6711293
?

PS: I knew this thread was gonna be full of "a function is just a prescription from input to output", and my intuitive response was to talk about intentionally different functions. The "pairs of data" definition of a function isn't even so old.

http://en.wikipedia.org/wiki/Extensionality#Example

[Anonymous No.6711433]

>>6711425
Except none of that is part of standard definitions. A map is generally a function or morphism, depending on context. Partial functions aren't functions and something is a total function iff it is a function.

[Anonymous No.6711440]

>>6711433
It's standard at least at my university. One just notes that some authors use the convention that everything is a total function because that's the only type of structure they actually deal with.

[Anonymous No.6711448]

>>6711440
>It's standard at least at my university
Except that in no way implies that it's standard.

[Anonymous No.6711451]

>>6711448
No, but what is your justification for a standard?

[Anonymous No.6711458]

>>6710621
>>6711426

[Anonymous No.6711459]

>>6710371
>"hardcore math"
do you even category theory?
functions that accept functions with only one argument (i.e. curried) are way more interesting than single parameter functions.

[Anonymous No.6711460]

>>6710621
>>6711426
The mapping from the bottom to the middle picture specifies a graph.
(I've used a set with numbers, but the sets are just names of the vertices or edges, they don't matter - only the sizes of the sets counts.)

[Anonymous No.6711479]

>>6711460
>>6711458

This is basically one of the formal definitions of a directed graph (there are some other equivalent definitions as well as some not equivalent ones for dealing with special cases), it just took me a while to understand what you were saying since it's so very strange.

A graph G can be defined as a 4-tuple
G=(V, E, o, t)
where:
V is a set of vertices
E is a set of edges
o is a function o:E-->V that maps each edge to it's "origin" vertex
t is a function t:E--V that maps each edge to it's "target" vertex

This is equivalent to the more common definition
G=(V, E, p)
where p:E-->VxV

It is also common for others to include two extra sets and functions for the edge and vertex labels. Both definitions above are suitable for multigraphs, loops, and digraphs. Unfortunately I don't think it's actually suitable for undirected graphs.

[Anonymous No.6711496]

>>6710822
>using namespace std;
like a god damn pleb

[Anonymous No.6711504]

>>6710822
Not newlining before the initial curly brace.
Is this how shitty STEM has become?

[Anonymous No.6711541]

>>6711205
god you're awful.

I don't like the understanding of "companies" as "process" something.
That suggests that something happens, some sort of work, but that's not what companies do. They actually don't do anything, they're just there, static and unmoving.

See how ridiculous you sound ? Yes functions are abstract, but some of them can be seen as processes.

[Anonymous No.6711624]

>>6711541
>See how ridiculous you sound ?
uh, no. your rephrasing doesnt compare to what i said in any way because in companies, stuff happens over time. in functions, it doesnt.

[Anonymous No.6711628]

>>6711624
oh yeah ? how about a function that depends on a parameter.

[Anonymous No.6711631]

>>6711628
thats just a function with two inputs

[Anonymous No.6711636]

>>6711504
Gross.
Why would you do that?

[Anonymous No.6711640]

i like your enthusiasm op.
I think you're confusing a function with the algorithm to compute it. Of course in order to "know" the value of a function for a particular input object you need to carry out the algorithm for computing it. which is, a step by step thing...
step
step
step...

(church turing thesis)

[Anonymous No.6711645]

>>6711479
>This is equivalent to the more common definition
I think a much more common definition is G=(V,E) where E contains two element subsets or ordered pairs, for undirected and directed graphs respectively, of V.

[Anonymous No.6711662]

>>6711479
Where you writing this explanatory text back to me?

The definition is strange because it provides more data that the direct definition. For one, there will be arrows connecting the graphs (natural transformations). And the fact that we know its a topos implies lots of things: E.g. we know a priori that there is a notion of product of graphs. (In fact, all limits taken over finite index categories exist.)

>>6711645
In the end, it's a matter of applications, but this defintion is "the bad one", I'd say. It might not matter sometimes that your edges don't have names, but (I think it was pointed out) this definition also doesn't generalize to graphs with several edges connecting the vertices.

[Anonymous No.6711665]

>>6711662
>this definition also doesn't generalize to graphs with several edges connecting the vertices.
if you replace E by a multiset it does

[Anonymous No.6711667]

>>6711645
You're right in that this is a common definition but it is only sufficient for defining simple graphs (not multigraphs). It's actually a special case of the above definition, here the edge set is given implicitly.

Consider this example:
You have two vertices, <span class="math">v_1[/spoiler] and <span class="math">v_2[/spoiler].
You have a directed edge <span class="math">e_1[/spoiler] connecting <span class="math">v_1[/spoiler] to <span class="math">v_2[/spoiler].
You have another directed edge <span class="math">e_2[/spoiler] connecting <span class="math">v_1[/spoiler] to <span class="math">v_2[/spoiler].

Using your definition <span class="math">G=(V,E)[/spoiler] this would be:
<span class="math">G={{v_1, v_2}, {(v_1, v_2)}[/spoiler]
Note that your edge set <span class="math">E[/spoiler] can only contain a single instance of the edge <span class="math">(v_1, v_2)[/spoiler] because it is a <span class="math">set[/spoiler]. Some intro courses and text books will at times instead rely on the notion of a <span class="math">multiset[/spoiler] and say that the edge set <span class="math">E[/spoiler] is a <span class="math">multiset[/spoiler], but it's not the best solution in my opinion.

Using the above definition <span class="math">G={V,E,p}[/spoiler] it would be:
<span class="math">G={{v_1, v_2}, {e_1, e_2}, {(e_1, (v_1, v_2)), (e_2, (v_1, v_2))}}[/spoiler]

>>6711662
No, I was explaining that the way you did it is identical to the definition I provided. However, it is not sufficient for dealing with undirected graphs.

[Anonymous No.6711670]

>>6711667
Crap, I forgot my backslashes.

<span class="math">G = \{\{v_1, v_2\}, \{(v_1, v_2)\}\}[/spoiler]

and
<span class="math">G = \{\{v_1, v_2\}, \{e_1, e_2\}, \{(e_1, (v_1, v_2)), (e_2, (v_1, v_2))\}\}[/spoiler]

[Anonymous No.6711711]

>>6710399
Yes, I lately realized what simplicity means. This guy for example does really good in explaining some math concept. When you can relate abstract things to real world it is a lot more fun and interesting. Just spiting out some abstract forms without knowing what does it mean I start to feel like damn autist.


http://betterexplained.com/archives/

[Anonymous No.6711824]

>>6711711
>http://betterexplained.com/archives/
that thing about e is pretty interesting to read. its not anything new if you already understand it, but it gives pretty good intuition of what its about, at least for real numbers

[Anonymous No.6711838]

>>6711711
>>6711824
his stuff on infinity on his old princeton website is a bit bad though. talking about infinity in finite terms only doesnt seem right to me

[Anonymous No.6713036]

>>6711425
Nigga (it seems to me that) you have this shit all sideways.

Are you translating from a different language? Because (in my limited experience) it is highly nonstandard to use 'map' for (what I have seen to be called) 'relation', 'function' for 'injective _', and 'total _' for 'surjective _'.

Seriously, it's wigging me out.

[Anonymous No.6713047]

>>6713036
It's a subtle difference but the definition for a map is different from that of a relation. A map can be from any set to any set, but a relation is a map from a set to itself.
M:A-->B is a map
M:A-->A is a map and a relation
Every relation is a map, just as every function is a map. Also, total isn't the same as surjective, we're talking about the domain and the pre-image whereas surjective talks about the codomain and the image. I'm really surprised more people aren't familiar with this, the textbooks we use at my university and all the professors seem to be onboard.

[Anonymous No.6713068]

>>6713047
Oh, right, my mistake. I feel a little silly.

I understand your analogies now.

I recognize your usage of 'total', but the rest is a bit weird to me.

In case it's relevant, I'm a grad student of mathematics in North America, and I've only ever studied in Canada and Washington.

[Anonymous No.6713521]

>>6713047
>It's a subtle difference but the definition for a map is different from that of a relation. A map can be from any set to any set, but a relation is a map from a set to itself.
anywhere i look i see relations defined as subsets of cartesian products of arbitrary sets. this is the first time ive ever seen someone define a relation as a subsets of $A^2$ for some $A$. also, its the first time if seen someone define a relation as a function, since $A \rightarrow A$ suggests that the thing is tubular, i.e. for every x in A, there is at most one y in A such that (x,y) is in the thing.

[Anonymous No.6713522]

>>6713047
>It's a subtle difference but the definition for a map is different from that of a relation. A map can be from any set to any set, but a relation is a map from a set to itself.
anywhere i look i see relations defined as subsets of cartesian products of arbitrary sets. this is the first time ive ever seen someone define a relation as a subsets of <span class="math">A^2[/spoiler] for some <span class="math">A[/spoiler]. also, its the first time if seen someone define a relation as a function, since <span class="math">A \rightarrow A[/spoiler] suggests that the thing is tubular, i.e. for every x in A, there is at most one y in A such that (x,y) is in the thing.

[Anonymous No.6713538]

>>6713522
I've been doing some digging around. I'd never run into the notion of a relation between distinct sets (we always just say function or map). It seems like the definition we use for map is identical to the definition of a binary relation on the wiki page. We use all of the other definitions defined under "special types of binary relations" as well.

http://en.wikipedia.org/wiki/Binary_relation#Special_types_of_binary_relations

The notion that a relation must be between the same set was something we learned in one of our intro courses early on, so it's possible that it could've just been a convention we used then. In the past I had found our definitions on a wiki page, I think under "map" or something, but I can't find them anymore. In fact everywhere I look seems to have a map defined as a synonym for function.

At any rate, I'm satisfied with replacing my definition of a map with the more standard definition of a binary function (between distinct sets).

[Anonymous No.6713555]

>>6713538
The following is standard in set theory and most of mathematics:
'map' and 'function' are synonyms
a binary relation <span class="math"> R [/spoiler] is a subset <span class="math"> R \subset A \times B [/spoiler] for some sets <span class="math"> A,B [/spoiler], i.e. it is a set of pairs of elements
a function is a binary relation <span class="math"> f \subset A \times B [/spoiler] such that for each <span class="math"> x \in A [/spoiler] such that there is some <span class="math"> y \in B [/spoiler] with <span class="math"> (x,y) \in f [/spoiler], this <span class="math"> y [/spoiler] is unique.
Notice that usually we demand that <span class="math"> A [/spoiler] is the domain of <span class="math"> f [/spoiler]

[Anonymous No.6713556]

>>6713538
where do you study? ive never come across those terms as being defined the way youre used to. this >>6713555 is what everyone i know of uses

[Anonymous No.6713571]

>>6710439
h(f, g) = f . g

[Anonymous No.6713575]

>>6710526
>stack overflow
>never returns
3/10

[Anonymous No.6713600]

>>6710371

>Fibonacci sequence
>Mandelbrot set
>Julia sets
>iterated functions
>functions
>...hardcore math

[Anonymous No.6713602]

>>6713556
I don't want to spread a bad perception of my university so I'll just say it's a university in Canada.

>>6713555
The only big difference between our "map" definition and the binary relation definition is that in the context of maps we also talk about the domain, co-domain, pre-image, and image.

[Anonymous No.6713606]

>>6713602
thats not a difference, those terms apply to relations.

[Anonymous No.6713626]

>>6713606
Oh, good then. The wiki article hadn't mentioned them and it was making me think otherwise.

[Anonymous No.6713629]

>>6710439
Differentiation is the most familiar example; it's a function from the set of differentiable functions <span class="math">\mathbf{R} \to \mathbf{R}[/spoiler] to the set of all functions <span class="math">\mathbf{R} \to \mathbf{R}[/spoiler], defined by sending each differentiable function to its derivative.

Another example: given sets X, Y, and Z, function composition is a function that takes two functions <span class="math">f: X \to Y[/spoiler] and <span class="math">g: Y \to Z[/spoiler] as input, and outputs their composition <span class="math">g \circ f: X \to Z[/spoiler].

[Anonymous No.6713630]

>>6713626
wut? http://en.wikipedia.org/wiki/Image_%28mathematics%29 explicitly says "Image and inverse image may also be defined for general binary relations, not just functions."

[Anonymous No.6713632]

>>6710371
Mensa is but a glorified circle jerk.

[Anonymous No.6713638]

>>6713630
Thanks, I was just looking at the binary relation page.

[Anonymous No.6713647]

>>6710371

Functions are most certainly babby tier mathematics, I don't know where this "hardcore" impression comes from.

[Anonymous No.6713656]

>>6713647
>Stroke's Theorem

[Anonymous No.6713667]

>>6713647
>triple integrals not at the very bottom

[Anonymous No.6714086]

>>6713647
that picture is all messed up. how do you do arbitrary powers without logarithms, how do you do logarithms without limits, how do you do limits without convergence? half that thing is upside down

[Anonymous No.6715853]

>>6713647

>one-time pad decryption
>math

[Anonymous No.6715874]

>>6715853
>one time pad
>can be broken
bunch of morons made that pic

[Anonymous No.6717433]

>>6710812
>is it his enthusiasm?
omfg I can't stop laughing. Just go to yahoo answers. That's where I go for help.

[Anonymous No.6718834]

>>6714086
>how do you do logarithms without limits

Introduce them as the opposite of e^x, which can be vaguely understood as an abstraction of multiplication. You're looking at the concepts too formally, the chart is about where they're introduced.

[Anonymous No.6718966]

>>6718834
>too formally
pic related.

[Anonymous No.6718976]

I picked up a book on the universal mandelbrot set about 2/3 the way through my semester and nearly went insane. I didn't get enough sleep to wrap my head around it.