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Fax uncountable ordered

after having dealt with the real numbers last time we're going to now redirect our attention to the complex numbers and what we did last time was show the set of all real numbers R is uncountably infinite that is it's not possible to put the set of real numbers into a bioactive relationship with the natural numbers or the integers or the rational numbers and it was one of two sets that we demonstrated to be uncountable the second set being the set of all infinite binary sequences and to summarize what we've observed in the past say four or five videos in terms of classifying some infinite sets we have the category of countably infinite sets that is these are the sets we can put into one-to-one correspondence with the natural numbers the trivial case we have that the natural numbers were countably infinite there's no deep reason why this is true this is really true just by definition more interestingly we showed that the set Omega times Omega the set of all ordered pairs of natural numbers is also countable and we demonstrated that through that Cantor pairing function we also showed that the set of all integers Z is also commonly infinite and very interesting way we show that the set of all rational numbers is also countable and then we had the second category of infinite sets the uncountable sets and as I say we observed last time that they set Omega to the set of all infini binary sequences is uncountably infinite we also showed this is a sort of intermediate step in showing that the real numbers are uncountable is that the unit interval the interval of real numbers between 0 & 1 not including 0 & 1 is also uncountably infinite and again we told that the real numbers are on combo and what we're going to do in this video is examine the by tractive relations to and from the complex numbers and what we're gonna do is we're going to show that the complex numbers even though they're constructed from the real numbers are still equinumerous to the real numbers that is if we can show this that the cotton that the complex numbers are equanimous to the real numbers then the complex numbers are also uncountably infinite now I made this supplementary video part 14 a where I talked a bit about the complex numbers and one can only talk a bit about the complex numbers since it's such a huge field but all of we essentially need to know for this video is that the complex numbers are defined by taking ordered pairs of real numbers and this is why they're often written as R squared or R times R and the complex numbers are essentially those those are how you build the objects in the set of all complex numbers and then we add some algebraic properties which tells you how to add and multiply the complex numbers and some examples of objects within the set C would be the ordered pair 1 0 or 1 and 0 are understood to be the real number is 1 0 and this object served as the multiplicative identity and we had this other object 0 & 1 which also want to see and this this serve that is the element I that we informally write it and we need to only know for this video that the complex numbers are objects that look like this just ordered pairs of real numbers and this is all we're gonna need to show this this theorem that C is equal numbers to R and really the only thing we have to do in this video is to come up with one new by ejection for the reason that the by trick ssin that we showed last time between the set of infinite binary sequences and the real numbers is going to take care of a lot of the problems that we may run into in this video now to outline the proof strategy what we're gonna do is we're going to start with this set of complex numbers our times R I'll enter change these R times R is going to be the same as see red start with our times R we're going to demonstrate first a by ejection between R times R and Omega 2 2 times Omega 2 now what is this set this is just the set of all ordered pairs of infinite binary sequences so you give me to infinite binary sequences and I stick them into an ordered pair it's gonna be a member of this set here so first I'm going to show a by direction between those two sets then this is the new step in this video I'm going to show that there's your by direction between Omega to 2 times Omega to 2 and just Omega to 2 that is there's a bi ejection between the set of all ordered pairs of infinite minor e sequences and the set of all binary infinite binary sequences and that's going to be the interesting part of this video really and then this last up showing that there's a bi direction between Omega 2 to 2 R that's what we took we took care of last time so that's why I say there's only really one new stop here and just to allow you to visualize what is so looking like we start with some ordered pair of real numbers let's call them X and y first we're going to convert those to some bitstream some infinite binary sequence and it's going to be a pair of though it's gonna be a pairing of two infinite binary sequences and these don't really have any meaning I just pick some random ones in tutorials but again it's gonna be this infinite pair of sorry that's the ordered pair of two infinite binary sequences and then what I'm gonna do is I'm going to take that pair I'm gonna collapse it into a single infinite binary sequence and that's gonna be this step here then finally I'm going to regurgitate what I did last time I'm going to take an infinite minor your sequence and map it to a real number so this is gonna be our proof strategy here and again for emphasis the new step is going to be this step here the step between the pairing of binary sequences and that single binary sequence and from notation purposes in this video the letter H is gonna correspond to that by junction so don't confuse that with the H I used last time so H is gonna stand for this new by direction that we're gonna want to create this means we have three by junctions to work on in this video and we're going to deal with that new by direction first the one between the the pairing of infinite binary sequences and the binary sequences first now we'll a restatement of what we have to do is we have to take the data contain in the two binary sequences in that pairing and store it into a single binary sequence that is if I had some infinite binary sequence so let's call it X made up of 0 1 0 1 0 and so on and another one y which is made up for example 1 1 1 0 1 and so on what I want to do is create some Z some new infinite binary sequence made upload the Z 0 Z 1 Z 2 Z 3 and so on which stores the data of x and y inside it and I want that first property that's going to store the data and furthermore if I present you with Z I would like to be able to retrieve x and y uniquely that is there should be some way of decoding the data stored within Z into x and y so there's got to be some unique way there's got to be some well-defined a way of taking x and y 2z the pairing to z and taking the single sequence z and taking it back to the pairing uniquely and that what what that saying is that this process has to be left invertible or injective the process has to be injective and at this point I encourage you to think about this problem since I think you can I think a lot of people can solve this very easily just by thinking about how you would take the data stored an ax and y and store it into Z if like what strategy you would use that is give me a formula for what Z 0 Z 1 Z 2 and so on I need Z K would be in terms of X's and Y's and I think a lot of you can solve this so if you'd like to have a stab at this problem pause the video and think about it a bit as a historical know I'd believe Kantor also thought of this but in my opinion this isn't a profound idea or anything and actually it's very well known to engineers where I would expect it's very well known to engineers and the fundamental insight behind solving this problem of storing the data of ax and y into Z is to systematically interleave the bits that is the zeros and ones from X and y2 formzee and this is the key statement to interleave the bits that is if I add X again this is the same as up here I'm just coloring it red and had that Y just calling it black but the same as up here the fundamental strategy here well one way we can solve this problem is to create Z but interleave the bits that is I'm going to form Z by reading each bit one by one alternating between the two sequences the Z is going to be formed by in this case 0 1 1 1 0 1 1 and 0 0 1 and I've color-coded those bits and where they came from just to show you what this is looking like that we interleave the bits and you can see this is going to solve our problem it's fairly obvious that's going to solve a problem because it's systematic how I'm encoding the x and y pair into Z it preserves the order of x and y there's no way it can be confused with Y and X and furthermore if I present you with Z you can retrieve x and y for me if I present you with just 0 1 1 1 0 1 1 0 0 1 you can tell me about X and wire if I told you that we're using this encoding algorithm and you just do that by reading off every other bit so by reading off every other bit I know that X is 0 1 0 1 0 and by starting at that next position and breaking off every other bit from that I have 1 1 1 0 1 from Y and I'll emphasize why it is obvious that this strategy is going to take care of a problem just using the terminology of search action and injection it should be obvious why this strategy is surjective that is taking a pairing to a single bit stream is surjective because if I present you with any bit stream if I present you with any Z you can tell me some x and y ordered pair that got mapped to Z that is every element in the range as a corresponding element that maps to it in the domain and that proves that it's surjective that any infinite bitstream can be decomposed into two bit streams and furthermore that the retrieval process the process of going from z to the single order single binary sequence to the pairing is unique it's injective and there's no way that you can go back there's no way that you can decompose it into two bit streams in other words there's no way that a pairing of binary sequences that two different ordered pairs get mapped to the same infinite binary sequence again there are lots of ways to think of injective I'd like to think about its left invertible you might think of the typical way of no two domain elements can map to the same range of value to be rigorous about what we're doing here let's introduce all the formalism all of the symbols now what we're going to do is suppose we have some ordered pair of infinite binary sequences let's call them x and y as we have been doing X is going to be given by some bit stream what I call bit stream and it's gonna be made up of X 0 X 1 X 2 X 3 X 4 and so on and Y is gonna be similarly labeled y1 y0 y1 y2 y3 and so on so just any two infinite binary sequences placed into an ordered pair and we're going to define that by judgin H which is going to take that pairing to the single bit stream and I'm gonna use this notation Z it is going to be H it's gonna be output it's me a thing that gets mapped to by the ordered pair X and y and z 0 C 1 C 2 Z 3 Z 4 and so on as the following and this is just what I'm doing before I'm gonna create this new Z by interleaving the bits I'm gonna have the first two bits be X is zero Y zero the next you're gonna be x1 y1 then actually gonna be x2 y2 x3 y3 and so on and this is just what I have written here again the first you are x0 y0 the next two are x1 y1 then x2 y2 and so on and to be precise about the indices here suppose I have some Z sub k k is gonna be the index of the bit and z so this could be z1000 z 5 million whatever z sub k is gonna be the same as X sub K over 2 if K is an even number and hopefully this makes sense because if I take a look at let's say Z 6 what is Z 6 it's even so it's gonna be an element from X from this first bit stream and it's going to be the third element it's gonna be nifty now not the third element with the element in the three position so Z six is equal to X of six over two or three and if you examine the bitstreams here this should be obvious here and similarly Z sub K is going to be equal to Y of K minus 1 over 2 if K is odd and just by examining this it should be pretty clear for example if if I look at the z77 tune odd number it's gonna get mapped to some element in Y and which element and why I take 7 minus 1 that's 6 divided by 2 and I get 3 so that tells me that Z 7 is going to be Y 3 and it's important to write out these relationships here because if I ask you what is the 84 element what is Z 84 that's very easy to tell because you can see that 84 is even so I use this first equation here and z84 is gonna be X and it's gonna be X not Y it's gonna be X of 84 divided by 2 which is 42 likewise if I present you with Z 501 501 is an odd number so you you apply the second equation here Z 501 is going to be Y of 501 minus 1 which is 500 then divided by 2 which is 250 so we can tell I can present you with any Z sub K if I tell you if K is even or odd you can tell me what Z sub K is going to be just to remind you of where we are in our proof sketch we just took care of this by Junction which I call H between the ordered pairs of infinite binary sequences and the infinite minor sequences and to remind you once more we've already taken care of this last up from the ejection between the infinite minor sequences and the real numbers we already have some by Junction and we talked about that in the previous video and I'll call that F for this video so the last thing we have to take care of is this first by ejection here mapping at the set of ordered pairs of real numbers to the set of ordered pairs of infinite binary sequences and let me call such a buy action G so that's what we have to define right now and just to allow you to visualize what G is doing G is taking some ordered pair of real number so let's call that X Y the pairing X Y and it's mapping it to a pairing of bitstreams and again I just picked these at random they don't really have any meaning but this is essentially what it's doing now since we already have a bye Junction taking binary sequences to real numbers what you call F that's that last up here binary sequences to real numbers all we have to do is to take each of those real numbers each of those components and map it to some unique infinite binary sequence and I claim since we have a function f which maps in the forward Direction the binary sequences to the real numbers to map the real numbers to the infant binary sequences we can just apply f inverse to each of the components and let that be the function G and to be precise about this G is again the complex numbers to the pairing of infinite binary sequences and G of some ordered pair X Y is going to be defined by the ordered pair with its first component F inverse of X and the second component going to be F inverse of Y and hopefully you've demonstrated some time up to this point that if F is a bijection f inverse is also by junction so we don't have to worry about that so I just apply F inverse component wise and again concerns about repeating once have already been taken care of by the construction of F and we took care of that in part 19 because this is something you have to worry about when you talk about things like this you have to talk about the repeating once problem but we've already taken care of that so since we've already taken care of that we may as well exploit that let me summarize what we've done in this video the last step by the way and all that stuff hopefully it's obvious why Chi is a bi ejection because it inherits the by directive properties of F so the logical reason goes F is a bijection therefore f inverse is a bijection and G is just applying F inverse to both components so that's gonna be by directive too so I hope we can see why G is gonna be my directive and and it's mapping the complex numbers to the pairing of infinite binary sequences so how would we map a complex number to a real number that's what we're trying to do after all so let's say we have some complex number a B or we can write this as a plus bi I'm gonna call that X first we apply G G is that function taking the complex number choose some pairing of infinite binary sequences and again that's applying f inverse to each of the components so the first bit stream is gonna be f inverse of a and the second bit stream is gonna be f inverse of B secondly I apply H and H is gonna collapse those binary sequences the pairing of binary sequences into a single binary sequence let me called see and I will note 8c by H of G of X and just for emphasis C is a single binary sequence and then finally I apply F to that see that single binary sequence to get a single real number which I'll call Y and Y just to notate it once more it's going to be F of C or F of H of G of X so we start with some complex number X and what we get out of it is a single real number Y and again it's a theorem that if each of the chains of this function composition are themselves by ejective that is if F H and G are by themselves by objective then the function composition treated as its own function is also by directive so this is indeed a bi direction between the complex numbers R times R and R and for emphasis we see that by action that we want to set up is the function composition F of H of G and then maps to complex numbers to the real numbers and since that's a bi action between the two sets we say that the complex numbers are equally numerous to R and since the real numbers are uncountable the complex numbers are also uncountable and along the way in this video we've also shown that the set of all ordered pairs of in front of binary sequences Omega to 2 times Omega 2 is also equally numerous 2 Omega to 2 and we derive that conclusion from creating that by ejection H that is the set of all pairings of bitstreams of infinite bit streams is equinumerous or it can be put into one to one correspondence with the set of all single bit streams all single infinite bit streams and more generally if you're an engineer we investigated how to encode multiple bits into a single bitstream and this technique generalizes to encoding let's say three or four different vid streams into a single bit stream by what's called multiplexing the engineers use the term multiplexing to do this the strategy is basically what we did in this video and here are the two practice problems for this video in the first I asked you to show that r3 and r4 our force also called the quaternions if we attach an algebraic structure it's called a quaternions are also equinumerous to art that is we can not only add the complex numbers to the set of on composites we can also add r3 and r4 if you if you can prove this and secondly I ask you to do some sample calculations with these function compositions for example what does 0 0 map to 0 0 is understood to be the complex number 0 0 what is that map to what real number is that map to and relatedly what does the complex number point one point one or 0.1 plus 0.1 I map to which real numbers of the same map 2 and this is a pretty interesting question I think if you want investigate some some continuity aspects of this by ejection and that'll wrap it up for this video thanks for watching and if you enjoyed the video feel free to like the video and subscribe and stay tuned for more videos