One important topic in higher mathematics is known as equivalence relations and equivalence classes, so we'll introduce both of those here. So, to begin with, I can talk about what's called a binary relation, and the idea is I have two objects, and I want to some way of comparing them. So, for example, if I have two numbers, A and B, we might have something like this binary relation, which says that A and B have some sort of relationship to each other. Or maybe I have this symbol, and again the idea is that A and B have a specific type of relationship to each other. Now, this symbol and this symbol have very specific meanings that everybody understands, but maybe I need a new type of relation between the two things. So, I'll invent a new symbol, squiggle, A squiggle B, and because nobody knows what this means, I do have to clarify when I can write down A squiggle B, and maybe it's that I can write this whenever A minus B is an even number, and so here's a new binary relation. And so, I can invent binary relations that happen to do whatever purpose I have in mind. So, just because you can invent something doesn't necessarily mean it's useful, the most useful of these binary relations are those that are called equivalence relations. And they are defined as follows. An equivalence relation is a binary relation, and we need three key properties. One is that it has to be reflexive. Something has to be related to itself. It's also nice if our binary relation is symmetric. A related to B implies that B is related to A. I can reverse the order of the term. Similarly, if our binary relation happens to be transitive, well, that's a great big bonus. It tells me that if A is related to B and B is related to C, I can drop out the middle man and I have A related to C. And if my binary relation is reflexive, symmetric, and transitive, it gets promoted to what's called an equivalence relation. So, for example, let's consider the set of integers once again, and we have two binary relations. This one we read as our ordinary equals. This one we read as our ordinary greater than or equal. And let's see which of these two is an equivalence relation. Now, remember, as a mathematical question, that could mean that both are equivalence relations. One of them is an equivalence relation, or maybe neither is an equivalence relation. So, we will have to check out both of them. So, I'm going to take a bunch of things in our set of integers and check out our three required properties. So, for equals, for A equal to A. Yeah, I'll buy that. That seems like our relation satisfies reflexivity. A equals B implies B equals A. Yeah, I agree with that. That sounds good. And let's see, transitive A equals B, B equals C. Can I drop out the middle man? A equals C. Thinking, thinking. Yeah, I think I agree with all of these things. So, this equals is reflexive, symmetric, and transitive, so we get to call it an equivalence relation. Well, how about the other one? So again, I'm going to take a bunch of things in our set Z, and I'll check out is this greater than or equal to reflexive. Is it true that A is greater than or equal to A? Yeah, okay, that sounds good. I like that. How about symmetric A greater than or equal to B implies B is great. Wait a minute, I don't believe that's true. So, that does not work. Greater than or equal to fail symmetry. And because of that, it cannot be called an equivalence relation. Now, what makes equivalence relations so useful is that when I have an equivalence relation, I can produce what's called an equivalence class. And these equivalence classes emerge as follows. What I'm going to do is I'm going to take an element in our set, and I'm going to form the equivalence class of that element by finding everything in the set that is equivalent to our element. And one bit of grammar here, we might say that the equivalence class induces a set of equivalence classes that forms a partition of our set. Now, it should be fairly easy to prove. You should prove them. In other words, the following. If I have something in an equivalence class, then the equivalence class of that something is the same as the equivalence class that it's in. One of two things also happens. Either two equivalence classes are identical or they are completely disjoint. They have nothing in common. This is why they actually form that partition. So both of these should be relatively easy to prove, and you should take the time to prove them as they do remind us about some of the key properties of what an equivalence relation is. For example, let's take a look at this. So I have two things, and these are ordered pairs in N to the set of natural numbers, and I'll define my equivalence relation as follows. AB is equivalent to CD whenever A plus D, the sum of the outer terms, is equal to B plus C, the sum of the inner terms. Now, remember, all of these are in the set of natural numbers, so plus makes sense in the set of natural numbers. And let's see if we can find three distinct equivalence classes and then find a simpler description of the corresponding sets. So that is an equivalence relation. Are you sure about that? Because if it's not an equivalence relation, equivalence classes don't exist, and you'll waste a lot of time looking for the snark. So we can create an equivalence class by choosing an element of N2. So I'll pick one. How about 5, 3? So here I've chosen an element of N2, 5, 3, and it produces the equivalence class, square brackets, around that element. So this is our shorthand notation for the entire equivalence class produced by the element 5, 3. Let's see. Well, I want to produce three distinct equivalence classes, so I'll pick another element. How about 2, 1? So this element 2, 1 corresponds to the equivalence class 2, 1. By 2, I have to make sure that these two equivalence classes are not the same thing. And finally, 3, 2 corresponds to this equivalence class 3, 2. And let's do a quick check here. The next thing that we do notice here is 3, 2, and 2, 1 are equivalent. Remember, they are equivalent whenever the sum of the outer terms is equal to the sum of the inner terms. So 3 plus 1 is 4, 2 plus 2 is 4. These two terms are equivalent. So this has to be in the equivalence class formed by 2, 1, which means that the equivalence class formed by 3, 2 is the same as the equivalence class formed by 2, 1. These are not distinct equivalence classes, so I don't need the second one. What else can I do? Well, I'll pick something else. So the element 1, 1 corresponds to the equivalence class 1, 1. And you should verify that these are actually three distinct equivalence classes. This is not in either of the other two. This is not in either of the other two, and so on. Now, let's consider any of those equivalence classes. We do want to see if we can find a simple description of the corresponding sets. For example, let's take 5, 3. Now, again, our definition, 5, 3, is the equivalence class where we're looking at the things that are equivalent to 5, 3, where x and y are drawn from the set of natural numbers. Now, 2 ordered pairs are equivalent whenever the outer sum is equal to the inner sum. So that's x plus 3 is equal to y plus 5. And maybe I can rewrite that. This is x minus y equals 2. So I could rewrite this equivalence class, 5, 3, to be the set of ordered pairs where the difference is 2, x and y are in the set of natural numbers. Now, you might ask, well, why didn't we just start with this? Why didn't we just start with this and not go through all of this tedium here about talking about equivalence classes? The answer to that is here with this equivalence relation defined, I'm doing things that I can do with any natural numbers that I want. Given any two natural numbers, I can add them, I can add them, and then I can compare them. The reason we didn't start with this is that here, this x minus y, I can't always subtract two natural numbers. I can't subtract 1 minus 7, for example. I'll have to introduce something new for that. And if we're trying to build up our mathematics rigorously, we do have to be careful that we don't leave the yard. So here, my yard is the set of natural numbers. I want to make sure that everything I do can be done within the confines of the set of natural numbers. So this makes sense to us, but that's only because we've learned a lot of mathematics. We've developed a lot of mathematics. At a higher level, we do want to be careful that what we're defining isn't something beyond what we're allowed to define.


Hi and welcome to the start of day number two, Taka Bambina 2015. Here in Fitianga in Blacksmith Lane, the cars are about to head out onto the first stage this morning back over to Tai Rua. From there we head to Waihi through Matamata and by the end of day we're back in Rotorua tonight to end day number two. So there's 136 kilometres of special stages today, over seven stages. So there's still lots to do. It's still very, very early in the event with day three obviously tomorrow. So we look forward to bringing you more target tales throughout the day. I hope you've enjoyed the coverage so far. We look forward to talking to you at the lunchtime service. See you then.


Okay, gentlemen, thank you for waiting this setup. I'm happy to be here to present to you the Collabora Online that is based on LibreOffs Online. I was expecting to have, it's an introductory text and introductory presentation. I see a lot of developers, some of my colleagues, so if anything goes wrong, I have the support here. But basically the idea is just a simple introduction to this software that we are developing in Collabora. And to give you a glimpse on what are we preparing in terms of LibreOffs Online. Okay, so basically the idea, as I said, was to make a small, a short demonstration of the features that we have already implemented. So basically what we have here is an installation of on cloud with the files that you have in your space area. And if you want to edit or display one of the files, inside the Collabora Online, you just click on it. I will click on this file here. And then we have Google Chrome is really sometime boring. So what you see here is a presentation in Collabora Online. And you can have, you have the right pane with the thumbnails. You have your main edit area. You have a toolbar on the top. And you have also some status bar on the bottom. And then you can either edit your presentation or run it your presentation. I will just show you to the presentation as it is. It is basically about Collabora Online, LibreOffs Online. And this is running into one of our servers. And you saw the transition of the slides. So we have some implementations of the transitions. And when you just hit your click in the mouse, there is the presentation, okay? So it takes the full screen. It's the same rendering as the desktop. So you have a quite faithful reproduction of your application. Why is it so? Because of course LibreOffs is running behind all these interface for the web that we have, okay? So basically this is what we have. Collabora productivity is as you know, the version produced by Collabora of LibreOffs. And then what we have basically in terms of the company, we are the leaders in open source consulting. We have more than nine years and 80 collaborators. We formed the Collabora productivity division three years ago, dedicated to Enterprise LibreOffs. And we provide level three support for code issues and to clients including some of the customers. And we have the architects of Microsoft OpenXML filters and for Android and online, okay? This is the commercial part, right? But what we have so shortly in terms of LibreOffs online, we have Writer, Caulk and Impress applications with full content edition of Microsoft Format and ODF formats. And we have also Collabora editing, collaborative editing, which means at the time of this presentation we have one editor and multiple viewers, okay? Later on, in short time, we will have the possibility of having multiples editors, okay? Well, so this is what we have. I will now show you how we work with the Collabora online. Okay, so we'll escape from these modes and turn out into the edit modes of the application. So what is the interface? We have, as I said, these thumbnails on the left, we have on the bottom here some four icons, which means icons for presentation to add a new slide, to duplicate a slide, and to remove a slide. So if I want to duplicate the introductory slide, then I have now two slides with the same content, okay? And of course, I can select it here and then remove by clicking this button. Of course, Collabora online detects the local of your browser. And for one reason, I don't know, this is in Portuguese because possibly my account in on cloud is in Portuguese. So I have the message in Portuguese, okay? So I can delete it here. And there we go. We have deleted a slide. Also, basically what we can also do is of course edit the content. So I can here, oops, sorry, edit contents. And now I can type whatever text I want and you will automatically store it in your file. So I can type whatever I want and you see the response time is quite good for the keyboard because this is being sent to the server and back to the browser. Okay, it's not a local edition. And also, you have seen in other presentation that actually this area here where the slide is is an image. And whatever I touch, we touch and we edit. This image is divided into tiles. Each modified tiles is validator or invalidated and is refreshed by the underlying system in the server. Okay, you correct me if I'm wrong, guys. I use my layman terms. Okay, so here is a four edition. You can of course use one of the top buttons and the list box to display your, to modify your text. Okay, so you can also, for example, do what you do in your desktop. You can select text, right? And you have these two handles to make it sure that you can grab visually what you are selecting. And you can change with direct editing. You can put into italic. You can change colors. Just doing some exercise here to, for example, I may put this background. Okay, so I have full direct edition of the text. I also can change of course the size of the font and also the font itself and make alignments and other things. I also can, because the screen is a bit short, so I have a continuation button here. I have further, I have others commands. I can insert images and I can also make some alignments and create a table, okay? To help me to control visualization, I can also use the zooming. Okay, we have a zooming possibility here, so I will just click a bit and you can see how the tiles are redrawn to accommodate the amount of enlargement that you want. Right? So this is very smooth to use and very nice also. So you can shrink your presentation and your slide if you want. Or you can also click here and restore to 100% of editing, right? So it's pretty nice, very effective and you can do a lot of things with all the implementation done so far. Okay? I'd like to show you also how is spreadsheet rendered. And so I will close this presentation and I will open a spreadsheet, right? Here we are. I'm closing it and I'm opening a spreadsheet. So here we are in terms of rendering. Let me just take this chair elsewhere. I don't... So basically you can do a lot of other edition. You will recognize that you have on the top a formula bar. So if I click on a cell, okay, I click on a cell, I get the formula that is in the cell in the formula bar for the edition, right? This layout is exactly the layout that I have in my desktop version of LibreOffice and the same kind of edition. I can also do, for example, now because it's a spreadsheet, I can change the currency, remove the currency symbol. We can change the percentage. The number of leading and zeros or trading zeros, okay? So all this kind of edition is immediately available for a cell and also you can do selection, okay? Selection of cells. And here we are with, oops, because it's a combination. Let me take another sample. So I can select cells this way and apply commands into the selection, right? In our status bar, we reproduce some of the information that we have in the desktop and we can do a lot of things. For example, there is here the number of spread of sheets, the number of columns and lines and rows selected, the kind of selection. Also, we implemented a nice feature available on the desktop, which is what to do with the numbers that you've selected. So currently, by default, it's a sum, but you can have all the other operations if you want, right? We also can activate the zooming. I can here shrink a little bit and also restore 200%. And I also have one search bar here that allows me to locate specific string, okay? That's too easy, no? Is anybody using spreadsheets? Or, okay. What you can do is I put this yellow box here, what you can do with the current version. Row and column size and adjustment is already available, but in the next update that we do, we can insert and remove columns. For example, here, I can with the button, with the context menu of the, we can insert a column before. So I have now shifted everything because I inserted a column. I can also delete the column. And the same for rows, okay? I can do selection of data with cut paste and copy. Also do formatting, formula addition, zoom adjustment for full view screen. This is, I can increase here if you want more detail on your zooming, right? And also do and redo, undo, redo, insert image, insert and view and hide comments. For example, I have comments in the spreadsheets. Okay, so the context menu here, the text and a comment. And I can show the comments here, sorry. Here. Here is the comment on the cell E or F2, right? Pretty effective. That kind of implementation address a lot of use of simple use of spreadsheets. And of course, as soon as the product is evolving, we will have more and more functions. But of course, we need customers that ask for that, okay? Customers are very important. So we are able to display more than one spreadsheet. And this is a file that you can obviously upload from your desktop and use in the online edition, right? Very well. We have also in all the modules, the possibility to save, to print and to save as if you want to or download. You can download the file as a PDF. You can download as ODF or Microsoft Formats. These are the three formats that we are using at the moment. And it covers perhaps 95, 99% of all the customers demand, right? Okay, any questions? No, no question. When the feature XYZ will be implemented? Is this what? Yes, how do we? The editing on slow connections? Yes, because. Well, I live in Brazil. When I use Collabora Online, I really don't see absolutely nothing disturbing in terms of delays. I have the same, what you see here is the same I see almost in the other side of the world. So of course, my connection may be fast, but the connection is not actually the main issue for the time being, okay? But of course, if we have a slow modem, there's no way. Because in Egypt, most people are running less than 500 kilopytes per second connections. How much? 500 kilopytes per second. Less than that, most people are not sure. Yeah. That's something that we optimize it quite a lot. We spent a lot of energy optimizing the way to refresh the tiles, to optimize the, to send only the necessary tiles to be updated. But we cannot go further with a very slow network. That's an issue, of course. But it's an issue for Google Docs and for other solutions that are on the cloud, right? Not sure if I, Michael, maybe you want to. So yeah, we require something like 100 kilobits per person per second. So 10 kilobytes a second per person. So, you know, 100 people on a, well, 1,000 people on a 10 megabit connection. 100 people on 10 megabit connection? Something like this. So, you know, I think in terms of any realistic use, that's quite a high bandwidth, right? If you look at NX, they're arguing that they're 20 kilobits. So I think there's significant room for improvement. And we have some pretty funky ideas to do that, to reduce the bandwidth. But it's good to have a product that you can ship. That's very useful for a subset, and then you can optimize it.  And if you're interested, you know, there's a whole lot of good text, things that we want to be doing. The joy of the bitmaps are that, of course, you have your full text layout, or you're shaping all of your complex text, all of your Arabic, Thai, Northern, whatever. Martin, shoot. Ask about high latency, like 0.4 seconds. Yeah, so high latency is also a problem, just because of the way it's done. But again, as we start to move text, rendering into the client, and glyph stamping into the client, I hope we'll be able to hide a lot of that by letting you appear to carry on typing somewhere. But I mean, ultimately, each character you press can relay out the thing. So yes, it's on the roadmap. We're aware of it at the moment with a 0.4. We're looking at 150 milliseconds, sort of, you know, to get something reasonable. Make sense? Did that help, Olivia? Yeah, I hope so. Did it answer your question, Khaled? Yeah, cool. So, time is running. I have, I'm showing you what is the rendering of a document in the online, callable online. So you see exactly the exact layout that you have on your desktop. We, you see that we have comments. And if I, just to make sure if I want to type something, you will see how it is fast, and it is update. Sometimes you see the image refreshed to be a bit shuffling, you know? But this is the sending and receiving of updated tides, okay? So basically, what I would like to show you, let me see if I can show you is to display two sessions. Okay. And see, oops, okay. So just to put the two side by side. And then when you, one is the editor, this, the piece on the part on the left, on the right here, sorry, on the left is editing. So I can insert text here, and you will see how, oh, there is a bit. Okay. And you see here how you have the addition displayed on both sessions, okay? These are separate sessions. One is a session that is in a private mode and the other one is normal. So I really have two access, different access on the same document, okay? So this is what I wanted to show to you, what is being cooked by Colabora. And we will have very soon some new implementations and everything is available. All the development of LibreOffs Online is done within the TDF infrastructure. I mean, the repository, the commits are in TDF, okay? Questions, no questions. Yes. Thank you. Is there an estimate which infrastructure should I use if I have a company with say 10 people, 100 people, 1,000 people? Oh, this kind of value is still being evaluated. Our engineers, of course, have some ideas about that. But, well, everything is being measured and monitored but I don't have myself numbers. Maybe the team of Colabora can give better numbers, okay? So I'll say that again.  We need 10 users per CPU thread. So if you consider your laptop there, probably has four threads, so that's 40 people. And 100 megabytes of RAM, your laptop there probably has easily enough RAM for that. And 100 kilobits. And again, you know, you probably have a gigabit network connection in your laptop.  That's that. Any further questions? Over there, let me close if you want. Can I have two different views of the same document in the two windows? So, for example, on one side is the annotations displayed on the other, not. As in comments versus not comments. So two different views of the same document, currently not. So you have to see those for all of them. Yeah. That obviously means that you render the view once, which saves quite a lot of time. So there are some compromises like that that make things much quicker. Possibly Ash will, you know, say something like that. Yeah. Makes sense? Okay. Well, thank you to the guys for their attention. And let's see if next week we come up with much more. Thank you.


Really? The one episode I don't watch on time is the good one? Ooh! That's a hot mug, guys! Hey guys, this is my review for Supernatural Season 15 Episode 17. You might be kind of wondering what's with the get-up here. Well, it was Halloween work day and I dressed up as my Mad Max warboy combination and I haven't had a chance to wipe it off because I literally got home just a few hours ago and I've just finished this episode and some people were saying this is the best episode of the season so far. It's not hard to say that that ain't false. This episode has the brothers kind of coming to a head. We have Amara kind of coercing, trying to convince God to go back on his decision while Dean is taking Jack off to... turns out to be Adam, like original Adam, who actually in fact the actor who's in it was in Battlestar Galactica. I knew I recognized him but he was in Battlestar Galactica which is like over 15 years ago which is crazy. They get his rib because apparently his rib has absolute incredible divine power. All the while Sam is trying to find out another way. I do like the interactions between the brothers in this episode are the most part. The first argument that they have is actually pretty solid. They're writing isn't bad. Dean isn't a cartoon. It's actually pretty solid and I like how while the whole Jack walking into the end is kind of stereotypical I do like the dialogue between the two and then while the whole what was Sergey looking for the key to death's door kind of MacGuffin thing is a bit far-fetched. I do like the entire sequence of when they go in. I like how Sam comes in and there's bodies everywhere and the door closes and I first thought it was a visual but it was actually just a really well done practical effect. And then Meg is there once again. Cool to see this actress again. Admittedly though I do feel bad for her because you can see she's fighting it.   And it was very nice to see her again. We find out what Billy's plan has been this whole time. Apparently she just wants to reset everything. Which to be honest I kind of thought that's what was kind of on the table already. When you kill the two main deities you have to kind of hit a reset button. That's because I've been watching The Good Place too much. Maybe that might be why. The whole twist of Billy being not what she appears to be isn't exactly a big surprise considering the lacking amount of interaction we've had with this character in this final season and just how repetitive and boring she has been in terms of what her presence is. The fact that she doesn't appear in this episode at first kind of niggled me but in the end I actually don't mind that. I think that's a good idea. And then they come back and there's a great slash horrible slash good conversation. Two-way conversation between God and Amara and Sam and Dean. Dean brings Jack into the building with the bone marrow bomb getting ready to start off in him and Sam tries to convince Dean. He has the book now. He has all of the plans of what Billy is trying to do but Dean doesn't want to listen to him. He wants to just put Jack in there and set off the bomb. Amara is trying to convince God, hey we have one more last chance. You can join us but in the end she gets coerced by God. So it's a reversal of conversations whereas Sam is trying to and admittedly succeeds in changing Dean's mind. Amara fails and in the end succumbs to God or Chuck's ideas.  For instance when he just lets out that big top-notch quality writing there guys. I'm kind of at the point now where I'm accepted that this is this new character. It's not the same Chuck but theory that happened at the end of season 14 where they thought that Chuck was actually the empty. That's making a hell of a lot more sense than this is even though it's clearly not what's going on but I did enjoy the more mature conversations and then while the one with Sam and Dean did get a bit ridiculous where he pulled out the gun and they started fighting each other it slowly progresses into the brothers actually making amends with each other and coming to believe and trust one another. The dialogue that Jared Padalecki gives his delivery is really really good. Dean almost dips into that cartoon level but thankfully he's bought right back and the editing is great too. The editing back and forth is very very well done in terms of concept, in terms of actual forward based story. This is in fact the best episode of this season. Sure you got to accept all the ridiculous redundancies that this show has done to its characters, to the story, to the lore itself but on its own basis this is the best episode of the season so far.  I think it was in this season. It was after Rowena died. Those three are so far the best episodes of this season. Those are the three that I will commend. Three. Seventeen. Not a good ratio guys. But I do enjoy this episode. Is it as solid as it could have been? There are a few moments that are just a little bit odd. The whole Adam Eve thing. The rock test. The whole childness of God in certain situations. Them going to what I assumed was heaven but nothing about the whole fact that heaven dying thing. None of that. Still don't want to touch that. But it was decent. I was entertained. So I'm going to give episode 17 a five out of seven. I actually thought this was an okay episode. However, the end part did get me chuckling because of like hey guess what you're going to watch him die. I was like oh cool. Let's watch him die for a fourth time. Or is it a third? I can't remember. Jax died too many times.  Anyways guys I hope you enjoyed this review. Sorry for it being late but like I said late night last night and I just didn't want to do it. This might not come out until Saturday morning but we'll see. Anyways I hope you guys enjoyed the review. If you did leave a like and if you're interested in more subscribe. Give me your guys's comments. We're now in the final three episodes. So make sure to give me your comments on what you guys thought about this episode and I'll do a video talking about your guys's comments soon. Thank you guys. Happy Halloween. See you guys next time.