Wel i gynllun o Tafffi TV, mae'n gweithio'r awsiau Everson o'r Arsynol 530 Gwdysun Park yn ystod yn gweithio'n gwneud o Andy the Piv, ond mae'n gweithio ym mhwy fydd yn ei gael. Rwy'n gweithio'n gweithio'n gweithio'n Tafffi TV, rwy'n gweithio'n gweithio'r clubau yn y Llywodraeth Prif. Rwy'n gweithio'r Gwyddaeth Ffwrddol, ac y gallai yma mae'n gweithio'n llun gyda'r ar draws i unrhyw. Yn y gweithio, mae'n i gweithio'n gweithio'n gweithio. Rwy'n gweithio'n gweithio'n gweithio'n gweithio e-taro, yn eu cwmhau'r gweithio'i ddiwedzol i gydag ffyrdd o'r fwyllt. A byddai'r digwydd o ddiddordeb yn ganodd. So on to the football Arsenal Goodison Park, tough game, and now they are 15th place at the moment and not doing what they should be doing, giving some of the quality of their players, but still represented tough game, doesn't it, forever? The quality players, I know they're not particularly cohesive, playing as a good side at the moment, they've got their issues off the pitch as well as on, but you can score goals, and you can see last season how integral he was when you scored goals for them, so it's still tough game no matter what form Arsenal is in, so it's a bit of an interesting one really. Will be an interest on, I mean, the last four seasons of Goodison Heaven was one two drum, one last one against Arsenal, and a couple of clean cheats I think in the last two, then you won 1-0 and drew 0-0 last year against them, but always a difficult game, isn't it, they've, and now mainly it's away from Goodison where they normally hammer us at the end of it, but it's still a tough game. You look at our tether, it's been in there a year now, as we're coming up to a year, say, in Meshcarlo, as one in FI Cup, so you've won a trophy all the way before them, but obviously this season, I don't think anyone would have predicted it'd be as bad as it is for Arsenal, the 15th, you know, they've got 14 points, they've scored 11 goals, it's not going to plan, is it? No, you know, we'll talk about XG, but they're only under-performed by three goals, really, which is not a lot considering other teams who won the title last year was much bigger, but there's fundamental problems with the team, and I think the type of person is, he's tried to change things quite massively by leaving Ozil and Socrates out, and I think they're big presences, and it's hard, I think he's tried to go too fast too soon in a lot of ways, and you need to sort of balance that. But I think Arsenal's a club anyway, we've got a lot of problems in terms of off the pitch, the recruitment, the relationship, which I don't know if I haven't, I've also got a relationship with Keir Gerabchen, but Arsenal's is much more in the pockets of the number of players they've signed, Louise and Cedric and others, so yeah, I think it's been a tough year for them, but you look at the difference stream in Meshcarlo since last season, when they both joined the box in days, I think Meshcarlo's now nine points ahead overall in terms of the number of points they've accrued, so we're going one way, and it seems to be stagnating. I don't know, as you say, they did win the FA Cup, but the season's just not clicked at all, and they're still not creating enough chances, they weren't creating just about enough last season, Obama Young obviously with his finishing was getting them out of a lot of trouble, but it just doesn't seem to have taken that next step that they wanted to. Yeah, I mean, you look at the team, they have got some good players, Obama Young's incredible football, they're brilliant, and he's scored his first goal this week since October, which was a penalty, the last goal he'd scored, but he's scored against Southampton in the week.   Mae hi'r trogyn nhw'n ddim caen y blaes, i am cair yn tain, he's a decent left foot, you know, party, you're looking through and going well. Some of the parts are there for the money, but it's, I guess for me, looking at Arsenal from the outside, we've been, not brought up, but we've been used to a very attacking team for so many years under Arsenal and Venga, they were brilliant under Venga, now, listen, Arsenal fans are telling me better. It's easy to look at a club from the outside, but 24 years on the run in the Champions League under Venga, lots of goals, lots of trophies, okay, they never won the Premier League, you know, as often as they probably should have done, but they still won trophies, they still play great football. And obviously then, when they left, it was always going to be difficult going in after them, em, em, they come in and it was okay and got them to the Europa League final and they had spells, didn't they, where they win like 11 games on the run and then they wouldn't win again for it, so he couldn't quite get the balance right. And he decided to replace him, and obviously he'd gone with McElr Tether, and a year in, okay, I guess he hasn't gone the way he hoped he'd go. But it's still only a year, isn't it, and we're only basically a third of the season gone out, so is it about being patient? It depends really, I think their fan base is much more tricky than ours, you know, one of the teams you've, you know, sort of want to get back to where they used to be. I think, you know, you know, the, as a club, they seem to have two different plans going on. I think when you bring artists and you think you've got a lot of young players coming through, we want them to work with them, coach them and develop them. But then they've also got this little, quite an agent squad in a lot of way, where they've given a lot of long-term contracts, and they've obviously brought William in and some Cedric. They've had Mustafi, they've had, they've got Louise, who's on the contract at the end of the season, Ozil or Bani Angan. It's a bit of a mismatch there, and I think for a first-time coach as well, they're quite big personalities to deal with. And I think, you know, the way Arsenal got about it is maybe they either need patience or they need to go a completely different way in bringing someone who's much more experienced. You can handle those big personalities. So for me, I'd stick it with them. I think, said on Twitter last night, I think they've got a lot of contracts coming off the bill at the end of the season, which means that they didn't qualify for Europe. They probably saved them enough. Those contracts would be the same as they would in Europa League. So it's one of those, but it's how long, you know, and obviously if they get in relegation trouble in January, it's the board start getting nervous then. So, yeah, they're in a bit of a weird quagmire, but much like us, they've got a bit of an unbalanced squad. I think, obviously, Carla was a bit more experienced in getting the way, you know, smoking mirrors and getting the best out of players. It's trying to impress on them a system of control through possession. I just don't think they've got the players for it, to be honest. No, I mean, I do look at the, you know, look at the team the other night that they had, you know, Lenow and Gold, they had Holding and Gabrielle Centre back in the Elmeni, Kebalos, they had Saca, Maitland Niles, Edinacette, Abamiang, Pepe. You know, we've got good players there that, if they can get it right, should be doing better on the bench like I said, William. You know, again, two very attacking players who can win games, neither of them came on against Southampton, which is strange. But, yeah, they're in a little bit of a sticky situation. Arsenal, I've seen people, you know, tongue in cheek going on, they could be in a relegation battle. Arsenal, it's far too good. I'm going to be involved anyway in that. They'll be more likely to be fighting for Europe in places. You don't get relegates at world class strikers. No, that's the answer. But it is, they are in a sticky situation. So, from an Everton perspective, which is obviously what we're looking at, it's a good time to play them, isn't it? Obviously, the only lingering thing is that it felt like this when we had Man United. I've seen a lot of stories today, you know, of Everton winning a tether, I'll be shocked on Saturday night and all that. We had all that with Solshire, that Man United came and spanked us, so that's your only thing. What I would say, the difference between then and now is that we look like we've got a little bit of belief and solidity back where, when we played United, we were still letting too many goals and we were still poor as well. Yeah, we played like the way we played against United, against Arsenal, it made the game much harder for us. I think if we play like the way we've been playing the last two games, I think that's where Arsenal fall into trouble this season, when teams have got behind the ball and got organised, because they try and break them down, they try and put too much impetus and try and get against the final there and score goals, which leaves them open at the back. I know Gabrielle's done well this season, beholding David Louise, the famous steak pro when he's played, and they haven't really got a defence in the field. The other party was both, but he's not really that sitting midfielder, so they haven't really got the players to play the way they play and it leaves them quite open. I think for a team like us, without that piece on the counter and that ability, if we can force them to cross it, you give us a good, you don't worry about me and can score heathers, but you give us, you think against Chelsea would be fine. They're quite hard to break us down, so we put ourselves in a good position to win the game, and I think that's solidity, making Arsenal way harder. I think they're much like Tottenham, we're at quite the stage of the Tottenham squad in terms of Harry Kane and Tony Cavill who has done well this season, but we can use that sort of way of playing, sort of cause Arsenal problems, and I think we definitely can win the game. Is that where we're likely to win it then, by letting them have the ball and us trying to counter attack, particularly the make-up of our defence at the moment? Because obviously we've got no Llew Cyddi, we've got no Sheamus Coleman, we've got Alan out now, James Rodriguez maybe in the squad, we don't know Sheamus Coleman maybe in the squad, we don't know. So what do you think Everton's best way of approaching this one? The best way is they haven't asked to make mistakes on getting those turnovers and putting them, then obviously getting those man advances up the pitch. I think without Gabriel, they'll probably be holding much too short of Louise's back or not. Yeah, they come on. So you'll be Louise. So, you know, at Fancy Donner Carver, Llewyn against either of them, and you're with Charleston as well. I think Alex will, we can cause Kearney problems if he plays or Hammers, whoever it may be. So yeah, I think we can have joy there. You know, I think the way, if we keep that place at the player of high intensity as well, I think they'll struggle with that because I think they tend to hold on to the ball too much. Obviously, the way we're going to play is we want to force them to cross it because obviously Obama isn't going to be on against me, they're going to key, and that's what you want to avoid. So that's what we need to really avoid. So yeah, I think there's a clear path and plan to win the game. It's just executing and I think if we do it the same way we did against Charleston, I think we can win. It's just letting them in the game. I think they were getting back into the game the night against Southampton before Shach had got injured because I got sent off. I think I was saying to you last night that that. I think I said to Gabby that was Shach. Shach was last week. So the average does it sending off every seven games under that set, which is crazy really. That's a lot of sendings off really. So there's a bit of ill discipline in the team and I think we can force them. If they force them into areas they'll get frustrated and then we can start getting our set pieces which I think we'll do well. So there's a number of ways we can win the game but I think keeping that same base of solidarity and forcing them to make mistakes themselves is how we'll win the game. I guess from my perspective the key thing will be first and foremost you expect Jordan to expect Pickford to be back in the team. Robin Olson obviously coming against Leicester kept the clean sheet. I'd expect so. I think yeah I'd expect so whether that's the right choice or not. That's up to the manager. He sees them all the time. Again I don't think Arsenal are going to be putting Pickford under a lot of pressure from crosses and obviously a couple of players can shoot from the distance which is another thing. But no I think I don't specifically see this as a game with a good target. Pickford's weakness is over all just he has to play mistake free football and go on. If he can do that we can win the game of course but I think I'd expect to see Pickford back and go on. And then obviously the other key area is the centre of the park really because obviously like I said before losing Alan is a cushion blow to Everton really. He was just playing. He's been excellent since he came in from Napoli and was brilliant the other night until his hamstring went so. The manager's got decisions there. It's whether Tom Davis comes in and plays in the six. Whether he just puts the corey there like he did at Leicester and puts someone in there. Go home as in Sigarton he decides of them. I don't know whether I'd be that happy about it or whether it's a different make-up. Whether it's Coleman back in and Hogan in the field. So it's going to be key getting the centre of the park right to keep the balance. I think personally if he keeps the back for the way it is. Then he gives us a good chance of staying solid because them lads like to defend first and foremost. Then it means we don't have to go as high up the pitch because obviously we don't want a bamiang running off key and mean it.  Try to cut the space down behind us. And then from that perspective then it's really it's about can you break from midfield. Now if he goes with the corey as the whole midfielder. Whoever he puts in that midfield with him isn't going to be quick enough to break from midfield. And now it isn't always about speed because it's about timing. Sigarton you'd expect or hope could get up alongside them. But I think if he put Tom Davis and sat him like he did against Brighton. The corey then would have the option for breaking into the box but also covering. Then if it's gilfy or if it's Gomez they have got a bit of a responsibility to drop in on Tom Davis as well. I mean that's a key area for me. I worry when you say Gomez because I don't think he can do that. I think we need people to come in the ball back. I think Tom's will say an extent obviously to corey is really good at that. I think getting those turnovers in the middle of the pitch as they're trying to break up onto us. And then getting the man advantage on the counter. It's actually key against Arsenal. Because I think that 1v1 you'd fancy with Charleston. I don't want to cause them problems. It won't be the place. So yeah I think Andrew Gomez is fine when he's got a terminal. But it's not going to be that type of game. I don't think we need to be teams that pass it forward early and quickly. I think Tom's good at doing that. Obviously he needs to make those right decisions that's key for him. But I think he would be the better choice on it too. And also I guess Davis plays with a little bit more intensity than Gomez and Gilfy. That's what you need. Sometimes you need those in those games. It's putting players under pressure to force them into mistakes. I think Andrew Gomez is a nice player in all. Sometimes he doesn't do those things that we need him to do. After the last night's game I think Tom should be in the frame really. I mean if you look at that, if you look at the midfield with Dave. Are you confident that Dave isn't the core he could do that? And to cause Arsenal's midfield problems? I think so. I think when Tom's coming this season he hasn't let us down. I think he's raised, you know, hopefully making the next step in his career. He's getting to that age where similar to Holgate and Dom last season where you know 22-23. Now he needs to take that step forward and hopefully this is the season. He just needs opportunities to do so. And I think the following game we played right wing back, I thought he was fine. I thought he was decent and brightly did well. Leeds. Sorry Leeds, my apologies. Yes, sorry. Come on, you're normally 100% accurate. Yeah, sorry. No, that was good. We proved he showed his ability. That's, you know, I think if he can continue to do that then he'll be in the manager's trust and get back on the team which is what we want. I guess we're seeing it with a woby api. I'll have to struggle really, let's be honest, he joined from Arsenal. He's had some, showed some good sides, showed some bad stuff. He's injured at the start of the season, but because he's playing in the team now people are starting to appreciate what he's got and looking at him and going, actually you know what, this lad's all right. He's working hard, he's a threat when he's got the ball and that literally is just the case of playing games isn't it? Something that Davis has never really been afforded and don't get me wrong, he played a lot against us the end of the season and he didn't do himself any favours. We have to say that he was poor towards the end of the season. You know, but now, I suppose in new season he's in there with the corey, bit of a different make-up of the team, someone who's got the legs that maybe he hasn't got so. What I thought he did well against Brighton was he just stayed in his position as the sixth and just sat and he broke the stuff up and he breaks the game and just passes it to someone else and lets them do it. I think he'd be fine because he's done that for England under 21 so I know it's a different level, of course, but he's up against young, quicker midfield players often at that level who can run and he always seems to hold his own so maybe he could do that for everything. You'd hope so, I think, as you say, the level's not great from the spin. It's understanding the nuances of the role and the positions where you need to be the discipline and the role and I think playing there certainly helped him and I think, I just think he's ready for a chance. I think as I go on with the show he's probably not in certain games that can be good but in other games not so much I think in this game, I prefer Tom Davis. Yeah, it's going to be a tough game, like you say Arsenal got some really good players so we do have to be wary of that. But from our perspective looking at it, we're playing a team that's lost 7 out of 13 games so they're not scoring loads of goals. You've got to, again, like show-off on in the Premier League the first goal is going to be key in this can we get it, can we break them down and get the goal and then can we build on it and it comes from that, you know. If we can do that and win this game then it'll be a huge boost because like you said, I expect Arsenal will improve. There's no question about that, they'll get better. For us just right now, I think a win or the win will be incredible for us and really if we could win it, be three wins on the spin we'd be going in, I know we've got United before Christmas but we'd be going into that Christmas into Christmas in a great position in the Premier League with points on the board which is massively important isn't it at this stage? Isn't Arsenal a team that we would have expected for a Europa League place?  I don't want them to get on the run because of us. I'd rather put them back by gap. You want to build the gap. It's a mentality boost isn't it that we're around where we should be in the Christmas period. It's been a while since that's been the case. Obviously with January coming I don't know whether we're going to do any deals or not but the option then becomes you can have a look at it because in a certain position it opens up options so it's a key. I think ending the year on a high is definitely a key. Obviously the Manchester United game in the Carabao Cups big as well and hopefully you're heading to the new year with things to look forward to. Definitely, definitely. I'll go through the depression bit which is the stats around the game. The last time we kept three was 1966-69. So data break records isn't it. Arsenal have won more Premier League games. 34 scored more Premier League goals. 110 against Everton. In the longest half. Then they have versus any other opponent in the competition. Indeed there are 110 goals against the Toffees more than any other side has netted against another in Premier League history. Everton haven't won their final match before Christmas in any day last six seasons since we last beat last one against Swansea 2-1 away when they lost Barclay and James Bowman. That was the last time we won before Christmas so we need to sort that out don't we. Since beating Fulham 3-0 on the opening day Arsenal have scored just twice in their last five away games. Win them one draw and one losing three. Arsenal have lost eight of day last 12 Premier League matches kicking off our five. Although one of their victories was a five one win over Everton. Big Sam. He gave them that. Since Miquel Artheta took charge of Arsenal last December the gunners have won more matches in cup competitions than they had in the Premier League. They've won 14 out of 17 in the cup and they've won 13 out of 33 in the Premier League. Carlo Antillotti has lost his last three matches against Arsenal in all competitions in his managerial career. So we need to sort that out don't we. Batman Young has scored four goals in four games against Everton. Follow all of the goals that have come with the Emirates and Everton's ritualising his luck on to score in consecutive Premier League appearances for the first time since February 2020 when the second goal in that run came against. Arsenal. There you go. There you go. There's the stats to read them. Anyway you want at the end of the day they're all historical stuff and it means nothing going into this game. But I look at it and just one of the stats I said to you before is that the last four years are good and we've won two-drome on loss once so I mean the one we lost was a heavy thrashing on the Ronald Cewman was the 5-3 which saw the end of Cewman's time didn't it but you know so we've got to be keep doing what we're doing nice and tight and put the pressure on them because they will come into the game the manager's under pressure the players aren't playing well they've lost Gabrielli's are big being a big part of their defence this season they're not scoring a lot of goals at the minute so for us we've got to make all of them things can't we? Yeah they've just got consistency so let's see what happens there you go let us know what you think in the comments section below Everton the Arsenal Saturday big game and a big thanks to Andy make sure you subscribe give the video a thumbs up join us on Peyton if you want more videos and don't forget the e-toro link in the description below if you want to check out their coffee trade the stuff click on the link below see you later


I promised earlier that in lecture 11, we would explain why an order geometry is called ordered. What order are we referring to? First of all, we have to make sure we understand what an order is in the first place. It's an order, it's more specifically a strict order to not be confused with a partial order, although the two notions are related. I'll explain the relation just a second. We say that a strict order or a strict ordering, it's a relation on a set X with itself. By definition, a relation is a subset of X cross X there. And a strict order satisfies two assumptions, two axioms. The first one is the irreflexive property. That is, we never have that an element X is related to itself. So this is like the antithesis of the reflexive property that we see with equivalence relations. We don't allow an element to be related to itself. We also require that it be transitive. And this is the same transitivity axiom that you see with an equivalence relation. If X is related to Y and Y is related to Z, then it must be the case that X is related to Z. And so this gives us what we call a strict order. And the way we write it, we often think of it as like less than, right? X is, if X is less than Y and Y is less than Z, then that implies that X is less than Z, all right? And you can't be less than yourself.  It doesn't really make any sense with that. Now, whenever you construct a strict order, there are other relations related to it that are also important to remember. So the way we've drawn this, this little symbol here, we're saying that it's less than, we're generalizing the notion of the less than symbol, it's axiomatics there, is that it's irreflexive and transitive. But we could also talk about greater than, right? So we can define a new relationship, call it greater than if this one's called less than, in which case we mean that X is greater than Y if and only if Y is less than X. So basically when it comes to an ordering, there's a direction to it, the direction matters, we can define its inverse relation by flipping it around. So we can talk about less than or greater than, okay? And so when one talks about a strict order, we're talking either about the less than or the greater than, you can call both of them the strict order because you have one, you have the other. Associated to every strict order, there's also the related partial order. So you see a symbol like this, it kind of looks like the less than or equal to symbol and that's what we're trying to do here. So we think of this as less than or equal to. So an element X is less than or equal to Y if either X is less than Y or X is equal to Y. Now notice it can't be both, this is an exclusive order in the situation because of the irreflexive property, but we can modify the relationship. So do we want it to be strictly less than or do we want it less than or equal to? And we can do the same thing to the greater than symbol, we can have a greater than or equal to which means X is greater than or equal to Y if X equals Y or X is greater than or equal to Y. And so this gives us a strict order, less than and greater than, but we can also have a partial order, less than or equal to or greater than or equal to. The idea of a partial order and a strict order are very related to each other like we were able to turn, we can turn greater than, integrate and they're equal to. You can also go the other way around by basically forbidding equality in that situation. So if you have one, you have the other. If you have one of these relations, you have all four of them. And so we often talk about all four of them together. Now, the reason I started off with a strict order is because to prove that lines have an ordering to them, it'll be easier to do a strict order because strict order only has two axioms, irreflexive and transitive, transitivity, as opposed to a partial order here. It has three axioms where it's reflexive, anti-symmetric and transitive. So the proof will actually be simpler by using a strict order in the situation. Now, we actually are gonna talk about a totally ordered set or sometimes called a linear ordered set. This is a strict order that we've attached the so-called totality, the totality axiom to it. So given three elements, excuse me, given two elements of the set, we say an order is totally ordered if either x is less than y, x equals y, or x is greater than y. So one of those three relationships happens. We often refer to this as trichotomy as well. It's like, okay, x is less than y or they're equal or x is greater than y. They're always comparable to some degree. And so this is the totality axiom. Not sure why I can't say that in this video. Sorry about that. This axiom is also sometimes, so if you have an ordered set, so a set with a strict order or partial order because you have one hand with the other. A set with an ordering is called a ordered set. It's sometimes referred to as a partial order because some elements might not be comparable. But of course, if you have a totally ordered set, that means every element is comparable by this trichotomy situation here. This is sometimes called a linearly ordered set. Why? It actually comes from the theorem that we're gonna be proving in this video here that all lines are gonna be linearly ordered in an order to geometry. That's why we call it order geometry because lines will always have a total order to it. And in fact, a total order you can think of as a generalization of the ordering's online. So we're gonna see in a little bit. And so I should point out, of course, that there are orders that we've seen before. If you take the real numbers, we then say that a real number a is less than or equal to another real number b if the difference b minus a is non-negative. So notice this is, I'm using the partial order here, the less than or equal to.  Because of a equals b, you'll be at b minus a, which is zero, that's non-negative, so that's okay. If you want to use the strict order, a is less than b, then you would swap this to say, oh, the two things are positive. It doesn't really matter which one you do. But the real numbers have a very natural ordering to it, okay? And this is also connected to other subsets of the real numbers, like the rational numbers, the integers, the natural numbers, they all have a total ordering. And when we think of the Euclidean plane, which the Euclidean plane, of course, is R squared, the points have an ordering on lines that is essentially this same ordering right here. And this is what we want to prove just right now, that every order geometry lines have a natural ordering to them. And it's essentially the same as the ordering of the points, because after all, you can think of the plane as like a ordered pair of points, kind of. At least in affine geometry, that's how the coordinates work, but in general, ordered geometries can order their lines hence the name. That then brings us to the main result for this lecture here, that every line has a total ordering. So this is the ordering of a line. Now it's not unique, you have some preferences you can make, right? So it's kind of like thinking of an Euclidean geometry, like where is the origin? You can make that choice, you can pick any point basically to be the reference, like this is X equals zero. So we have sort of that power as well, but really we get only one ordering up to isomorphism of regard to the ordering here. So given any line L, pick two distinct points, A and B on that line. Then there is a unique ordering on the line L such that A is less than B and given any between this relationship so that D is between point C and E, these are all points on the line, we're gonna either have that C is less than D less is less than E or C is greater than D greater than E. So you do have sort of like two choices there, which that's basically what we're trying to say here. So if you've given a line, you give three points on the line, right? C, D, E, do we want it to be ordered like this? So it's less than less than or do we want it to be ordered like this? So it's greater than greater than. You can pick the direction, but once you decide, oh, A is less than B, that enforces the direction of the line so that everyone's bigger on this side, basically choosing who's positive infinity, who's negative infinity. And so based upon your choice, A is less than B, that'll imply that this between this relationship gives you either this ordering or this ordering. You don't get both, but given the two points A, B you choose, you'll then have to, you have to make a choice as A less than B or the other way around. And it doesn't matter which two points you choose, you just have to choose a direction. That's really what you're doing here. And so every line has two total orderings. You know, if you think about it going, if it's like a horizontal line, are we going right to left or left to right? That's what we have to choose. Okay, so in order to do this, we have to construct our relation. So less than or equal to is a relationship. So we have to talk about how it relates between elements of the set. Now we don't have to relate everything. Not everything necessarily has to be comparable. Because of the irreflexive axiom, we actually cannot let an element be related to itself. We won't ever do that. So by construction, the relationship we're gonna make right now will be irreflexive, but we have to argue it's transitive and then it's total. That will take a little bit more effort. So the first thing is, let's consider the two points A and B. We are defining A to be less than B. That's just the choice we made. And this kind of has to be the case because we said we're gonna make a total order that does that.  Is it less than B or greater than B? We wanna make that argument. We haven't made that choice because for totality, every point, different points have to be related to each other, not the same point, but different points have to be related to each other. So we're gonna make a choice. So if A is between P and B, we will then say that P is less than A and P is less than B. If P is, if P, Papa here is between Alpha and Bravo, then we'll say that Alpha is less than Papa and Papa is less than Bravo. And likewise, if Bravo sits between Alpha and Papa, then we have that Alpha is less than Papa and Bravo is less than Papa. So I want you to be aware that in this situation, if you have a point P and you're related it to A or B, you'll get these statements because of these three between the statements. Now by trichotomy, one of these three things happens. It has to happen. P is related to A and B since they're all on the same line. So one of those three things happens and so then you get these three statements because of that. And so then looking at this right here, we do have these relationships going on here. So when you have this between this relationship, you're gonna get exactly that. So P is less than A and P is less than B, just like we were saying before, because of the between this relation. A is less than P and P will be less than B because of this one. And then finally A is less than P and B is less than B because of those. So we're building the ordering from the between this relation. At least that's how we're gonna do it for any point as it relates to A and B. So then the last heavy lift here in terms of the definition will prove it satisfies the axioms of this total order in just a moment. But we have to also define, well, if we have two different points, P and Q and these two points are not A and Q, excuse me, they're not A and B. P and Q are different than A and B right here. How are they related? Well, we're gonna say that P is less than Q if one of the following between this relations happen. So if we have P dash Q dash A dash B, then we say P is less than Q. If we have P dash A dash Q dash B, then we say that P is less than Q. And you can kinda see what we're trying to do here with these between this relations, right? Cause A is less than B, so we're forcing P to be less than Q in that situation. So then likewise, number three here, we say that P is less than Q if this happens. A or P Q B, let me say that correctly, P A B Q. We'll say that P is less than Q if we have the between this relation, alpha, papa, Quebec, bravo. We'll say that P is less than Q if we have alpha, papa, bravo, Quebec. And lastly, we'll say that P is less than Q if we have alpha, bravo, papa, Quebec. So if one of those six between this relationships hold, we say that P is less than Q. And so, yeah, so we have those statements there. And I should mention that the way we've defined relationships, we've never defined a relationship where P could be less than P. That was never a possibility. And so by construction, this relationship is irreflexive, just like we said it would be. Now, the totality axiom follows from basically trichotomy and the linearity rules. Like I mentioned before, that any point will have some between this relationship between A and B. Therefore, by construction, A is related to B. P will be related to A and related to B. So there's totality there. But then if we take two other points besides A and B, why are P and Q related to each other? Well, because if you look by trichotomy, there are forced relationships between points, right? So we could say something like, okay, so we know that P-A-B holds, right? Okay, so let's assume we're in that situation, which I think was the first one we had listed right here. Yes, absolutely. So let's say this is the relationship between P, A and B, which by trichotomy, one of these three things happens. So we have this relationship. Well, how does Q fit into all of this? Well, you could have something like, oh, A-B-Q, there is some trichotomy relationship between A and B. Now, if you put these two things together by linearity rules, you're gonna end up with relationship three right here. So, and then that would tell us that P is less than Q. You could have instead that Q is between A and B, which then by linearity rules, you're gonna have to have the relationship that P-A-Q-B happens, like so. So again, linearity replies. And then the other trichotomy relationship is that Q-A-B. Now, when we talk about the linearity rules, there was sort of a problem here. It's like, oh, that's not enough information to say what happens. Do we have a P-Q-A-B, or do we have Q-P-A-B, right? That's the possibility. And so the first one implies P implies Q, a lesson Q, excuse me. The other one then says that Q is less than P, so which one is it? Well, then, okay, so if we have these relationships, we also have to have a relationship between P, Q, and A. So by trichotomy, do we have this, or this, or something equivalent to that, like Q-A-P? Did I list all the possibilities? Nope, that's just the first one backwards. So we'll do this one. I don't think we did that one yet either. Yeah, so by trichotomy, one of these three relationships happens. Again, we're assuming these ones right here. So this one right here is incompatible with those. That's not possible. You get a contradiction, so you can remove that one. This one would imply that P is less than Q because that gives us condition one. This one would actually tell us that Q is less than P because we'd get kind of Q, this one, but it's Q and P. Like, so this is what I mean by using linearity rules and using trichotomy. You always are gonna get a relationship that looks something like this. So again, the P, Q, A, and B could be scrambled around, but as these are four points on a line, there's gonna be some foray between this relationship of those points, and there's only gonna be one of them, unless you reverse the direction, but that's beside the point. We're always gonna make it so that A is less than B.  So that gives us totality. So we basically, by construction, got that it's irreflexive and we have the total axiom. Transitivity is the hard one, but that's what we're gonna work for right now. So we're gonna break it up into some cases. So let's suppose that, well, okay, by transitivity, of course, we're assuming, right? We're gonna assume that P is less than Q. We're gonna assume that Q is less than R, and then what we want is that P is less than R. That's what we have to prove right here. And so we're gonna make some assumptions because the ordering kind of depends about it's relative to these points A and B. So what if A and B are two of the three points in consideration right now? So something like, what if this was A and this was B, then this would be B as well. And so we know, obviously that A is less than B, that's always the case here, but what if B is less than R, okay? So coming up to this original picture here, so if we know A is less than B, if B is less than R, we're sitting in this situation right here because the only possibility here where B is less than R would be this one. So if B is less than R, that implies that B is between A and P, which of course implies that A is less than R in that situation. There's no other situation where B would be less than R unless A is less than R as well. So that's exactly what we're looking for. A, which is P right now is less than R. So that's the first possibility. There's three possibilities we're considering of course. There's also the possibility that Q is A and B is R, right? Now in that same situation here, if you look at these three possibilities, there's only one that allows for P to be less than A. And so if P is less than A, that then forces in this situation that P is less than B. So that one also satisfies transitivity. So the last one that's a little bit harder is what if A is P and what if R is B? Now you might think there's some other possibilities, like why couldn't we do something like B is less than A? Well, I could stop you right there. By assumption A is less than B, we don't get the other way around. So that's a contradictory assumption there. So but we could have that P is A, Q is unspecified and R is B, something like this. So if A is less than Q, that's gonna give us these possibilities, excuse me, these possibilities like so. And if Q is less than B, that would give us this one, this one. So wherever they overlap, it's this one right here. But honestly, that consideration isn't even necessary. I want you to be aware here because A is by definition less than B. So the transitivity happens in a sort of a trivial manner. So I do want you to consider this very important but simple case that if two of the points coincide with P, Q or R, so if two of those points are A and B, then the way we defined the ordering, it's gonna be transitive in that situation. All right, so then we move on to the next situation. What if the overlap between the sets P, Q and R with this points A and B is only one, okay? What happens in that situation? So we could consider a few possibilities, right? So for example, what if the point R was actually the point A? So we wanna argue that P, if P is less than Q and Q is less than A, we want that P is less than A. How are we gonna get that? Well, let's focus on Q for a moment because we know that A is always less than B. That's by assumption. And we've already proven transitivity on points where both A and B are involved right here. So this implies that Q is gonna be less than B in this situation. Now, since Q is less than A and Q is less than B, then by how we define the relationship that only happens because A is between Q and B. So that's an important thing to pull off right here. We have that A is between Q and B. But we have also the assumption that P is less than Q, okay? So what type of between this relationship can we have between P, Q and A? We could have P, Q and A is like that. So Q is between A and B.  We can't have any other one because if we try to do that, that's gonna be in violation of this assumption. We can't have something like this either because again, that's in violation of this statement right here. We could try to do something like this, but this is just what we said earlier backward. So the only thing that's consistent with our assumptions about the ordering and how ordering is defined using between this, it has to be that P dash Q dash A happens. So then by linearity rules, we have P dash Q dash A dash B, which let's see is that on the screen right now, it's not scoot up a little bit. That's our very first one right here. And so this implies that P is less than Q. We already have that. So I mean, this is the only possibility that's consistent with our assumptions. Remember, we're trying to prove right now that P is less than A, for which of course, since we have this assumption, then we can look at this, we can imply from this that P dash A dash B holds, which then implies that P is less than A. It also implies that P is less than B, but we don't even need that one. We're trying to prove for transitivity to this one right here. Another possibility, of course, would be something like the following. Let's suppose that B, let's suppose that B is greater than Q, and then Q is also greater than some other point P. What's happening there? Well, by trichotomy, we have to consider what's going on here. We're gonna have that A dash Q dash B holds, or we're gonna have that Q dash A dash B holds. Trichotomy does allow for a third possibility, but since we know A is less than B, that other one doesn't happen, okay? We also have to relate what's going on right here. The P is less than Q. How does P relate to A? How does P relate to Q? As you try to work through all the details, there's gonna be only one of these statements that's consistent with your assumptions. And again, I'm not gonna go through all the details of this. The statements are very similar to what we've talked about already, and I'm gonna leave it as an exercise to the viewer to provide all those extra details. But I want you to be aware that there's six possibilities total, because basically A could be P or B could be P. That's a possibility. A could be Q or B could be Q, or we have A is R or B is R. And we talked a little about those already, but there's these six possibilities. And if you run through your assumptions and how those assumptions of the relation correspond to between this relations, then there's only gonna be, in each and every case, there's only one of these four area between this relationships that hold. And then from that, you can ascertain all of the statements you need. Okay? And so then the last possibility, again, I know this is a very technical theorem here, and I'm skipping a lot of the steps for the sake of time. The last possibility is what if none of the points P, P, Q or R coincide with A and B? So we have this idea that we're assuming P is less than Q and Q is less than R. So we then have to say, we wanna prove that P is less than R. Well, even though A and B are not related, we wanna fit them in there somehow, right? So by the assumptions here, we wanna deduce that P is less than R. How do A and B play around with these points, P, Q and R? It could be, right? It could be that we have P less than Q less than A less than B.  If that happens using linearity rules, we can, well, I guess it's just for the definition of things, not the linearity rules.  Then by linearity rules, we could then infer that P is less than R, which is less than A, which is less than B. In which case that then tells us that P is less than R. So again, play around with trichotomy, in this case, particularly linearity rules, we can then get something like that. Because after all, if P is less than Q, right? Coming back up to the definition here, if P is less than Q, that happens in this situation, or it happens in this situation, it could happen in this situation, this situation, this one and this one. So if P is less than Q, all of these things happen. Not all of them simultaneous. I should say one of them happens, right? That's what I'm saying. So which of those ones happen? But we also assume that Q is less than R. And so when you go through all the possibilities, there's a lot of possibilities, and there's gonna be, there's 10 possibilities here, because essentially what we're looking at is you're having basically something like this. You have P, Q and R, like so. Where do you squeeze in the A's and B's? You can squeeze them in, you can try to squeeze A right here, B right here, that's one. You could try to put B right here, that's two. You try to put B right here, that's three. You try to put B right there, that's four. You could put A right here, then B could go here, here or here. You could try to put A right here, and then B could be here and here, or you could just put them both over here. And so essentially there's 10 possibilities that you have to consider. And so while I'm not gonna drive all of them right now, one of the possibilities is you assume this relation and this relation, which these are relations that are compatible with the assumptions. Cause after all, like I said, if P is less than Q, then one of these six things is true. And if Q is less than R, then one of these six things is true where you of course replace P, Q with QR instead. But not all of those ones will be compatible with each other because how they interact with A and B. And so there's only gonna be 10 that work out. And so that's what I'm trying to say there. Those 10 possibilities have to be considered and each and every one of them will lead to a transitive relation. So this might be the most unsettling of all the proofs we do in this course, but it's really technical. There's a lot of cases, but they're all very simple. This proof is a tedious proof. It's not necessarily a difficult proof. It's all that matters is keeping track of things. So in particular, every line has a total order attached to it and we will utilize that in the future. It's a very powerful result because there's a lot we gotta check here. And again, we didn't check everything, but everything we didn't check is similar and does pan out. I've checked them all, even if I didn't include it all in this video. So every line has an ordering. And so that leads us to a definition that we're gonna end lecture 11 with because lines have orderings, right? We can talk about what it means to be less than or equal to something. We then define intervals and this is gonna match up with what you would see in calculus. We can talk about the closed interval from A to B. The closed interval for A to B is gonna be all points C on the line such that A is less than or equal to C and C is less than or equal to B. In particular, includes the points A and B themselves. We have the open interval A to B, which will take all lines C in, all lines C on, excuse me, all points C on the line such that A is less than C and C is less than B. All right, so if you look at those two just for a moment, I want you to be aware that the open interval from A to B is the collection of all points between A and B. And it doesn't even matter what order you do. You can have one way or the other, left order or right order, it doesn't matter. This interval, the open interval will get you all the points between A and B. The closed interval right here, this is just the line segment AB that we defined previously, but now we define it using orders instead of betweenness. But the two things are related to each other. We also have half open, half closed, sometimes called clopin intervals. So there's the close on the left interval, there's the close on the right interval, which if you're closed on the left, that means you take all points C on the line such that A is less than equal to C and C is less than B. And then if you're closed on the right, you take all points C such that A is less than C but C is less than equal to B. And so with these definitions, we will talk about intervals in the future. For my students, you'll check out a few homework questions about intervals, but we can define them using orders and so many things about betweenness become fundamentally easier when you use order and set your, when you use order to prove things. So I encourage anyone, if you're trying to prove something about an order geometry, trying to prove something about betweenness, use the ordering if appropriate. It's a very powerful tool. Don't let all of the technical details we discussed in this video go to waste, use them. If you learned anything about ordering here, in this video, in this lecture 11, please like these videos, subscribe to see more videos like this in the future as always. If you have any questions, which is very natural, especially when I skip lots of steps. If you have any questions, please post them in the comments below and I'll be glad to answer them when I can.


The migration to IP version 6 based NGN is a very important and desirable thing to discuss. The reason is IP version 6 is going to stay inshallah. We have been with IP version 4 for a very long time, but IP version 4 suffered from something known as the Roads problem that is running out address space. So we look at the need for IP version 6 migration to NGN as a specific focus, we look at the approaches that could possibly be taken and the phases in which this is going to happen. The need for migration cannot be overemphasized, it's very obvious that NGN is based on the packet communication, either IP version 4 or IP version 6, IP version 6 being more desirable. So it means the long-term NGN is expected to be based upon IP version 6. There are certain features which essentially turn out to be advantageous with regards to the operation in NGN. These include simplified header format, so many of the fields in IP version 4 were obviated. Then the addressing scheme was expanded from 32-bit to 128-bit. The auto-configuration such as a zero-conf or no-conf were introduced. This helps the IP version 6 network to be self-sustaining or self-configurable. Then unlike IP version 4, quality of service could be implemented in IP version 6 with relatively more ease because we have the concept of something known as a flow. Then we have the security implementation that has also been made convenient through IPsec, particularly the application header and encapsulation payload. Then using the concept of IP in IP, in fact, if you are using the option of next header IP in IP in IP in IP, you can think about providing mobility support with much more ease in IP version 6. So once we are planning to migrate to IP version 6, NGN, we can appreciate three relationships that we have to take care of. The first one is the relationship between the end user functions and the transport stratum. Since there are a variety of technologies which are quite heterogeneous, so we need interfaces belonging to each network type. Then we have the second relationship between the end user functions and the application services. So IPv6-based applications and services have to have a direct interface or interaction with the applications and services in NGN. And thirdly, we have the requirement to have an established explicit relationship between the core network or you can say the transport stratum as such and the application and services. This means that we can think about now some interesting approaches to achieve migration. The first one is probably quite redundant and heavy but easiest to implement and very straightforward, that is dual IP layer, IP version 4 and version 6 stacks are implemented both on each and every network equipment at the distribution side, the edge of the network and the end user equipment. The second option is basically translation that whenever a packet leaves or exits the IP version 4 network, 2 IP version 6 or vice versa, certain address or protocol translation has to take place. For that, different research proposals and prototypes were put forth, for instance, stateless IP, ICMP, network address translation, port translation and then something known as the bump in the stack which is a transport level interaction translation from IP version 4 to version 6 and vice versa. Then we have something known as the bump in API that is the application level translation that takes place above the IP version 4, version 6 placement. And lastly, the IP and IP concept of tunneling can be incorporated to allow the migration to take place from steadily from IP version 4 to version 6, essentially for NGN. Now the phases are going to be very intuitive. First of all, we can think about phase 0, which is let's say the starting point. Once we have no IP version 6 networks, so the NGN is only going to be based upon IP version 4. In phase 1, IP version 6 is introduced, although very sparingly and IP version 6 NGNs are connected across IP version 4 based NGNs. So it means the IP version 6 routers are on the stub side and the core and the distribution or the edge, if you may, are based upon IP version 4. Then we can think about moving to the dual stack implementation once IP version 4 and IP version 6 based networks are comparable in numbers. So what we can allow the end user equipment and NGNs is to help to facilitate them by looking at the logical partitioning of version 4 and version 6 stacks. So this is a logical implementation. It means each device is going to implement both of these. So NGN devices at any layer can access both of these transparently. And then we are going to move on to a better situation once we have an overwhelming number of NGN networks, which are IP version 6 based. And here slowly IP version 4 based routers are reducing in number. So we can expect the IP version 4 based NGNs to be on the across side or the stub side and on the core and the distribution side, we are expecting more IP version 6 NGNs to be present. And lastly, we can anticipate, we can hope for an all IP version 6 based NGN environment in which pull migration has taken place.


The individual that they were looking for was in the rear of the house. As the officers approached that particular part of the house, the individual fled out of the house. We'd been out of a rear window. The officers gave the suit. A confrontation struck occurred in the middle of the block here in the 2600 block of West Bates. At that particular time, the individual was fighting with one of the officers over his particular service weapon. A couple shots of fire, the individual was struck with the weapon announced at this particular time. Three officers that were involved in the struggle with him were all injured, the injuries were minor. One is at the hospital and the other one is in his way. Be mindful that this is a very preliminary. As we get more information, there's more to find. I'll be sure to have someone share it with you. So you can go back to the three officers who were involved in the struggle with the individual. In the midst of that struggle, the individual apparently was fighting over one of the officers' service weapons. At which time, shortly thereafter, someone stopped the fire, striking the individual, and we believe he was pronounced. At this time, I'm not going to release the name of the cease or the officers. The officers obviously will be placed in an administratively pending outcome of the investigation. And just again, the injuries to the officers, are all three of them injured? Yeah, three officers were injured. We believe all of them are minor, maybe with the exception of one, but none of them are life-threatened. What about the individual? Several ones. Again, I am unclear on exactly what the ones were. But I believe some were felony ones and some were misdemeanor ones. As we get that information, I'll be sure to share that with you also. Thank you very much. Thank you.


It's going on everyone Sam here with late sports cards ripping one box of 22 a panini immaculate baseball hobby for Greg M. So good luck here Greg. See what we got buddy Both of them nuts. Hey, man. No, you're not on your game. That's why Byron Buxton nice one there that is Player warren 17 out of 99 for the twins Joe Morgan 20 or 32 out of 49 game used for the open to ease Joe Morgan Thomas the Pucky 12 out of 37 For the mats right there Yeah, nice one there on the jersey auto on the acetate as well pretty cool one here We've got rookie patch autograph hollow silver is gonna be Michael Harris Rookie the year winner right there for the Braves. Congrats on that Greg. That'll be an awesome redemption when that comes in Very nice We've got going the Mets that's 13 out of 49 patch Jersey patch auto Tyler McGill Nice one there. I'm like this next one Nice one here Greg to end it. How about eight of eight? Lars new bar on the button Jersey auto Congrats on that. That is rookie as well. Let's go man Congrats on that las new baths new bar Congrats on that one Greg. That's awesome, dude That will do it for your break there, Greg. Thanks as always for the support my friend. We'll get those right out to you, buddy