/sci/ - Science & Math

Prove that for all sets, passing to the set of subsets preserved cardinal order.
I.e.

[math] |X| < |Y| \implies |{\mathcal{P}(X)}| < |{\mathcal{P}(Y)}| [/math]

Hint: For finite sets, the cardinality of the powerset is the natural number 2^|X|.

>>12224266
>>12224290
So does nobody have any input?

>>12229891
In what context do you even have to prove this?

What you want is "function extensionality" and apriori it's not a feature of each of Martin Löf's type theories. In HoTT iirc it follows.

https://ncatlab.org/homotopytypetheory/show/function+extensionality
https://ncatlab.org/nlab/show/function+extensionality
Hopefully there's a longer/proper article on this, but that's the keyword anyway

Of course they main/famous axiom to prove equality of two types is univalence, saying that A==B -> (A=B) where by == I mean type equivalence, which is when you get find functons (equivalences) putting A and B as images of another

>>12216477
>So if something false is true, then everything can be true. I do agree with you that it is arbitrary and could easily be the other way (if something false is true, then everything is false)

These are classically the same.
If everything becomes true, then P is true and notP is true.
If everything becomes false, then not P is true and notnotP is true as well.

>>12215131
Basically what this guy said. There are 22⋅2=16 binary boolean truth functions and one of them we happen to call "implies".
https://en.wikipedia.org/wiki/Boolean_function

There are many other alternatives for the logic of entailment, see
https://en.wikipedia.org/wiki/Relevance_logic
But these of course all have much harder semantics than plain boolean predicate logic.

See also
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication

>>12216477
>So if something false is true, then everything can be true. I do agree with you that it is arbitrary and could easily be the other way (if something false is true, then everything is false)

These are classically the same.
If everything becomes true true, then P is true and notP is true.
If everything becomes false, then P is true and notnotP is true as well.

>>12215131
Basically what this guy said. There are [math]2^{2\cdot 2}=16[/math] binary boolean truth functions and one of them we happen to call "implies".
See
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication

There are many other alternatives for the logic of entailment, see
https://en.wikipedia.org/wiki/Relevance_logic
But these of course all have much harder semantics than plain boolean predicate logic.

>>10972455
This is 4chan bro, you're not supposed to agree to criticism of your posts.

I work with polynomials in [math] {\mathbb F}_p [/math] (p=2 is the case I'm most interested in) and would like to read up one results in that direction. For polynomials, a large corpus of information surrounds the algebraic geometry situation and there e.g. with Hilberts Nullstellensatz, sadly they want polynomials over an algebraically closed field.

Now since I don't work in the rationals or the reals, and ans my finite field isn't algebraically closed, I'd like to know in which direction to go or where to read. Can I find the closure of [math] {\mathbb F}_2 [/math] to make such theorems apply and what is it?
I know there are those tasks of counting solutions over finite fields, the Weil theorems and such. Can you give me some pointers? Much appreciated.

>>9364511
You write
>cc/call : ((a->b) -> a) -> a
>dots : a->b
and then both
>cc/call(f)
and
>f(dots)

so it's not clear what type your f has. Is it of type "(a->b) -> a" or "a->b"?

In any case, the theorem ((a->b) -> a) -> a is certainly not intuitionistically provable, just as there is no function of that type within a functional language like Haskell itself.

But when you program, there's the notion of execution and this is where "continuation" pops up. And then some languages can internalize this.

Think of how "runtime" is not explicit in algebra. The functions
(a,b) \mapsto (a+b)*(a+b)
and
(a,b) \mapsto a*a+2*a*b+b*b
applied to (5,6) will execute differently (e.g. in the first case, two additions and then only one multiplication is involved), so you have different runtimes. But "runtime" isn't part of the language and so it's invisible.

cont 4

>>9132924
Yes, Bitcoin is slow and expensive. It has been improved upon and from a technical standpoint made obsolete already 4 year after the release.
You still make money if you buy now, though, because the market capitalization is growing. The current price of 4000$ is somewhat absurd and there is surely a bubble to be popped, but all the top 10 coins in market cap are held and traded so much that it can't quite fall too deep - at least without external regulations. Here's the market cap
https://coinmarketcap.com/

>>8885550
[math] \lim_{z\to 1} \left( \sum_{n=1}^\infty n^m z^n - (-1)^{m+1} \dfrac{m!}{\log(z)^{m+1}} \right) = -\dfrac{1}{m+1} B_{m+1} [/math]

where [math] B_{m+1} [/math] are the Bernoulli numbers
1/6, 0, -1/30, 0, 1/42, 0, -1/30, ...

So
[math] \sum_{n=1}^\infty n^m [/math]
gives
-1/2, 0, 1/120, 0, -1/252, ...

>>8885550
[math] \lim_{z\to 1} \left( \sum_{n=1}^\infty n^m z^n - (-1)^{m+1} \dfrac{m!}{\log(z)^{m+1}} \right) = -\dfrac{1}{m+1} B_{m+1} [/math]

where [math] B_{m+1} [/math] are the Bernoulli numbers
1/6, 0, -1/30, 0, 1/42, 0, -1/30, ...

So
[math] \sum_{n=1}^\infty n^m [/math]
gives
-1/2, 0, 1/120, 0, -1/252, ...

Also, does anybody know how well integration methods such as Runge Kutta to various degrees carry over to 3-dimensional system?
Does anybody have a reference for e.g. Runge-Kutta in those dimensions?

https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods

>>8581206
I take it you started with
[math] \sum_{x=0}^\infty \dfrac{1}{x!} = e [/math]
and passed over to
[math] \int_{x=0}^\infty \dfrac{1}{x!} [/math]

That's the right time for Ramanuja magic, telling you

[math] \int_{x=0}^\infty \dfrac{1}{x!} = e - \left( \dfrac{1}{2} - \dfrac{1}{12} \dfrac{d}{dx} \dfrac{1}{x!} + ... \right) |_{x=0} [/math]

where the dots are higher order derivatives. For the Gamma function as x!, the derivative is the Euler–Mascheroni constant and this first order approximation misses the numerical integral by 0.00015.

>>8581207
-0.597006

>>8581206
I take it you started with
[math] \sum_{x=0}^\infty \dfrac{1}{x!} = e [/math]
and passed over to
[math] \int_{x=0}^\infty \dfrac{1}{x!} [/math]

That's the right time for Ramanuja magic, telling you

[math] \int_{x=0}^\infty \dfrac{1}{x!} = e - \left( \dfrac{1}{2} - \dfrac{1}{12} \dfrac{d}{dx} \dfrac{1}{x!} + ... )|_{x=0} [/math]

where the dots are higher order derivatives. For the Gamma function as x!, the derivative is the Euler–Mascheroni constant and this first order approximation misses the numerical integral by 0.00015.

>>8581207
-0.597006

Hard to say what you really mean OP, but I think the answer is in the 4th semester, when I stopped playing Magic the Gathering (played it more or less seriously, flew to places like Netherlands or Italy for Grand Prix' etc.).
That's when I started to care, and when I started to read mathematics textbooks that were entirely unrelated to what I'd ever learn in university lectures in my program.

>>7594653
Random story from muh life: I (brown eyed) was at the house of a friend of a friend of a kid. I was alone with his mother in the kitchen at one point. Never seen this person before.
We talk for 1 minute, then she tells me she would never give a kid my name, as it's (appearently) the name of some dictator, and also: brown eyes people are all stupid, because cows have brown eyes.
>o-okay

Literally reddit: the question
>A couple Australian scientists tested whether farts could contaminate a surgery. They found that a clothed fart on a petri dish grew nothing while a naked fart on a petri dish grew a veriety of bacteria.
>So in theory you could obtain someone's DNA by having them strip and fart on a collector and you could distinguish between bacterial and human DNA. But you would probably do better by collecting some skin flakes from their underwear while they were busy farting for you.

>>7101822
playyaaaaaa
https://www.youtube.com/watch?v=oJG8cmlkPuw