Are functions used anywhere in upper level mathematics? or is it just for introducing more abstract shit...

are functions used anywhere in upper level mathematics? or is it just for introducing more abstract shit? is there a Function Theory?

essentially all of mathematics is the study of functions

or sets, depending on how you ask, because functions can be defined as sets

If you're underage and just learning functions, the answer is yes, functions are used everywhere. They never really disappear, and when they do, it's only because we are using more sophisticated terms to replace them

Almost all of mathematics involves looking at structures and functions between those structures: topology and homeomorphisms, analysis and continuous functions, differential geometry and diffeomorphisms, algebra and homomorphisms, category theory and functors, sets and in/sur/b-ijections, etc.

i thought functions were the things you used to connect sets. like you have a set X and a set Y and you use a function f to state what elements connect by showing whether or not they fall under the domain and range of f. im just confused about all the stuff like stated because it seems like none of it involves funcitons, yet functions is all i hear about in classes.

>functions can be defined as sets
Sets can be defined as identity functions.

More precisely, a function i is a set if its domain and codomain are equal, and for any function x:
-if x has the same codomain as i, then ix=x;
-if x has the same domain as i, then xi=x.

>Sets can be defined as identity functions.
then how do you define function?

Functions are primitive so they don't have a definition, just like how there's no definition of a set in set theory.

>the things you used to connect sets. like you have a set X and a set Y

This is true, but functions are themselves also sets too.
>and you use a function f to state what elements connect by showing whether or not they fall under the domain and range of f.

Yeah.

>because it seems like none of it involves funcitons, yet functions is all i hear about in classes.

What that guy is talking about is how after you know all about functions you can use them to do anything. For example. Suppose that you wanted to prove that the set of all natural numbers had or had not the same amount of elements as the rationals. How would you do it?

You would construct a function from the naturals to the rationals that was bijective. (Or would prove that such a function cannot exist).

Another example he gave were homomorphisms and that is a particularly useful application of functions so let me try to explain you what they are.

In mathematics we have different algebraic structures. An algebraic structure is simply a set (like, for example, the integers) and some operation to do algebra with the elements of that set (for example addition).

As I said, the integers with addition are an algebraic structure.
Matrices with matrix addition are an algebraic structure.
Modulo N numbers with modulo N addition is an algebraic structure.
etc.

Now, a homomorphism is a function that preserves algebraic structure. For example: Suppose you had two algebraic structures. One with operation * and the other one with operation %.

A homomorphism is basically a function such that
F(a*b) = F(a)%F(b)

Which means that you can do algebra through that function. In practice what that means is that if you know an algebraic structure very well, and you want to study similar algebraic structures you could very easily use a homomorphism to study the new structures with the structure you already know.
Pretty useful if you ask me.

>Functions are primitive so they don't have a definition
if you take set as a primitive then function can be defined as set of ordered pairs such that there are no pairs (x, y) and (x, z) for y not equal to z

that's still only a partial function, you need a pair (x,f(x)) for every x

Sure you can do that, and there's nothing technically wrong with that, but you'll need so many other axioms that no one except graduates and above specializing in set theory can recall them without looking them up.

On the other hand axiomatizing functions requires no more than the identity axiom and an axiom for associativity, which anyone with a lick of mathematical maturity can quote to you. Which is what mathematical axioms should be: simple and intuitive.

Actually I retract the statement that
>there's nothing technically wrong with that
There are issues with the ordered pair definition of function as you've presented it, though they can be corrected by incorporating the codomain into the definition and making it an ordered triple instead.
See e.g., the thread and especially the link in >

>Sure you can do that, and there's nothing technically wrong with that, but you'll need so many other axioms

Like what? The classic definition of a function only requires atomizing inclusion.

>axiomatizing functions requires no more than the identity axiom and an axiom for associativity

This is false; any relation obeys these axioms for instance.

There is a branch of math called functional analysis, whose historical raison d'être was to study various spaces of functions (e.g real-valued continuous functions on a segment, periodic functions, square-integrable functions, etc.).
More generally, functions are used everywhere as a way to connect sets. When you have sets with more structure (for example operations like addition), it is often informative to look at functions that preserve these operations (in the case of addition, additive functions, etc).

I was referring to the axiomatization of set theory versus axiomatizing functions instead.
>This is false; any relation obeys these axioms for instance.
Yeah this is true, I was operating under the more general CS definition where functions are allowed to have side effects.
Still, you can specialize to the usual mathematical definition by defining a function as an equality-preserving homomorphism (which is why the category of sets has functions as their morphisms) so it's not too much additional effort.

What the fuck are you saying? A function is a relation (subset of a Cartesian product) such that no two pairs every x has only 1 y asociated.

That definition isn't really wrong, it's just incomplete.
We can define function [math]f:X\to Y[/math] as [math]f\subset X\times Y[/math] such that for every [math]x[/math] in [math]X[/math] there exists pair [math](x,
y)\in f[/math] and for every [math](x_1, y_1), (x_2, y_2)\in f[/math] if [math]x_1=x_2[/math] then [math]y_1=y_2[/math].
There's no need to use ordered triples, and there are no issues with ordered pair definition, you can say there are issues with that particular statement of definition

Brainlets. A function is a morphism in the category of Sets.

We are discussing real mathematics, not meme category theory.

that's true, but that's not a logically coherent definition because you can't define the Category of Sets before you define a function between two sets.

YOU can't.

i'm very curious as to how you would define that category without first defining a function

a function is a lambda term, end of

If you can't grasp so simple an idea then I don't think I can help you

and category theorists wonder why no one likes them

>mfw maths is just people using words that don't mean anything

>Functions are primitive so they don't have a definition

No, functions are defined in terms of sets.

>functions can be defined as sets

This is correct.

why isnt this made obviously apparent?
why does this person have to get these answers from niggers who watch chinese cartoons on the internet?

how do people pay real money for education these days?

>is there a Function Theory?
...is there a Google to search for Function Theory? Is there a retard too stupid to use it? These, and other questions, continue to baffle Social Science.