So team keep it clean. We got some really, really great news to talk about with Mark Andrews, with Lamar Jackson, with Tom Brady. Let's start off with Mark Andrews, though. John Hardball, head coach of the Baltimore Ravens. We know he is extremely careful with his words when it comes to injuries. He don't just be putting anything out there just to say it, and he is very tactful when it comes to how he speak about players being hurt. And we know with Mark Andrews, in the game where he got hurt against the Bengals, initially it was feared Mark Andrews is gonna be done for the season. Mark Andrews is gonna be out for the rest of the year. And it was like, oh man, that is a huge blow. And it's still a huge blow. But John Hardball, he said it. He said it. He said that there was an outside chance that Mark Andrews could return this year. And we heard that. I was like, okay, we'll hold up now, buddy. Is it really? But then John Hardball took it up another step. Because we remember last week, they said Mark Andrews was in the Ravens locker room and he ain't have no crutches. So him being able to walk on his own power, him being able to move without being aided by crutches, that says a lot about his recovery. But John Hardball took it another step further yesterday in the press because he said that Mark Andrews is either on schedule or a bit ahead of it. Because he said that our job for Mark Andrews, the Baltimore Ravens job, he said our job is to get to the point where we're still playing. So what he's saying about that is saying that the Ravens need to get deep enough in the playoffs to where they would have the option of allowing Mark Andrews to be taken off of injured reserve and to suit up for the Baltimore Ravens for a playoff game. And that would be something beautiful right there. I would love that. I know we would all love that. And like I talked about earlier this year, like y'all know how I feel about injuries. I don't feel like players should risk injuries, should risk coming back too early when it's long term. But if Mark Andrews is 85%, even 80% Mark Andrews, but if he's 85%, let's go, let's get it. And obviously anything higher than that. Now, if he's 75%, maybe 70% and lower, if he would be a detriment more than help, then obviously that'd be a no go. Cause Isaiah likely has definitely been doing his thing for the Baltimore Ravens. Him and Lamar Jackson, they have developed very large trust for each other, which we love. But if Mark Andrews can come back, hey, the more, the merrier. So we'll see what happens with that. Now, team keeping clean before we get into the other news that we got, I gotta say I really appreciate y'all and I really thank y'all because we told y'all like, hey, subscribe to the channel. Subscribe to the channel, turn your notifications on. And we want you to still do that. And I appreciate everybody that has been doing that. Y'all been subscribing like crazy. So thank you because we all wanna grow together. That's really important. And I appreciate y'all for being a huge part of that. Whether you knew here, whether you've been around for years, either way, I appreciate it. But y'all been subscribing like crazy in a way that y'all killed the like button on the last couple of videos. Like y'all been destroying it. So I appreciate y'all so much because it helps out so much. It really helps get the word out there. So I love y'all, I really do. So y'all keep it up.  Winning at a crazy rate is Lamar Jackson. And an interest in stat came out about Lamar Jackson being a comparator, somebody who's, he's I'd say is pretty good being Tom Brady. And this is just regular season of course, but still it does say a lot about winning. Says that only two QBs since the merger have the exact same record of 57 and 19 through their first 76 stars. Now that is an extremely very detailed statistic. It's one of those random ESPN statistics that you see during games and stuff where they just like, oh, well, if this player, he washes hands on every Tuesday, then he wins the game. If they play on Thursday night, but you know how it goes. But anyway, and those two players are Tom Brady and Lamar Jackson. So through their first 76 starts, they are both 57 and 19. And obviously when any quarterback is compared to a Tom Brady, that is a beautiful thing. Now we wanted to get to the point where their Super Bowls could start being compared. So this got to be the year where Lamar Jackson gets number one. And we know of course with Lamar Jackson, Tom Brady like, he like really publicly praise Lamar Jackson on more occasions than one. But of course, y'all remember the infamous, you're next, where he put about Lamar Jackson. So my opinion, that says a lot. And Tom Brady, he's somebody that knows his stuff and we're getting there. So we'll see how it goes now. This was interesting too. Because we know that with Lamar Jackson, he is possibly, right now he's a favorite to win the NFL MVP this year. And he would join some very elite company if he were to get his second MVP. Let's look at some names who have two MVPs in their careers. Peyton Manning, he got five of them. But he got five, Aaron Rodgers, he got four. Tom Brady has three. Brett Favre has three. Jim Brown has three. Johnny Innis has three. Patrick Mahomes has two. Kurt Warner has two. Steve Young has two. Joe Montana has two. So you're looking at a list of either all Hall of Famers or future Hall of Famers. Because we know Aaron Rodgers, he's gonna obviously be in the Hall of Fame. We know Patrick Mahomes, he's gonna obviously be in the Hall of Fame. And Tom Brady, I know they're gonna give that, they're gonna give them the clear. Where they say, oh, nah, just go ahead and get the, you ain't gotta wait the five years. So all those guys are gonna make it. So if Lamar Jackson could join that company, that would be, that would be saying something serious. I mean, it's already saying something serious that he's even in the talk. But something that I was thinking about the other day, I was thinking about this the other day. I'm like, man, Lamar Jackson, he could possibly get his second MVP this year. But one thing to remember that literally every year where Lamar Jackson has been the starter, he's always been in the MVP conversation. Think about that. Every year where he's been the starter, from obviously 2019, that's when he started and he absolutely crushed it. But he was in the talks the following year, in 2020, 2021, 2020, he was always in the talks.  So when he got hurt, and he obviously missed the rest of the seasons, that ended up taking him out of the MVP race. But Lamar Jackson is somebody who's always right there in the mix. So that's something to think about. Now, speaking of Lamar Jackson, Lamar Jackson, we know that right now, mentality is everything. And these Baltimore Ravens, they have the mentality where they wanna be looked at as underdogs. Lamar Jackson has continued to say that publicly. And I'm sure you say it privately to them as well. Against the Dolphins, the Baltimore Ravens are the favorite. The Dolphins are a good team now, but the Baltimore Ravens are still a favorite, especially with them being at home and the record and just everything. But Lamar Jackson, he was asked about that. And shout out to Sarah Ellison for providing this quote from the Ravens presser yesterday. She said, Lamar Jackson was telling the media about how the team rises to the occasion when they are underdogs. He was then asked about what happens when they're not the underdogs like this week since the Ravens have the NFL's best record. And Lamar Jackson said, we're the underdogs. We're the underdogs. And I'm sure y'all have seen that clip of the presser where Lamar Jackson said, no, we're the underdogs. And somebody asked, why do you feel that way? And he said, cause we're the underdogs. So he wasn't changing his tune on that. His mentality is remaining the same on that. He's like, hey, no, we are the underdogs. He doesn't wanna be looked at like, oh, they're the best of the best, even though they are, but they're still remaining humble and they're remaining focused. And that's very important for the Baltimore Ravens to do if they wanna get to exactly where they need to go. So that was really, really good to say. Now, John Hubbell, speaking of these Miami Dolphins, John Hubbell talked about how the Ravens, they learned a lot from the 21 point lead that they lost last year in the fourth quarter against those Miami Dolphins. I really hope so. We all really hope so, because that game was looking like, oh yeah, Ravens got it in the bag. We straight, Dolphins are done. I remember, cause of course, y'all know I'm down here in Miami talking to some of my people.  Like I talked to one of my guys after the game, like I was after the game. And he was like, congrats on your Ravens whooping my dolphins. And I was like, what? And he's like, yeah, congrats on your Ravens. They beat my dolphins today.  He said, no, I was watching it. And he said, I turned it off. In like the third, fourth quarter, I turned it off. I said, oh, okay. All right, you need to look up the highlights on that game. And he looked it up and he looked down at his phone and his eye just got huge because he couldn't believe it. Now just imagine us as Ravens fans, how we couldn't believe it. But that really started a trend for the Baltimore Ravens last year when it came to just giving up those fourth quarter leads. But another conversation for another day. It's important that the Baltimore Ravens do learn from that lesson. And they learned from the lesson of last year and even games in the past too. There's even been some games this year as well, but it's important that Ravens, they close it out because this game right here. It's a huge game. My guy Spencer Schultz, he put it nice. He said, is Sunday the biggest regular season game in M&T Bank Stadium history? If the Baltimore Ravens win, they skip ahead to January 13th or 14th. So what he's saying is that they would have that bye week. So if they won the game on Sunday against the Dolphins, then they could have a lot of time off to get rejuvenated, to get rested, to get healthy. And they could be in much better shape both mentally and physically, even though a lot of Baltimore Ravens fans, they fear that. But anyway, back to what he was saying, he said, back in the week 17 to 2018-19, they beat the Browns to take the wild card. I remember that game, I was at that game, that game was crazy. I still have never, even at this Bengals game, this Bengals game, the Thursday night Bengals game, couple weeks ago, that was extremely loud. That was probably the second loudest I've ever heard M&T Bank Stadium, but the first loudest was that week 17 game against the Browns. I've never heard M&T Bank Stadium that loud before. But anyway, continuing, he said they lost to the Bengals in the week 17 to 2017 to miss the wild card. And we remember that, but hey, that game hurt, but it got us LeBard Jackson, so no complaints from me. He said, winning this game equates to winning a wild card and getting a buy to get healthy. And yeah, it does. Because if the Baltimore Ravens beat the Miami Dolphins this Sunday, then the game against the Steelers, it means nothing. Now, you still want to beat the Steelers, but you already would be set. You already would be the number one seed. So Ravens have another opportunity again this year, and this is the last one right here. This is the last one where things have lined up perfectly for the Baltimore Ravens this year. In the regular season where they have the opportunity to take care of their own business. If they beat the Miami Dolphins this week, they're the number one seed. No matter what happens anywhere else, no matter what other games, no matter the outcome of those other, they would be the number one seed and that would be a beautiful thing. Now, back to Lamar Jackson and the MVP voting, the MVP argument when it talks about the numbers. There's been a comparison going around when it comes to Peyton Manning and Lamar Jackson. Again, if you can compare to these guys, I think something's going right. But people wanted to compare the numbers. And this tweet also came from Sarah Ellison. Shout out to Sarah. She said Peyton Manning when he won the 2008 NFL MVP versus Lamar Jackson through just 15 games. Peyton Manning had 4,023 total yards, 28 touchdowns, 12 turnover, 66.9 completion percentage, a 95 QB rating and 12 and four record. And she said, note, Kurt Warren and Phillip Rivers had more yards and more touchdowns. But NFL still said, even with those guys having more yards and more touchdowns, Peyton Manning is the NFL's MVP. Even with those guys having more yards and more touchdowns. So they said it back then that it's not all about the yards, it's not all about the touchdowns. They play a big part in it now as they should. But I think they look past that. But let's look at Lamar's numbers this year. She said Lamar has 4,143 total yards. So more than Peyton Manning had in the whole season. He has 24 touchdowns. He has 13 turnovers. He has a 66.3 completion percentage. He has a 97.2 QB rating. He has a 12 and three record thus far. Thus far. Cause he ain't done yet. That record could end up being 14 and three that end up could be in 13. We'll see. We got a couple of games left. So we'll see how things shake out. But that was some very interesting numbers when it came to the comparison of Peyton Manning's MVP season versus Lamar's possible second MVP season. Now, we know a lot of people love fashion. They love what they wear, what they got on, how they looking. And when you love that, you also love what the Baltimore Ravens got on, what they wearing and how they looking. This week, they will be wearing the black jerseys with the purple pants. And it has been said that in that outfit, the Baltimore Ravens are three and O. They have never lost when they've worn that uniform. So hopefully this Sunday that keeps up, that stays the course and they can remain undefeated when they wear that. And hopefully they really remain undefeated all the way through February 12th.


We're going to start looking at arithmetic now, and we'll be working with the same four basic operations that you'd expect. We'll start with a review of arithmetic in decimal, and then we'll see how to perform these operations in arbitrary basis. Addition and subtraction are straightforward to do in any base, but multiplication and division end up being a lot harder. This is largely because we learned to do multiplication and division by memorizing a times table. Since we only learned the times table for base 10, we would have to build a times table for any other base that we want to perform these operations in. So for multiplication and division, we'll primarily be interested in performing the binary because we can use a couple of tricks to make the computation simple. We'll use this information later in the semester when we start building hardware to do arithmetic for us. We'll end up using these same algorithms that we use here, but with some minor twists to accommodate the structures that we have available to us.


Hello and welcome to the session. In this session, we discussed the following question that says, let z be the set of all integers and r be the relation on z defined as r is equal to the set containing the elements a being such that a and b belongs to z and a minus b is divisible by 5. So, we have to prove that r is an equivalence relation. So, we have given a relation r defined as this, we have to prove that r is an equivalence relation. First of all, we have a relation r in a set a is first is reflexive if a a belongs to r for every a belongs to the set a. Then the relation r is said to be symmetric if we have a1 a2 belongs to r implies that a2 a1 also belongs to for all a1 a2 belongs to the set a. The relation r transitive belongs to a2 a3 belongs to implies that a1 a3 also belongs to r for all a1 a2 a3 belongs to the set. The next is a relation r in a set a to be reflexive symmetric transitive. This is the key idea that we use in this question. Let's proceed with the solution now. We have given a relation r defined as a set a, b such that a and b belongs to z which is the set of integers and sb is divisible equivalence relation. As we know from the key idea that a relation r in a set a is said to be an equivalence relation if that relation is reflexive symmetric and transitive. And we already know the conditions for reflexive symmetric and transitive. So, let us see if this relation r defined as this is reflexive symmetric and transitive or not. For the check whether r is reflexive or not, r in a set a is said to be reflexive if a a belongs to the relation r for every a belongs to the set a. Now for any to the set of integers that we have is equal to 0 or you can say 0 into 5. So, this means that a minus a divisible by 5 the conditions of the relation r as it belongs to that and a minus a is also divisible by 5. So, this means that a a belongs to the relation and therefore we say that the relation r is reflexive. Now next we see if the relation r is symmetric or not. A relation r in a set a is said to be symmetric if we have a1 a2 belongs to r implies that a2 a1 also belongs to r where a1 and a2 are the elements of the set a. For this we take a and b belongs to the set of integers z and we assume that a b belongs to the relation r. So, this means that a minus b is divisible by 5. That is we can say that a minus b is equal to 5 p where this p belongs to the set of integers z. This means that b minus is equal to 5 into minus p. This shows that b minus a is divisible by a is divisible by 5. So, this means that b a belongs to and therefore we can say that the relation r is symmetric. Now next we have to check if the relation r is considered and see which belongs to the set z of integers and a relation r is said to be transitive if a1 a2 belongs to r, a2 a3 belongs to r implies that a1 a3 belongs to r where these a1 a2 a3 are the elements of the set a. So, if we take a b belongs to r this means that a minus b divisible by which shows that a minus b is equal to 5 p where this p is the element of the set z. Now again consider bc belongs to r. We have that b minus c is divisible by 5 that is b minus c is equal to 5 q where this q is an element of the set of integers z. So, now we have a minus b equal to 5 p and b minus c equal to 5 q then a minus b plus b minus c is equal to 5 p plus 5 q or you can say that a minus b plus b minus c is equal to 5 into p plus q the whole. Now b and minus b cancels so we have a minus c is equal to 5 into p plus q the whole which shows divisible by belongs to but as we have a minus c is divisible by 5. So, this shows that ac belongs to thus we have shown that if we have a b belongs to r bc belongs to this implies that ac belongs to r and c belongs to the set of integers z. This shows that the relation r is transitive. Now finally we have that the relation r is reflexive symmetric transitive therefore we can say that r is a relation. So, hence proved to be it is the session hope you understood the solution of this question.


Adolescent patients with chronic illness eventually need to make the transition from pediatric health care in which there is a strong presence of medical personnel and parents into adult health care, where they are fully responsible of their treatment. During this transition, patients need to develop enough autonomy and independence as to take the treatment into their own hands. They need to become aware of the choices to maintain a healthy life and the complications that may suffer if they do not pay attention to their health. But it is also during this period that patients are typically undergoing other important changes in their lives, such as going to college, getting a job, emotional changes, moving, etc. Unfortunately, there is not enough information about how young people go through this transition, and most importantly, which factors have the greatest influence in the process. The questions we try to answer in these projects are, can technology be used to support this transition, and if so, what are the most effective type of interactions? New generations have grown accustomed to the use of mobile phones to stay connected with friends and obtain most of the information they need. Could these devices be used to foster the acquisition of autonomy and independence? The proposed solution is a mobile application called TransitionMind that offers a bi-directional communicational channel between patients and medical personnel with different types of interactions to understand what is the best way to support patients to gain autonomy and independence. The app has been designed with a simple and intuitive user interface so that patients can handle it with barely no previous experience. In a preliminary phase, medical personnel uploads the information about the events occurring as part of the treatment for the patient. These events are then sent to the mobile device. When the patient starts the application, she initially sees a screen with the treatment events that will occur in the future. The user may edit each event and enter simple annotations about it. For example, if it was a doctor appointment, if the visit occurred, and some comments about it. The application also offers another screen showing what was reported during the last days and how close is this behavior to an ideal one. These two screens target the compliance aspect of the treatment. The application contains two additional interactions that make the patient self-reflect on her treatment. In the home screen, the user may write whatever is on her mind at the moment with respect to the treatment. Catching these impromptu comments may help understand the situations or aspects that affect the patient. The second interaction allows patients to write arbitrary memos with their annotations or thoughts. For example, some questions to ask in the next doctor appointment, a summary of a consultation, information obtained from other sources, etc. Self-reflection is then the underpinning mechanism of these two interactions. The third type of interaction provided by the tool is a screen to report their emotional state. With a single touch, patients report their energy and emotion levels. In the same screen, slider is used to report how they perceive their overall health, and at the bottom they may enter any comments about their emotional state. The application also includes an additional screen with information of medical personnel that can be contacted regarding different aspects of the illness. Which of these three interaction groups' compliance, reflection or emotional state is the most appropriate? Which one has a larger effect on making patients aware of their treatment needs? Could different interactions be used with different users? How often are patients willing to report information? Do they increase their notion of autonomy and independence? These and other questions are going to be answered with the help of transition might. The application will help us obtain clear understanding of this delicate transition, and the information will be extremely helpful to tailor the way to interact with patients, know the problems and challenges they face, and anticipate complications that can translate into an improved well-being.