How many functions are there?
ฝัง
- เผยแพร่เมื่อ 7 ก.พ. 2025
- The answer is: a lot of them! In this video, I show that F(R), the set of functions from R to R, has the same cardinality as P(R), the set of subsets of the real numbers, which, in a previous video, I’ve shown to be much bigger than R. This is set theory at its finest :)
Oh man, I just took a set theory exam today. Peyam, you should have made all these videos a couple months ago xD.
Also the proof that |ℝ| = |ℝ^2| that I received brought attention to the fact that most proofs online of this fail to form a direct bijection because of base funkiness, or at least the proofs that attempt to even form such a bijection. Of course, unleashing schroeder bernstein makes it a delight.
Also I've been pronouncing schroeder bernstein horrendously wrong :c.
Also what's perhaps more interesting is that |continuous functions from ℝ to ℝ| = |ℝ|. Continuity is quite the restriction!
Lol’d at when u said you’ve been pronouncing it wrong🤣
I have no idea how to pronounce it either cuz I’ve only read it!
(Also, sometimes online, I’ve seen it called Cantor-Schröder-Bernstein, which always makes me 😆 like.. can you make it longer?!)
I have that w symbols sometimes cuz in the book I use they don’t always tell the reader what they’re called and I’m doing this solo.
I think that Hilbert curve is a good explanation why |R|=|R^n|.
It's not only a curve but most importantly it's a function F:R^n->R or F:R->R^n.
Fantastic!! Absolutely love your videos. Particularly this one!
Thank you so much brother.. again ur saving my behind! I’m working through a math book and the author often doesn’t give the solution to problems cuz he probably assumes you’ll have a teacher or smth (which I do not, cuz I just got a book and started working through it.)
If it weren’t for you, I’d be so stuck! Really, really appreciate it!
Now just gotta keep playing w these proofs until they’re part of my ‘spleen’.
You da man Doc🙏🏼🤟🏼
I really liked it!! Please keep making videos of set theory :D
I took a course that introduced the concept of different sizes of infinity in my year 1 sem 1, everything seems blur. Now I am year 3, finally can appreciate the beauty of this. It's the most beautiful concept I have learned so far in uni.
Gotta love these videos about cardinalities!
Awesome video! Best content on TH-cam
Adàlia Ramon I agree!
more videos of general topology :) its getting interesting
Is it really necessary to find a bijection between the two sets? Wouldn't it be sufficient to find a surjection s:X->Y and another surjection s':Y->X, to show that they have the same cardinality?
At least for finite X and Y I think it would make sense, are there any trivial counterexamples for the infinite case?
It’s true but nontrivial, your argument relies on a variant of the Schroeder Bernstein theorem!
Dr. Payam - it might be helpful to number the videos dealing with a particular topic. This would make dependencies clear, and let people search for other videos they need to understand a particular video
That aside, thanks again. My Brain feels good after a workout like this! (It insists on uppercase 'B'. I have nothing to do with it.)
If one has a one to one relation in |R s.t. f(x) = y where f varies, does this mean that there is a countably infinite amount of possible functions to describe it (i.e. the cardinality is the same as the natural numbers)?
My god! That was beautiful.
Great !! Can you please make a video for |R×R| = |R| ? 😃
I’ll do once I find a simple enough proof! The one I’ve seen so far is very complicated :O
I can do half a proof. |R x R| >= |R x {1}| = |R|. I'll let you take the other half, Dr. Peyam.
Take the arctan(x)/pi + 0.5 of each number of each pair of real numbers, then construct a number 0.x1 y1 x2 y2... where xn is the nth digit of the first number, and yn is of the second. then multiply by pi, and subtract pi/2. You will get a number between -pi/2 and pi/2. Take the tan of that number, and you're done. proving the bijectiveness of each part is easy, hence the function is bijective.
Gilad Reti Yes, except there’s this tricky issue that 0.99999 = 1, which may pose problems to the bijectiveness!
Hmm... where? The first and the last part are just the standard bijective from R to (0,1) and vice versa, and I the middle we have only numbers from (0,1). Am I missing something?
Great video, man!!! :)
wonder, how about cardinality of function to R^n to R ? .its same or not?
Yes, since Rn and R have the same cardinality
Fun fact: Because of this, strictly speaking, multivariable calculus is as hard as single variable calculus :P
No idea what any of that meant.
I wrote a calculus exam today and played games for the rest of the day. My mental energy is depleted haha.
Is it possible to explicitly construct a 1-1 onto map from P(R) to F(R)?
Perhaps, but I’m not 100% sure how!
Ah, three videos to watch in sequence about sequences (or sets!).... this is a linear regression!! :)
Hahaha
sin(1/x) is not a function from R to R.
Tracy H If you define it to be 0 at 0 (for example) it becomes one :)
Also the cool thing is that even if your functions are undefined at some (even all) points, there are still as many such functions as P(R). :)
I'd like to see why |RxR| = |R|. I can't think of a bijective function.
Wouldn't a space-filling curve like a Hilbert or Peano curve be a bijective function linking RxR and R?
Cool.