/sci/ - Science & Math

>what is a function
An array of numbers.

>>15030950
What is an array?

>>15030952
An array is a series of elements of the same type placed in contiguous memory locations that can be individually referenced by adding an index to a unique identifier.

>>15030945
A function is a rule which associates each element of a set with another element of the same set

>>15030955
kys

>>15030950
this

>>15030945
you are posing a question (from the set of questions) and expect an answer (from the set of possible answers). This is a function. Don't pretend this is some esoteric incoherent shit.

>>15030945
a structural or semantic phenomenon that occurs predictably independent of the broader scope of whatever contains it

A function is a set of ordered pairs (a, b) for a in some set A and b in some set B.

It is a type of relation, that is, a set of ordered pairs from A x B that meet certain requirements, such as having only one such a in (a, b) for all b.

That's it. A subset of the cartesian product of two sets that obeys certain rules. Obviously you have an intuition for what this is, but it has a pretty simple set-theoretic construction.

>>15030955
So it doesn't have to be numbers or even the same set. Although we sometimes use 'mapping' when we're not dealing with numbers, but the domain and codomain definitely don't need to be the same set.

>>15030945
Fuck these explanations are terrible. Is it a meme to make them confusing as possible ?

>>15031076
>set
definition please

>>15031767
a collection of things.

>>15031767
define definition

>>15031076
>>15031781
pretty much

>>15030945
A function takes an input and produces an output
f(x) = x + 2
In the function above if the input x is 2 then the output f(x) is 4

>>15031790
define?

>>15030945

A systematic algorithm which provides a unique solution for every unique variable.

>>15031767
Learn to deal with the fact that mathematics is not some pure flawless divine construct. It's messy and indistinct when you get to the ground floor. Deal with it or be forever filtered. You're a grownup now, not a child and you must learn that the world not always satisfy you in the way you want.

>>15031790
define "define definition"

>>15031985
what about this ?

f(x) = |x| - |x| + 1

A function can do nothing or something. It can take an input, or not. It can have an output, or not. Functions always have a name.

>>15032027
>It's messy and indistinct when you get to the ground floor.
To add, we just have assumptions in-built that we don't even address until later (thinking here probability theory: we did the math with assumptions and then later came back to figure out measure theory).

>>15031985
>unique solution for every unique variable
No they don't need to be unique, sinusoids for instance are very much cyclical functions in their output.

It basically is like a party. It's a stupid name. Usual connotation is like it's a party but not fun.

>>15032027
>Learn to deal with the fact that mathematics is not some pure flawless divine construct.
Sure, it is a frauding pretending mess. Everything is over complicated, relies on person instead of function names and has limits nobody tells you who profit from. So question is why can i only learn that here instead in the science church?

>>15030945
A function is the performance or execution of a task

>>15031807
Damn this might sound really retarded but I never really considered how this is the exact same thing as writing functions in code. I mean it makes sense now that I think about it. I feel like I had an epiphany of something really obvious. I didn't make the connection even when I was studying programming and we'd have math classes. Like

function f(x) {
return x + 2
}

I'm a real dumdum

It's a reusable piece of computing code that does stuff.
Don't listed to those insane math people.
Now, sure, a set of numbers mapped to a set of numbers is a possible way a function can be, but how do you "assume an infinite set"? Are you going to type the numbers yourself forever? Of course not. What you do is describe logical operations that describe a behavior given the inputs. That way, you can have an "infinite" function. Using a method of describing arbitrarily big numbers, you can have it truly as infinite as your hardware can handle. But mathtards think they can just "assume" something to be infinite without describing how that stuff is supposed to work in the first place. They can't even prove that stuff to themselves! Oh and they'll get angry if you confront them on how their insane squiggles are meant to actually work in real life. So be careful out there lest you become like them.
t. your local computer fren

>>15032027
>scientist's discipline gets questioned
>"LEARN TO DEAL WITH THE FACT THAT SCIENCE IS NOT SOME PURE DIVINE CONSTRUCT DEAL IT WITH IT OR BE FOREVER FILTERED"
what a cope

>>15031076
>Obviously you have an intuition for what this is,
i don't

https://math.stackexchange.com/questions/1473169/

>>15030945
>what exactly "is" a function?

It's simply the following:
>restriction
>transformation
>pullback
>surjection
>pushforward
>morphism
>bijection
>lifting
>functional
>injection
>chart
>retraction
>section
>functor
etc.

>>15030945
an algorithm to convert an object from a set to another object of another set.

>>15033103
Lol, what a retard. Obviously a function is a surjection only in relation to some set.

a set is a mapping of elements from one set (called the domain) to another set (called the range).

For example, I could create a function called isFaggot() that maps from the set of all people to the set {Yes, No}. Applying isFaggot to OP evaluates to Yes.

>>15031076
You're describing the set theoretical model of the graph of the function.

>>15030945
There are multiple definitions.

In computer science: An explicit series of instructions with potential input, potential output, and potential side-effects.
You can run/execute a computer science function.
Every series of instructions (that "compiles") is a function.
No need to prove that what you write down is a function as long as each step makes sense.

In first-order logic: Either of
- One of the given function symbols from the language of the theory you are working in.
- An n-variable predicate such that for each assignment of the n-1 first variables, there exists a unique assignment for the last variable that makes the predicate true. The "value" of the function is that unique value that makes the predicate true.
For example, if P(x, y, z) is the predicate "x + z = y", then given any x and y, there's a unique z that makes P true. The only way that P(3, 10, z) is true is if z = 7. In this case, P is the function we call y - x.
Can't "run" a first-order logic function unless you find a computer science function that finds the unique assignment that makes the predicate true.
Not all predicates are functions. To be rigorous, before calling any predicate a function, you need to actually prove that uniqueness property about that last variable.

In math based on set theory: A set of ordered pairs such that no pair has the same first element.
Most set theories have an axiom similar to ZFC's axiom schema of replacement which allows defining a set using a first-order logic function. For example the set { ( (x, y), z ) : x, y, and z are integer, and x + z = y } is a well defined set in ZFC. That set is the function f(x, y) -> y - x.
Also can't "run" a set theory function unless you find a corresponding computer science function that finds the second element of the pair.
Not every "description" is a set. For example there is no such thing as "the set of all sets" in ZFC.
To be rigorous, you should prove that what you write down is actually a well defined set.

>>15030945
There is no solution

>>15031767
>definition please
Gladly. The semantical mainstream definition is:
A set is an individual in the Von Neumann universe
>What is this so-called universe?
https://en.wikipedia.org/wiki/Von_Neumann_universe

>>15033181
Which is sufficient to describe the function and can be used to render it in any interpretation you choose, duh

>>15031807
Can't there be functions without input requirements?
f() = 2 + 2
You could instead use f() instead of 4 whenever needed.

>>15034060
yeah it's a constant function

>>15034060
>>15034066
retard

>>15034071
why?

>>15032946
thats why "functional programming" is called "functional programming"

>>15034060
>Can't there be functions without input requirements?
As you see in this thread, the word "function" is overloaded in science.

In programming, yes there are "functions" without possible input.

>>15030953
What are series, elements, sameness, type, placement, contiguity, memory, location, individuality, reference, addition, index, uniqueness, and identification?

>>15030945
In practice: a mathematical abstraction f that allows you to pass from an object "X" to another object "f(X)".
Formally in mainstream set-theoretic framework: a collection of pairs (x,y) from X x Y such that every x in X appears exactly once in a pair (x,y)
>but what is an object? what is a set? what is a pair? what does appears mean? what does exactly once mean?
>t. 4 year old

>>15031765
I suspect there are a combination of factors that go into why math explanations are so terrible:
1. Literal autism. Autism is partly defined by the inability to get inside other peoples' heads, which is vital for knowing how to convey information in a way that will actually be comprehensible to the other person.
2. You get to results in math *via* extreme pedantry. But pedantry is mostly terrible as a *starting point* for understanding a topic.
3. It appears to me that people learn math *linguistically* in many or even most cases. It's a great shortcut for doing well on tests without actually understanding much (without understanding existing math in a way that enables mathematicians to make new math).
4. There's a lack of clarity about what math itself is. I would say math is, most basically, counting. And that without knowing this and keeping it in mind, explanations veer off into niche territory.
5. The dominance of set theory as an enframing for math, instead of counting. This creates an unintuitive basis for math, where counting would presumably work a lot better. Like, if I ask a famous topology question, it's a lot easier to just ask how many holes something with [x] property has, or how many ways [y] thing can be done if we defined what makes a seperate way carefully. The set shit tends to get in the way.
6. Muh seekrit club. Intentionally being obtuse. I think this is actually rarer than I believed at the start.

I have enjoyed using Matt Walsh to launch many of these threads.

168.19

>>15033552
>von Neumann universe[...]is a class
>a class is a collection of sets
What is a set?
Then, what is a collection?

>>15030953
An array is a sequence of positions that can be associated with a value.

>>15034345
keep it up anon. I enjoy these

>>15034987
>>15031767
>What is a set?
Can be left undefined as long as there are axioms stating how they interact. God, you're stupid.

>>15034345
>The dominance of set theory as an enframing for math
Set theory is a [math]viable[/math] theory of everything of math: Every concrete object is a set, every abstract one a set-theoretic construct.
My retarded friend, it's up to YOU to propose an alternative even remotely as powerful as set theory. Bourbaki tried that with the notion of a structure (which they also failed to define, ultimately, by the way. Notice a connection yet, retard?) and failed horrendously after being unable to formalize categories.
Autists here dogmatically reciting textbook defintions and axioms does not make your stance any less fucking braindead.

>>15035180
Just say so then, retard. Don't pretend that something is well defined just that you need to be a jewish set theorist to understand it when it obviously relies on axiomatic constructions.
It's like you're ashamed to admit axioms exist and when you're pressured into it, turns it was obvious all along and everyone else was a dumb goy who can't understand your pilpul.

One of my better professors in college started one of his books with a brief quote:
>Winged expressions like 'f of alpha' have by now became a part of our conceptual vocabulary
OP is not in that 'our,' apparently.

Holy moly, this board's IQ really has dropped considerably.

>>15035644
My advice to the pseud morons who think they know it all despite being unemployed former factory workers is to take a class on set theory and proofs, combined if possible. However, we all know you won't do that because it'll expose you 100iq dunning-kruger midwit retards for the pathetic wastes of oxygen that you are. Plus, you probably haven't set foot in a school since you dropped out of high school.

>>15034060
f(0)

>>15034345
>Like, if I ask a famous topology question, it's a lot easier to just ask how many holes something with [x] property has
the concept of "hole" seems much slipperier than the concept of "set"

>>15036445
Slipperier, perhaps, but also much more intuitive. And the notion can then be specificied/pedantised/"made rigorous".
A hole is also something from everyday life. I figure knowledge starts at everyday life, meanders through theoretisation/pedantry/logic, and then arrives back at everyday life.

>>15036685
>And the notion can then be specificied/pedantised/"made rigorous".
Without relying on a notion of set? How so?

is chatGPT-sama correct?

>>15036700
I have no problem with then *using set theory* to do things. It just seems sensible to start with the intuitive and move to the pedantic.

>>15036431
f(0) = 4

>>15036445
I'd set yo momma's slippery hole

>>15036707
That's how math is already done

>>15030955
>>15031076
Now that's a perfectly fine definition for finite sets. But it doesn't make any sense for types of objects in which there is not a finite number. It doesn't even make any logical sense to refer to any "collection" or "set" of those types of objects.

>>15037557
>It doesn't even make any logical sense to refer to any "collection" or "set" of those types of objects.
Why not?

>>15032146
If [math] x_1 = x_2 [/math] , then [math] f(x_1) = |x_1| - |x_1| + 1 = 1 = |x_2| - |x_2| + 1 = f(x_2) [/math]
Therefore [math] f(x) = |x| - |x| + 1 [/math] is a function.

>>15037559
There are several reasons. First, the notion of a "set" or "collection" is purely intuitive. That is, it's based on being able to collect things or list things. The able to list or collect things being a human action is limited therefore to only finite things. When you assert that there's such a thing as an "infinite set" you're pretending that you can do something that you can't and therefore the result of this logical contradiction are paradoxes. And contrary to what is appealed to when referring to ZFC, these paradoxes still exist and have not been correct since mathematicians like to pretend that introducing another set of undefined words that are even less intuitive than the alleged object which lead to these paradoxes doesn't fix the problem. It's just sophistry. It's utterly meaningless and any definition from such silly word games are therefore meaningless.

>>15037570
>The able to list or collect things being a human action is limited therefore to only finite things.
Does it make logical sense to refer to finite sets that contain more elements than the number of particles in the observable universe?

>>15037574
Nope. And I would agree that you should not imagine that there is such a number or that a number that cannot be written down in a reasonable amount of time should not be considered either.

>>15037598
Logically speaking, is there a largest number which exists?

>>15037603
>Logically speaking
At this point it doesn't have much to do with logic. It's just "what if?", "what about?", etc. without any purpose. It approaches sophistry quickly.
>is there a largest number which exists?
This is a question that cannot be answered since you need to answer which base you're writing the number in; Base 2, Base 10, Base 16, etc. Second, it's difficult to consider such a question as the largest number is contingent on a few factors like computing power and computer memory. At this point in time the total amount of computer memory possibly available is not known. However, it is reasonable to assume that the total computing power available is finite.

>>15031767
Start with the empty set. You can write it on a piece of paper as "{}". It's just a symbol. Then, define the axioms of set theory and you can use them to construct more sets. It's that simple.

>>15037676
the empty set refers to your balls

>>15037647
>At this point it doesn't have much to do with logic.
No, I keep referring back to logic because of the original claim
>It doesn't even make any logical sense to refer to any "collection" or "set" of those types of objects.
so I'm trying to sus out what mathematical concepts make logical sense to you.
>This is a question that cannot be answered since you need to answer which base you're writing the number in; Base 2, Base 10, Base 16, etc.
Base 10
>Second, it's difficult to consider such a question as the largest number is contingent on a few factors like computing power and computer memory. At this point in time the total amount of computer memory possibly available is not known. However, it is reasonable to assume that the total computing power available is finite.
What was the largest possible number before computers were invented? Not asking for an exact answer, just a rough order of magnitude. I predict you'll say you don't know, but just give me your best guess -- no penalty for being wrong

>15037696
>No, I keep referring back to logic because of the original claim
No. It doesn't have anything to do with that at all. You're posting solely for the sake of posting. You're not as smart as you think you are.
>What was the largest possible number before computers were invented?
This is an even more stupid question than before. I was being polite earlier by entertaining the first question. But you're asking a lot of really stupid questions.

>>15037718
they are stupid questions because ultrafinitism is stupid. If you were able to answer my questions in a non-stupid way you would -- but you can't quite bring yourself to admit the existence of a number n which exists while n+1 does not

>>15030950
It's actually two arrays of numbers. Input and output.

>>15037560
i was more talking about the claim that
>[a function] provides a unique solution for every unique variable
and in the example
>f(x) = |x| - |x| + 1
any x value produces the same output of 1

>>15038293
yes, for any x value, there is a unique f(x) produced, the number 1

>>15030945
the emergent property of structure

>15037761
>but you can't quite bring yourself to admit the existence of a number n which exists while n+1 does not
If you can write down n, yet do not have the ability to compute n+1, then n+1 doesn't exist. It doesn't matter how much you want to pretend it does.

>>15038552
If you can write down n then you can compute n+1 easily by the algorithm we all learned in elementary school

>>15030966
That's an example of a function, but it doesn't tell me what a function is.

>>15039750
No, every function works like that.

>>15030945
>what exactly "is" a function?
in my mind any fiction is real

>>15030945
>what exactly "is" a function?
function is composed by fun-action, that means, the funny action of something in act

An action, but don't ask me to define it or anything because I can't. I just know it's something that you do, which is an action.

>>15035654
>unemployed former factory workers
does retirement really make you seethe so much?

>>15030945
lambda expressions

it a tuple [math] (X, Y, \mathcal{G} ) [/math] where [math] X [/math] and [math] Y [/math] are sets and [math] \mathcal{G} [/math] a subset of [math] X \times Y [/math] with the condition that for all [math] x \in X [/math] there there is only one [math] y \in Y [/math] such that [math] (x, y) \in \mathcal{G}[/math]