Thread 16837525

A function maps one input to one output... but why don't we make this bi-directional? A function must map an output to exactly one input.

[math]f(X) = f(Y) \iff X = Y[/math]

This seems like it would make it much easier to reason about things if outputs are unique.

That's called a bijection

>>16837531
Why don't we throw out functions and just have bijections? It seems like they're much easier to reason about for the same reason it's easier to reason about functions when inputs map to exactly one output.

>>16837525 (OP)
because that would solve too many problems and most mathematicians would be out of a job

>>16837535
>but why not bijections
One is the onliest number that you'll ever have.
Two would be just like One,
Cuz between them there exists a bijection.
Oh, yeah.

>>16837535
Because few self-inverse functions exist, and fewer serve any practical purpose.

>>16837525 (OP)
that's a cool idea. We should call it an injectinger because it's like you're injecting inputs into all the outputs instead of spreading them around

>>16837535
>>16837525 (OP)
A bijection between two sets exists if and only if their cardinality is the same.

Congratulation, you just limited yourself to what are essentially permutations of set elements for no reason.

>>16837525 (OP)
I think we should make it bi-directional too but not like that. One input should be able to map to multiple outputs, the way multiple inputs can map to one output

>>16838273
That’s called a relation, anon.

>>16838274
What about one input can go to multiple outputs but each output only has a single input?

>>16838273
As the other anon mentions, that's a relation. Actually we have a foundation where functions are defined as relations instead of mappings, but number theory schizos and combinatorics sickos are allergic to type theory because of how rigorous it is. Also while we have formal logics that trivialize ambiguity in the first place, academics are reactionary autistic retards and will lose their shit if you suggest that paraconsistent logic is the future.

The thread is two weeks old and nobody finds, or is bothered to point out that the property is fulfilled by any injection, not just bijections.
(The injector jester got there I guess, but that was opaque to others I think.)

[math]P \Leftrightarrow Q[/math] is usually taken to be a defined connective and just means [math]P \to Q[/math] AND [math]Q \to P[/math]. The direction
[math](X=Y)\to f(X)=(Y)[/math]
is generally taken to be a defining property of functions. There's also formalization of operatons (e.g. random sampling) where this isn't given, but then it's better to just not use the word function.
Anyway, the direction
[math](f(X)=(Y)) \to X=Y[/math]
is exactly injectivity (in the better - because constructively stronger - formulation without negations.)

E.g. the identity map from N to R validates this without being a surjection.

The thread is a day old and nobody finds, or is bothered to point out that the property is fulfilled by any injection, not just bijections.
(The injector jester got there I guess, but that was opaque to others I think.)

PQ is usually taken to be a defined connective and just means PQ AND QP.

The direction
(X = Y) f(X) = f(Y)

is generally taken to be a defining property of functions. There's also formalization of operations (e.g. random sampling) where this isn't given, but then it's better to just not use the word function.

Anyway, the direction

(f(X) = f(Y)) X = Y

is exactly injectivity (in the better - because constructively stronger - formulation without negations.)

E.g. the identity map from N to R validates this without being a surjection.

Seems the textbox swallowed my <-> and ->

>>16838586
>Actually we have a foundation where functions are defined as relations instead of mappings
What gobbledygook is that? Relations are relations, functions are a type of relation (restricted via the Axiom schema of specification), and a mapping is literally just a synonym for a function.

What you're trying to mumble out is that we have a category [math]\mathbf{Set}[/math] and a category [math]\mathbf{Rel}[/math] and an obvious faithul functor from [math]\mathbf{Set}[/math] to [math]\mathbf{Rel}[/math]. But just because [math]\mathbf{Set}[/math] is a (not even full) subcategory of [math]\mathbf{Rel}[/math] doesn't mean it's somehow "worse" or "inferior".

>>16837525 (OP)
Because the whole point of a function is "for given input, this is the output" and sometimes more than one input can have the same result, even in the real world.
This logic does not follow in inverse because a given input can't have two simultaneous and distinct putputs in the real world as that implies another variable is missing from your function yhat determines which output you should get.

>>16838586
Functions are primitive objects in type theory, they're not defined in terms of anything.

>>16839001
There are no sets or categories in type theory.

>>16839032
Ok, something-something type theory. You just made a semantic redefinition and now call a function a relation despite it still being a function. Or is it the other way around? Either way, what you're saying is nonsense.
>what if we call doggos animals instead of puppers?
Ehhh idk and I don't care.

different concepts have different uses. we have concepts that are both stricter (injections, bijections) and laxer (relations) than functions.

>>16837525 (OP)
>f(x) = 0 is not a real function b-b-because it isn't, ok?
Horizontal lines BTFO.

>>16839054
Different inference system anon, different mode of reason. It's a substantial shift like any other foundation, not "semantic redefinition".

>>16837525 (OP)
There are relations that are many-to-one.
Consider the mapping "poster on 4chan -> poster name is Anonymous"
Now, this is obviously many-to-one. It is a function where several different input values can have the same output value. A function is something that predictably associates something in the domain set with a concrete value in the image/target set. It doesn't need to be reversible, as already touched upon. That's a more specific subset.

>>16839147
Lay off the weed.

>>16839247
My point being, we elide one-to-many mappings because it makes things easier. We should probably do it for many-to-one mappings for the same reason!

>>16837525 (OP)
I also want to know

>>16837525 (OP)
You did it, you disproved your precalculus teacher. You're the smartest boy!
Now everyone on 4chan knows that you didn't drop out for being a brainlet, you dropped out because you were too smart for The Establishment to handle!

>>16838002
>>16838271
This. It limits you to an extremely tiny and not-always-particularly-useful subset of all the possible ways of mapping sets to other sets.

>>16842769
But can't we also say this exact thing about rejecting mappings from one to many? We can clearly recognize we're better off without them... I don't see why one-to-many is any preferable just because it's slightly nicer for our chosen inference rules.

>>16842877
>I don't see why one-to-many
many to one*
it's early morning

>>16842877
>we're better off without them
you can't just throw out an area of mathematics from the top down because you don't like the sound of it, everything in mathematics was invented because it fulfilled some need for someone
imagine someone wants to know the closest integer for any real number and study properties of this procedure, now they can't use a function according to you but sure as hell they will just invent some new construct called a shmunction and carry on just the same

>>16837525 (OP)
what is an "input"? you need another theory for that. then one can complain about conventions there as well.

>>16842886
"you" can throw out math that doesn't suit you, but the alternative will still produce semantically equivalent results just as there are multiple proofs for the fundamental theorem of algebra.

>>16842886
>they will just invent some new construct called a shmunction and carry on just the same
shmuck you, shmaggot

>>16839389
Nope.