That's the big news today. Grinkowski told the patients he wanted to come back but he wanted to come back for the bucks with his buddy Tom Brady. So they made the trade. You heard what the specifics are. And now let's bring in our good friend Bart Scott. Part of the duo of Bart and former friend of the show Alan Han. You can hear them one to three every single day right before our show on 98 70 SPN Bart. How you doing my friend. Oh no. Couldn't be better. I'm here bro. We all good. There he is. There you go. Multi-tasking this dad time. I'm trying to season chicken. Get the other one to take the trash down. Get the other one to do the dishes. They're not used to working in tandem because they're used to the bait. They're used to the nannies and everybody coming clean the house.  That's amazing Bart. You're able to deal with your family. Multi-task and answer a phone call from the Michael Keisha. Well I always got time for the Keisha. Yeah. Remember where it all began. You know if we get all the other stuff the other company I worked for. Remember where it started. That's right. Started as a call in. We all come from humble beginnings.  You know I'm standing up in the sports world. All right. So what was what's what's your initial take on on Grinkowski going to the box. I mean it's just like like most people probably assume his body was hurting. What happens is you try and do other things in your life and nothing gives you the satisfaction of playing football and then the opportunity to kind of play and let your hair down with a coach that's going to provide a different type of atmosphere. I think everybody that plays in New England for a long time long for the opportunity to be able to play and just have fun and come to environment that the environment is fun and everybody's loose because there are more than one way to win. Now they've been successful in winning the way that they have over the years. But you heard what Philadelphia when they won and you talk about you know long Kyle Long and talking about not Kyle I'll forgive which one it is so many of them. I mean but you know long and what he said about you know his experience in New England and then the joy in which you had just the environment and the looseness of playing you know for Philadelphia. So I think this is an opportunity for Grunt to kind of go and be loosey goosey and have fun.  So you don't think he comes back if Brady stays in the way. Oh not at all not at all. I mean this is about him coming back and playing with a friend in a fun environment. Now from a Tampa Bay standpoint now you have some decisions to make because I believe OJ the OJ the juice man is down there and you also have Cameron break so they have their three tight end sets. And if they're going to come and try and absorb that $10 million salary I think you know they're going to have to let somebody go because they don't have a lot of cash to begin with. So you know watch for either Cameron break or I can doubt if it's going to be OJ the juice man because he's a younger of the two kind of being on the block because you know you can't have a three tight end set even though they have tremendous playmakers all around to kind of you know be able to compliment that you know you really when you go three tight end set you're basically playing with one receiver with what Gowen did last year and what Evan's done is no way you're taking old guys off the field. I'll tell you what Bart if you look at that offense that might be one of the most weapon heavy offenses that Tom Brady's ever played in. Well I think the one that was the most difficult hardly enough to stop was the one that had you know Gronk Hernandez and algae crumpler because that personnel package forces you to go heavy and when you go heavy they basically had two receivers playing tight end which when you look at it they will have three tight ends playing you know receiver so to speak because OJ is not a great blocker but we know Gronk is a tremendous inline blocker and also you know that Cameron break is a South blocker so it could pose a lot of questions but they're not going to walk away from God when we're at a career year and they're not going to work away from Evan's who's one of the top ten receivers in the game. That just tells you exactly what they're going to do on the offensive side they got to go left tackle now because you can have all those great players you put Gronk on the right side he's going to be tremendous in the run game as well that allows you to protect Tom Brady's backside and give some opportunity to process and go through all different types of reason giving him an opportunity to burp the baby and now you have these tremendous receivers on the outside. I mean I could see them if they decide to stay with this whole package with the three tight ends I can see them going with a personnel with no running back in the game and daring you to figure out what to do. Wow. What does burp the baby mean I'm sorry. I need help. Well burping the baby is when a quarterback has time to sit back in the pocket he's weighing his options so he's looking at the ball like burping the baby. Yup got it. I know you got a dog. I know you got a dog. I'm slow but I get it now. God okay. And the Jets have a quarterback at Sam Donald didn't have a lot of time to burp the baby. So at eleven are you going. He dropped the baby. He dropped the baby and ran for high.  You had you definitely go offensive line because you look at you know the hardest thing to get in free agency and it was rare that Conklin was out there. The hardest thing to get especially in this league is a tackle and what's even harder than getting a great tackle is getting a great less tackle. I feel like depending on what the Giants do the Jets will have a tremendous opportunity to upgrade that position which allows you to be able to evaluate Sam Donald and see if he is the truth.  This is an opportunity for him to be able to really solidify his position as a franchise quarterback. So if you go get the left tackle I mean you can get great receivers as I've been breaking this drive down. You can get great receivers in the third round. I mean I've never seen such a deep pool of receivers and they got big ones. They got fast ones. They got slot guys. They got run after the catch guys. You can think about a guy like I you or you know he's a guy that reminds me of Quincy Anoula. You look at the guy like a man from from Penn State.  So it's just it says you can get whatever you want there. I think this is the deepest position there. We've seen some of the great receivers rather than Michael Thomas you know whether it's you know whether it's a B these guys come in the later rounds. You can find guys in the later rounds but don't miss on the hardest position to pay in the hardest position to drive in the left tackle. All right. So then the question about the top four they're going to have their pick of any of the offensive line in the top four or Isaiah Simmons. Who would you go with? No way I'm going with Isaiah Simmons. Isaiah Simmons to me is the jack of all trades in the master of none. You got to have a clear and decisive plan for a kid like that because is he the best safety. No. Is he the best inside line backer. I can argue that I'll pick three inside linebackers that you know actually my favorite inside linebacker right now is Willie Gay from from Mississippi. I mean this guy is tremendous. Runs a 4 4 40 and he's one of these guys man that hits with the bad intent. Now I know I don't think we have any concerns. It's been better. I know I spent plenty of time talking with him. Oh no. That was a misfire. Sorry Bart. Cut. Sorry Bart. Okay. Okay. Okay. Well yeah. I was thinking about the zoom stuff going on. I thought maybe we got zoom bombs or something. Well you know Willie Gay and also the kid Jordan Brooks I believe from Texas Tech. So whoever gets idea of Simmons has to identify him as a certain position. I think he can be a lead safety in the camp chancellor mode. But if you get into his mode of thinking that he's an inside linebacker and you're going to move them all around. He's not going to be able to master any of these positions and learn to my new details because every position that you put him in whether you want to rush him from the outside you want to put him in the inside you want to put him in safety he's going to have to you can have he can see three plays totally different depending on how he's lined up because what his keys are. So I mean it's great to be great at a lot of stuff but you got to be or good at a lot of stuff but you got to be great at something in this league because everybody's good at something they wouldn't be. Well what I guess the big compliment I hear from him and certainly what I loved in the three four and I'm not comparing him to Lawrence Taylor is is that you could put him put him anywhere in the three four and that the feeling seems to be in this modern NFL defenses this is the player that that jack of all trades is perfect for the way teams run defenses you don't buy that in this day and age. No I do but I wouldn't offer five different positions to this kid you know what I mean I think he can be he can be a weapon but you can you can get your offensive tackle from the Giants you can get your offensive tackle. You can go to the second round which is a high pick and I know people are going to say well this guy goes to Southern Illinois so that's why you like him but if you look at Jeremy Chen he offers you the exact same thing from a safety come down and play your dime linebacker and a tremendous blitzer you know for a second round draft pick so you can get the best of both worlds. Now this kid is actually six four as well just like Isaiah Simmons he's two twenty four so that means if you need him to add on you know seven more pounds he's two thirty so he's perfect for a wheel linebacker slash safety but he ran at four four as well so he is that's why government has to do is do diligence and go through the process and understand what's the best combinations of players I can get not just what's the best player I can get because at first round you can get a great player but you should be able to try and find a great player in the second round as well and players that compliment what Daniel Jones do but also offers what you can get from a standpoint of being dominant in the three four and I feel like you can get that from Jeremy Chen because he offers you the same things that Isaiah Simmons does but he offers you that at a cheaper price in the second round. What's up with Bart Scott one half of Barton Han you can hear them one to three right before our show every single Monday through Friday from 9870 SPN so would you look at two atagabaloa and say it's a huge gamble of taking because of the hip injury and they haven't been able to really look at him medically. Yeah but no risk no reward you know I mean you know you have to sometimes evaluate and see where you're picking it depends on where you're picking and what your backup quarterback situation is do you have a guy in place like a Ryan Swiss Patrick like a Tyra Taylor that if you miss see it's not the penalty that it used to be if you strike out on a on a quarterback used to be the quarterback came in he was picked in the top five top three he was the highest paid player arguably in his position still Sam Bradford when he came in he became the highest paid player in the position now you can get out of these contracts relatively easy you know not not without being unscathed but the talent is there and you think even if he comes back and he gives you four great years it gives you opportunity to build a team around him at a very cheap cost you know that's how teams are winning now teams are winning because they're either one they have a franchise quarterback that's a force multiplier that can make up the holes like a Russell Wilson like a Tom Brady or you're going with a guy like Mahomes that although he is a franchise quarterback or you look at Carson Wins who on a rookie contract that allows you to go out and get hired guns and put so many weapons around him that his chances of failing are low it's all about risk management so whoever is going to pick a tool could get a great player and maybe he's a great player for 10 years or maybe he's a good player for five years but if I'm a general manager that's good for me because my life expectancy in the league if I don't win it's probably about three to four years how much of a disadvantage are some of those late round players by the way that they are doing the draft now not getting a chance to talk to these guys not having the pro days are there going to be some guys that are just going to miss the boat and not get drafted because of what's going on right now yes it's brutal and you know I was one of those guys right I was the guy that got the invite but teams were able to come on and come to my pro day and see what I was and see what I could be and put me on the board and talk to me I look at these small schools of guys that didn't get invited to the combine they're not going to have the opportunity to have an interview to get opportunity to even know that they exist they're not going to get opportunity to show how did that game wait to show that hey they played faster than what it looks like on film to be able to put a time there they can't put a 40 up they can send videos and video chat and you know we're talking to Joe Douglas and talk to some other guys you know just in talking you know we feel that that's going to be the people that get left out and without the XFL even being in existence they don't even have a place to go prove themselves you know maybe the Canadian League to prove themselves to try and show what they can do with elevated competition. I'm curious before we let you go Bart so it seems like Brady wanted no part of Belichick and Grunkowski wanted no part of Belichick would you have rather been on a team and had a good time with a relaxed atmosphere a good team or would you rather have been miserable and want to super ball with Belichick miserable and want to super ball with Belichick. The thing is all these guys have gotten that so now it's like one of those things once you've got your titles now you can do what you want to do and have fun and try and go at both right. You can you both can be possible you can have fun and win championships. I mean just look at that 2000 Baltimore Ravens team you saw you saw the you saw hard knocks before the season started they had a blast. You know you saw we saw that underdogs Philadelphia team they had a blast but when you've already gotten it you know that's why you can look at Kevin Durant and he say you know I got my titles so now my resume is complete. I mean is Brady really going to up his resume by going to win a Super Bowl with Tampa.  He's still a first ball at Hall of Famer. He's still in the top 100. So now he's like man I can go actually have fun with my friend and I know that we're going to be successful and it's a team that possibly could win. I mean that's all the boxes and taxes. Do you think with guys like that yeah exactly with the taxes but do you think with guys like that there's an excitement of like we can actually go down there and rule the rule the roost it'll be our team in a way that you just can't feel in New England because it's always Bella checks team. Absolutely I mean I can actually show my personality off. I know Grunk always did but I actually can say what I feel.  You ain't got time to waste time. You tell people exactly how you feel your direct you do what you want to do. I think Brady's at that point where I do what I want to do. I'm going to have fun.  This isn't going to go the way you want because you need me more than I need you. See I thought it was maybe different. It was always a hey maybe I don't need you as much as I need you. Maybe I can get Jimmy G. Or maybe I can get somebody else. I don't maybe it was questions there but I think he wants to go there as well and we've seen listen if anything the last dance was taught us is that being successful is about managing egos is usually somebody pride and ego that gets in the way that wants more credit than what they deserve or they feel like they're not getting somebody feel slight and I feel like Brady feels like he's been slighted and he feels like you know Bill thought that he wasn't that important. Well guess what Bill go ahead and try and win with Jerry still good luck with that. And then guess what if it doesn't go your way because before me you won't win in no titles. Right before me you weren't having a success you matter of fact before me your job was on the line so how about let me prove to everybody and let me end the debate. Let me end and let me prove it to myself that I can go somewhere else and be successful and you know Bill you're later in age so you only got a couple more swings at Apple. You can't afford another 10 years and I went into the title like you went between our first dynasty and then our second one. So then if I win it proves no different from why why Shaq left Kobe right egos always get in the way. I'm curious about one thing you know everybody in the NFL are you friends with people that played for New England and is it no fun whatsoever Bart. It is the worst thing ever I know plenty of people I can name them as Marquise Cole. I can name names. I got receipts of people that I played with other places and Adelias won the Super Bowl with the Ravens. He went to New England and he played well in the Super Bowl they lost. He absolutely hated it. Really what's bad about it. Well it's miserable. It's no fun. They do just berate you. I mean they got conditioning afterwards.  The only reason you can do that in that environment in New England is because the best player on the team buys in and allows you to do that. No different than how Tim Duncan and David Robinson allow Poppins to do that to them.  You know what about peace. How about that. I'm curious what you think of Joe judge then you know I've made a big thing out of fact that he will not say I'll check without the trophy. So what do you think of something like that. What are the players think when the coach refuses to utter their name. I mean thank you think he's a clown.  We don't even know if you if you qualify and we don't even know if you deserve our respect you have to earn our respect. At least some of those other guys were coordinators that had success at doing certain things. So like at least they had success. The players noticed it. They want him to say Daniel Joe's name.   Right. Come on man. Quick wasting our time wasting time. Let's focus on the stuff we gotta worry about about like actually trying to win some games build a team learn offense really. Who do you think you are Yoda.  Literal not literal. But right. I understand that but what would you know obviously what he's trying to do is he's trying to say that all players are created equal but did you want that in the environment or did you want the pecking order of hey there's a starter there's a star there's a leader that's a guy that I want to get to that level or did you want to walk into a room believing that you were looked upon the same way as everybody else on the team. Bro don't lie to me when keeping it real goes wrong. Brian really was you know people people don't understand how great Brian Billy was right. He was his offensive genius that couldn't get offense in Baltimore but he still found the way to win. How did he find a way to win because he knew how to manage egos.  So listen our success is intertwined so I want you guys to understand that we're going to create an environment a competition a accountability and we're going to keep it real I ain't going to lie to you bro if you if you want to ask me why you ain't playing because you suck or if you want to ask me why you're playing that's like me coming to the Baltimore Raven and I'm a rookie undrafted from Baltimore and they're like hey man there's an open competition between you and Ray Lewis but you know what I can't even do I can't even do with this one because y'all if y'all are lying to me like that y'all lie to me and cut me like I'm not even going to fall for that banana and to tell producers I'd rather go somewhere else right here on 98.7 so Bart we love you thank you for coming on man hey man I appreciate it man you know and congratulations on number one you know what I'm saying you guys are the women these are our wings you and your family stay tuned buddy you keep doing it buddy I don't know no doubt no doubt have a great day alright thank you that's Bart Scott and tomorrow we're going to have on Mike Tannenbaum at five o'clock so the other half of Bart and Mike


Another method of finding the inverse of a matrix is known as the adjoint method. The adjoint method is based on the following theorem. Let A be an invertible N by N matrix. Then the inverse of A is 1 over the determinant times the transpose of some matrix M, where the components of M are the cofactors of the original matrix. Remember, cofactors showed up when we tried to find the determinant. We might say that the inverse is the transpose of the matrix of cofactors scaled by the determinant. For example, consider the 2 by 2 matrix. We'll find the inverse using the adjoint method. For that, we need the determinant, which we found earlier. Now we'll form the matrix of cofactors. Since the cofactors will be the minors of each entry multiplied by plus or minus 1, depending on where in the matrix they're located, it's convenient to set down the pluses and minuses first, and then compute the determinants. So we'll set down our checkerboard pattern of plus and minus, and then we'll find our minors. So the first row, first column entry, we'll cross out that row and column, and the minor will be the determinant of what's left over, which will just be negative 5. The first row, second column entry, we'll cross out that row and column, and the minor will be the determinant of what's left over 4. For the second row, first column will cross out that row and column, and the minor will be the determinant of what's left over negative 3. And for the second row, second column entry, we'll cross out that row and column, and the minor will be what's left over 1. Now we'll incorporate the signs to get the cofactors, and we'll copy down this step, transpose this matrix of cofactors, and at this point we can either write the inverse as 1 over the determinant times the transpose matrix of cofactors, or we can go one further step and multiply the terms of the matrix by 1 over the determinant to get an expression for the inverse matrix. What if we want to find the inverse of a 3 by 3 matrix? As before, the inverse will be the transpose of the matrix of cofactors multiplied by the reciprocal of the determinant. For reasons that'll become apparent, we'll calculate that determinant last. We'll set up our checkerboard of plus and minus, and find our minors. So for the first row, first column, we'll cross out that row and column, and the minor will be the determinant of what's left over. For the first row, second column, we'll cross out the first row, second column to get the minor. First row, third column, we'll cross out the first row, third column, and the determinant of what's left over will be the minor. We'll do the same thing for the entries in the second row and for the entries in the third row. We'll evaluate all of the determinants, take care of the signs to get our cofactors, and then transpose the resulting matrix. Remember, we can think about the inverse as being the transpose of the matrix of cofactors scaled by the determinant. So if we can figure out what that scaling factor is, we'll know what the determinant is. So, notice that if we multiply the original matrix times our transpose of the matrix of cofactors, the first row of the original times the first column of the transpose of the matrix of cofactors is going to give us an entry of 10. And what that means is that the determinant of the original matrix must have been 10 if it's anything at all. And so we'll include that as a factor, which gives us the inverse of the original matrix. It's probably a good idea to verify that this actually is the inverse. But you don't really need to because this is on the internet and it must be true.


Oh, and welcome to the session in which we will discuss the qualifying child as well as the qualifying relative. Now, let's take a look at this tax form to understand the big picture. Why do we need to learn about this? In the prior session, we looked at the filing status of the individual, such as single, married filing jointly, married filing separately, head of a household, qualifying surviving, qualifying surviving spouse. And we stated that you have to have some sort of a dependent if you're ahead of a household and that dependent could be a child or that dependent could be a qualifying relative. So we talked about qualifying child QC and qualifying relative. So notice here the dependent after we list the dependent after we list their name or social security or relationship to you, we need to see if this dependent qualify you for the child tax credit or does it qualify you credit for other dependence which is qualifying relative. Now, obviously you want the qualifying child because it's higher, there's more money in there. So what we need to learn about here is when we come up to this dependent, whether it's a qualifying child or a qualifying relative, how do we make this determination? Before we proceed any further, I have a public announcement about my company Farhatlectures.com Farhat Accounting Lectures is a supplemental educational tool that's gonna help you with your CPA exam preparation as well as your accounting courses. My CPA material is aligned with your CPA review course such as Becker, Roger, Wiley, Gleam, Miles. My accounting courses are aligned with your accounting courses broken down by chapter and topics. My resources consist of lectures, multiple choice questions, true-false questions, as well as exercises. Go ahead, start your free trial today, no obligation, no credit card required. So dependent, there are two types of dependent, qualifying child or qualifying relative and unique tests apply to each type of dependent. So why it's important? So you might be saying, why do we have to know what's a qualifying child? Well, for one thing, for the purpose of the head of a household, we need to know if that individual is for the head of a household, whether it's a qualifying child or qualifying relative. Also, we're gonna learn later on about earned income tax credit, child and dependent tax credit, credit for child and dependent care expenses. So we need, we're gonna be using those terms later on when we talk about different credits. So that's why we need to know what is a qualifying child, what's a qualifying relative. Starting with a qualifying child, we have five tests to go through. We have the relationship test, residence test, age test, support test, and joint return. So as you know, anything about FARHAT, once I have a list, I'm gonna go through each of these lists in the tails to explain the concept. Starting with the relationship test. Now the first thing that comes to mind, if it's a relationship test about a qualifying child, so the child must be your son or daughter. That's the first thing you would think about, of course. But they could also be your step son or step daughter, your brother or sister. They can qualify as a qualifying child as long as they need other requirement, which we will see later. Step brother or step sister, half a brother or a half sister. And a standard of such individual, grandchildren, nephews and nieces. So notice, if you notice anything, it's you, it's the taxpayer, and notice it's gonna be people not younger below you. Like you cannot have your parents or your grandparents. Do you see this? It's all descendant of such individual. Now, a child who has been adopted or whose adoption is pending qualifies as well. A foster child may also qualify. Now, let's talk about the residency test, residence test. Qualifying child must live with the taxpayer for more than half of the year. Remember to qualify you as the head of a household. Remember the age when I spoke about head of a household, they have to live with you. Temporary absences is okay. For example, they could be in school and college or they could be have an illness in a medical institution that's fine. Those they get be waived. They must be a US citizen or a resident of the US, Canada or Mexico. Now, those same rules, they're gonna be applies to the qualifying relative as well. Same test applies to qualifying relative because we're gonna be discussing qualifying relative later. Age test. Remember we talked about brother, sister, so on and so forth. There is an age test. So the child must be under the age of 19, well, or under the age of 24, in case there were full-time student during any five-month of the year. Why five-month? Because in the US, when you go to school, each semester is four to five months. So even if they were one semester in college, so let's assume they were from January till May, January till May, that's five months. So once they attend one full-time semester, it doesn't matter. Now the age goes up to 24. Now they must be younger than you. So the younger than you, younger than you the taxpayer. If you're claiming someone as a qualifying child, that individual must be younger than you. So if we're talking about your brother and sister, that's what I said, they have to be younger than you. However, individuals who are disabled are not subject to the age test. So if the individual that you are qualifying as a qualifying child and that individual is considered disabled, you forget about the age test. Now let's discuss the support test. To be an eligible child, qualifying child, the individual must not be self-supporting. And the individual means the qualifying child. In other words, let's think about Macaulay Cokin, who is a famous actor, he was a child. You know, he's known for home alone. The point is they can be millionaires. The child can be a millionaire as long as they don't provide more than half of their own support. In other words, they did not spend their money on their own support. They could be making this money and don't spend it. They could put it in a bank account, in a trust account. So there is no gross income test. We have to understand this, this is important because we're gonna have another support test when it comes to the qualifying relative and we have to know the difference. So a child cannot provide more than half of his or her own support. So they can be millionaire, nevertheless, they're not spending that money. Someone else is spending the money on them, their parents, then that's fine. That's fine. Now, joint return test. The dependent cannot file a joint return with a spouse unless the only reason they're doing so is to get a refund. Now you might be saying, how can a child get married and have a spouse? Well, think about it.  So it's possible that, for example, my son get married to someone in college and they work part-time. And as a result, they have some money, some taxes withheld from their paycheck and they wanted to file a return just to get a refund, just to get the tax withheld. They have no tax liability. Both of them, they have no tax liability. So what they would do, they will file a joint return and that's that. Otherwise, there should not be filing a joint return. And this same test also apply to a qualifying relative. So those are the tests for a qualifying child. Now let's take a look at a qualifying relative. Well, qualifying relative can be dependent if the following tests are met. The relationship test, and notice here, or that's important, member of a household test. And we have to explain this a little bit more on the next slide. Gross income test, which is totally different. You're gonna see this. Support test, little bit different. Little, you have to be very careful. We have the residency test, which is the same as qualifying child and the joint return test, the same as qualifying child. So I'm gonna go over those very briefly. Let's start with the relationship test or a household member test. Well, the relationship test for a qualifying relative is more expensive. You can have more people than a qualifying child. And this includes the following relatives. Now you can go, you remember I told you it has to be people below you with a qualifying child. Now it can be your parents, your grandparents. You can consider them as qualifying relative, qualifying relative to be qualified as a dependent. Uncles and aunts, also, certain in-laws, son-in-law, daughter-in-law, father-in-law, mother-in-law, brother-in-law, sister-in-law, why not? The relationship test also include unrelated parties. And here we're talking about boyfriend, girlfriend, cousins who live with the taxpayer the entire year. So here you have a different test, member of a household. You remember when I talked about her of a household, I said boyfriend, girlfriend, your lazy friend, your irresponsible friend, they cannot qualify you ahead of a household. But they can be considered your dependent for the dependency purposes, but those cannot qualify you for ahead of a household. Just wanna make sure this is clear. The relationship test, gross income test, this is different. If the individual, if that individual that you're trying to claim as a dependent make above a certain amount, what is that amount? That amount changes every year. But there is a logic behind that amount. It's called an exemption. And this used to exist before 2018. Each individual will have a personal exemption. Now the personal exemption for 2022 is 4,400. This is given by the IRS, and it changes from year to year subject to inflation. So if that individual earns more than 4,400, that's it. It means they are making enough money. You can, they cannot be your qualifying relative, okay? Remember, this does not apply to a qualifying child. A child, they could be making millions as long as they're not spending it to support themselves, then that's fine. There is no gross income test for the qualifying child. There's a gross income test for the qualifying relative. Now the income does not include, it's not counted. So you don't count their income if their income is tax exempt income. So if it's a minibund interest income, if it's social security income, any sort of tax exempt income will not be counted. So you will not count this income. Let's assume someone invested money and they're only earning minibund interest. Then no worries, we don't have to count this income. So they could still qualify, even though they might be making $10,000 in interest income from minibund, that's fine, because it doesn't count against that 4,400. A joint return test, we talked about this, the same thing as qualifying child. If they are filing a joint return, only to get a refund. Citizenship or residency test, the same thing as qualifying child, same concept. Support test. Now we have to be careful. Remember at the support test, I took my time to told you to be aware of it. Now the taxpayer, notice here, the taxpayer must provide more than 50% of the qualifying relative support. Under the child, the taxpayer did not have to do that. Under the qualifying child, the child did not provide more than 50% of their support. Under the qualifying relative, the taxpayer must.  It's you are spending the money. You have to do that for the support test and you only count the actual money spent, okay? Now scholarship don't count, for example, don't count for them if you're counting their support. Now two exceptions apply to the support test. Sometimes there's a multiple support agreement or children of divorced parents, we have to be aware of those rules. But remember under the support test, the taxpayer themselves, they have to be spending money on them, spending money on them. The child, as long as the child did not provide more than 50% of their own support. Now multiple support agreement, what does that mean? It means when you have several individuals together contributed the 50% plus. Well what happened? They agree together and they would allow one member of the group to be considered that they provided more than 50%. Remember you have to provide more than 50%. So let's assume we have three individuals. One's providing 45, the other individual, this is the first individual, the second individual is providing 48. So that's one, that's two, what's the third? So we're missing seven, I guess, two, seven. Seven that will make it 100%, right, 80. This is 100%. Now notice not one person is contributing more than 50%. But they can agree, they can agree this is A, B, C. They can agree, but however the party that we agree to cannot be providing less than 10%. So C cannot be, so it's either A or B. So A, B and C they can say, okay, A for year one you will be the person claiming this dependent, B in year two, or whatever they agreed on. Each eligible party must meet all other dependency requirements. Of course just the dependency requirement would still apply. A good example will be allowing children of elderly parent to claim exemption for parent where none individually meet the 50% support. I remember my grandma, I had like five individuals and as uncles and aunts and what they do and none of them meet the 50% test, they'll just agree amongst each other who's gonna claim her from year to year. Children of divorced parent, again the general rule here is the parent having custody, the custodial parent of the child between they spend more than 50% of the time with them, usually will claim it. This is called the time test. The support test here is ignored, it doesn't matter whether that there's a support test or not as well as the divorced agree. If the custody is 50-50, the parent with the higher adjusted gross income. Now the general rule does not apply if there's a multiple support agreement is in effect they're agreeing with each other or the custodial parent, basically they qualify to have it and said, you know what, I would let you, I would let you have the dependent child, that's fine. Now if both parties are not the parent, the person with the higher adjusted gross income, what should you do now? We should go to Farhat Lectures and look at MCQs through false questions that's gonna help you understand this concept. Qualifying child, it's gonna appear again and again and again later on when we talk about credit. It's something that you want to be familiar with. What's the difference between quantifying child, qualifying relative? Definitely a testable easy topic on the CPA exam, easy topic for the enrolled agent as if you are an accounting students taking the stack scores, you should know it. Good luck, study hard, good luck and stay safe.


And the Chief Justice of Nigeria, Justice Stanko Mohammed, has again passionately pleaded with the striking judiciary workers in the country to accept federal government's offer and call off their indefinite nationwide strike. Justice Mohammed's call followed a meeting held with him by the Minister of Labor and Employment, Chris Ngige, at the Supreme Court complex over the ongoing strike in Batupon by the Judiciary Staff Union of Nigeria, Jusung, to demand for financial autonomy for the judiciary in the country. Upon being briefed by the Minister on the negotiation level, so far the CJN requested Jusung to call off the strike in the interest of the nation and in larger interest of justice. The CJN added that it would also allow opportunity for further negotiations towards the resolution of the dispute. So you get notified about fresh news updates.


Hey everyone, welcome to this weekend's video update. Today's Thursday, April 9th. Obviously, the market's closed tomorrow, so we are at the end of our week. Let's review all of the trade alerts, all the positions. But first, let's talk about who got caught being hot in the community. This week goes to our friend Dennis Price. Congrats, Dennis. Dennis had some technical issues with with Thinkorswim and shared with the community first to ask questions and then went with Thinkorswim support to get everything figured out. Maybe not the solution he exactly wanted, but it was it was awesome to see just the detail that he shared with the community because I know it helped some others as well. So congrats Dennis. Keep up the heat. You got caught being hot. I appreciate all the comments and feedback from the community to Dennis and his question as well. It's great. It's always great to see how the community comes together to help each other. All right, so let's jump into the alerts for the week and I'll talk about the market overall as we kind of go through these. Starting with the 6th Monday opening trade. So we added a double calendar in SPX. At this point there at the front, we've only had three days to expiration. I talked about this in length in another video and in the community about why we chose that three days. But essentially the vol differential, the volatility differential in these expiration cycles was attractive and that's why we added to this position with with less days than normal. And so we closed that out today. I had to take a loss on it. So I'll talk about that here in a second when we get to that alert. Next trade, rolling adjusting trade in ZB. So we've got two sets of ZB short strangles. This one was down to 18 days to expiration. So we rolled that out to 46 and adjusted our puts up a little bit. So let's pull up ZB. ZB's been coming down nicely for us since we've since we've done these rolls. If we take a look, you know, we've had this nice little down move in bonds. And so that's really helped our positions. In fact, if we look at both of these together, let me make this a little bigger for you. If we look at both of these positions together, we're up. We've gotten back $4,500 in profit just since the roll. So coming in nicely. So if we take a look at these each separately, here's one of them. Price is, like I said, coming down. We could use a little bit even more downside action. And then the other one right here, getting almost to a point of delta neutrality in that one. So what we'll probably do, well, it depends on where we're at with everything, but there's a possibility we may just close one of these out and then just continue to manage the other one with the ability to potentially add back in, you know, if price bounces around too much or volatility spikes again. So that's the plan. As of now, we're going to continue on with both of these. We're both of them are in the 43 day expiration cycle. So we've got a good amount of time. But if this one, for example, makes a quick move lower and we're well over 50% of max profit, we will either just close this or roll it out to the next cycle, even if we're not down to that 21 day period. So look for that in bonds. Next trade closing trade in SPY. So we had an iron duck with prices running higher. We booked a beak profit on our spy duck. Next trade opening adjusting trade in GC in gold. So we added an iron condor in gold and then actually today we took off the put vertical side of our other one. So we've got two different pieces here. Let me start with the alert. That was this full iron condor here. And you can see we're up a couple hundred bucks since we put that on. And that one is out in July in the June cycle with 47 days to expiration. And we still had an iron condor in this cycle, which currently has 18 days. Price ran higher. So we ended up taking closing the untested side. And so we now still have the call vertical side. So we need a little bit of downside action in gold to get back into range there. So you can see we've had a pretty stout move to the upside in gold. So if we get a little bit of a retracement back into range, we'll close that one out and still continue to manage our full iron condor. Next trade opening adjusting trade in ZW. So we added an iron condor in wheat and then right after that closed our other one. So it was essentially like a roll. We don't technically roll iron condors, but we basically closed one out open to do it in the next cycle. And so let's check that out. Wheat's been bouncing around pretty good, but this is pretty well centered up a little bit since we put it on, just waiting for some more time to pass in wheat. Next trade opening adjusting trade in IWM. So we added a long put vertical. So what I'm missing an M right there should be IWM. So we, as this market has been rallying over the last couple of weeks, we've been adding in some short delta. In this case, we added in IWM. So if we take a look at that, so we've got two different long put verticals here. This is the one we put on. I mean, IWM is really raised higher even just since we put this on. So it's a little bit out of range. And then the other one is further out of range. So we've gotten punished a little bit on this up move on some of our current short delta positions, but let's just talk about the market overall at this point. I think it's a good, good point to talk about it. So, and I've talked about this in some of the trade hacker updates throughout the week, but, you know, I just, I don't believe this rally. I just, I can't, I can't see this market just continuing to rip higher in the face of what we've got going on. And I, one of the videos I put out titled, don't get sucked into the stimulus. Yes, we have a ton of money coming into the system. Yes, rates are at zero. Yes, the money, you know, the government is trying to do everything they can for small business and individuals and send them money. But man, I just, you know, this thing could linger on longer. I mean, you know, the state home order right now is through April 30th. And who knows if that's going to hold, right? I mean, are we going to, are we really going to go back into the workforce full, full force when, when the coronavirus is still very prevalent? And, and there's, you know, the, the clinical trials for the for the vaccine are going to take 18 months before they're available.  I just, I don't know. And so I just, I just don't see. I think this is just a little bit of a relief rally and we're going to roll back over.  And by the way, we are not overly short at all. I mean, we are more, we're pretty close to Delta neutral at this point, even in the face of adding these short Delta positions. I'll talk about the bunker strategies we've added and some of these verticals. So, yeah, we're taking a little heat on the upside, but, you know, you're never going to pick the tops, never going to pick the bottoms. And so I am completely cool. In fact, I like where we're sitting as far as our overall portfolio positioning. So that's IWM. Next trade, opening trade in SPY. So we added a bunker in SPY. And if we take a look, we did this in July with the 99 day options. So let's take a look at that. I've got SPY. Now we've got several different positions on in SPY. And so I try not to do this too much because it can be confusing. However, these are all in different expiration cycles and all with different numbers of contracts. So it should be easily trackable in thinkorswim or whatever platform you're using. And so let's take a look at the bunker that we just put on. And it's pretty, pretty close to where we put it on. Price is hanging out right here. So, you know, we've got a market rips higher. We've got an upside risk of a 400 to 500 bucks, depending on where price is. And again, we'll close out of this about 60 days before the options expire. So just looking for a potential big downside move. And that's where we benefit from this thing. So, you know, if we get back down to what was the low in SPY, about 218. I mean, if we think about it, if we get back to that level, you know, we're talking about being up a couple of thousand dollars on this trade. Now, it depends on how quickly it happens and all that stuff. But that's kind of the goal of putting this on. We've got very little risk to the upside, but we've got massive potential to the downside. So I like those odds right now. Next trade, closing trade in GC. So that's the one I mentioned. We closed out the put vertical side on our 18 day to expiration iron condor. Did an opening trade in SPY. So we added a new weekly double calendar in SPY. And let's take a look at that. So on this one, we did this with the options. Did this with the options, uh, with eight days to expiration and the back week had 11. And the reason we chose that is obviously it's kind of in our wheelhouse for the front week days to expiration, but also you've got a situation where the front week of all is a little bit higher than the back week fall. Now, nothing like it was before for sure, but still still sets up well for this type of trade. So anytime you can sell the options with a little bit higher and buy the back week with a little bit lower, that's that's typically going to set up well. You know, if you look at some of these others, they've got higher 39, 41, you know, so you just got to sometimes it works out. Sometimes it doesn't. If we take a look at what that looks like, um, click off the bunker here and click on our double calendar. I also did this in SPY as opposed to SPX, uh, just to help out some of our account holders who have smaller accounts. And because you can really do this at any level. We did 10 contracts, uh, and it's, you know, using about 11, 1200 bucks in buying power, starting off with about a 60% pop, which again, you know, just a couple of weeks ago, putting these on, we were having 70, 75%, 80 plus, pop at initiation. Now, obviously these break evens can fluctuate during, uh, during the life of the trade that the max profit can go up and down based on volatility swings. So this is just at initiation, but hopefully this one works out well for us. We've got eight days to expiration on this. We'll get out kind of closer to expiration, either the day of or the day before. And then, um, and then while we're on SPY, we've also got this iron condor. So this one's got five contracts. You can see price with this up move price is hanging out up here. Squeeze this together a little bit. Price is hanging out right up here in the upper end of the range, but still well within our, well within our range. So just waiting for some more time to pass there. And then lastly, the last two alerts were the SPX weekly double calendar. So we had one contract in one had two contracts in the other. And, you know, that's the one thing I hate about these short duration trades is you get your, you can get yourself in a position where obviously the gamma accelerates, meaning the, the P and L swings as you get closer to, uh, expiration accelerates.  And so that's what we're seeing here. Let me get rid of these. Um, but, you know, what you're seeing here is, you know, we, we, we were on the kind of the edge of our range, you know, yesterday near the close and then it opened up, gapped up, went a little bit higher. We, you know, came back down. And so we, we had to take losses on these trades. Uh, and that's, that's the risk that you have with the short duration trades, you know, with more duration, you have more time to extend duration. You have more time to let price move around. When you get to these short duration trades, you've got to make a decision in that last day or two, as far as when you're going to take it off. Obviously in hindsight, we wish we took, took it off yesterday morning. We would have booked a profit at least on the, on the two contract one. Of course, price ran higher and so we ended up taking a loss on those, but you're still, still obviously a great strategy. We just, um, you know, sometimes the market wins and this time it got us. So all good though. Uh, our account was actually still PNL wise, still, uh, I think pretty even for the day, even after taking those, those couple losses and those, so very, very, uh, good diversification in our portfolio. Uh, let's see. And then, so those are all the trade alerts. Uh, let's take a look at some of the other positions starting with six E, six E, making a little bit of a bounce higher, which is helping our, uh, what's currently a short straddle was a strangle. So making back some profits there. We've got how much time we have, we've got 29 days. So in the next couple of weeks here, we'll be rolling that out to June or potentially closing dependent, depending on where we're at with everything. Uh, CL oil was up today and then is that pretty good up here? And then it, it kind of tailed off, uh, sold off pretty hard the last, last part of the day, but we've got two different pieces in here. We've got this short strangle, uh, which, um, seems like we've been in for quite a while, but we've had very little theta decay right when we put this on Ivy spiked, which pushed this PNL line down. And so now we've just been getting back up to, to even and so still well within range here, just waiting for some more time to pass in that one. And then our other piece, which is this adjusted strangle, since we've done the role, we've made back about four grand, uh, on that.  And we'll continue to manage these similar to what I was saying in bonds. You know, we may cut one of these, one of these loose to, uh, lower our exposure in this symbol. Uh, you know, these are big contracts. So having two different pieces, uh, for the size of portfolio is a pretty big exposure as far as overall allocation.  You know, make back about seven grand on this piece, we may cut that loose, continue to manage the other one, giving us the ability to jump back in with another piece in a different cycle. So we'll see what happens. Uh, but right now we'll just continue to manage these as they are. Uh, ES, we've got a long put vertical price moved way out of the range. So we need a nice down move to get back into range there. And that's going to be kind of the theme with some of these verticals, uh, GC already mentioned that one. NG, Natty gas, uh, just kind of managing. Oh yeah, I was going to mention this. I kind of posted this in the community this morning. We've got 18 days to expiration. So we typically want to roll out of these things to the next cycle by 21. But I mean, this thing is really holding its theta. I mean, this is it, it got some, uh, acceleration of theta today, but. Uh, man, this thing really hasn't moved with 18 days left. And there's still this much profit potential in this thing. Uh, we're going to get a pretty decent IV contraction at some point that's going to really pop this P and L line up. So I wanted to give this over the weekend and early next week, we'll probably roll, um, you know, this P and L line is not too steep either. Uh, as far as the gamma risk, but, uh, we'll, we'll definitely roll this next week, but just wanted to give it a little bit more time in case we do get a little pop higher. And we do get a real nice move up in the P and L line as we get closer to expiration, uh, didn't want to miss out on that. So that's the reasoning for holding that one a little bit longer. Uh, ZW I mentioned that Apple. So this is one of our long put verticals. This was one of our remaining positions still in April and hoping for a little bit of down movement before we do anything. This will probably roll that one out, but we'd love to see a down movement before we do that. De John Deere, this one has gotten smoked to the upside real steep rally in John Deere. So we've got, we've got two different pieces here. This one is way out of range. And then this one, uh, is, is a little ways out of range here too. So just looking for some downside in DE same with DIA prices moved out of range on that piece. And we've got a second set, uh, that prices moved well out of range there too. Now, if you look at the expected move, which is the gray box here, the edges right here, um, you know, it's the probability of it getting back in this, in this, uh, cycle is not very good. So instead of letting this theta decay all the way down early next week, we'll probably roll one of these. Now we're in, um, you know, we've, we've got 36 days. So we may even roll out to the June cycle with more duration. Uh, we'll be down into the sixties of days to expiration. So we'll probably at least, uh, move one of those out in time just to keep that as a positive theta position. Uh, I mentioned IWM QQQ, kind of a similar situation, uh, prices moved at a range. We've got two different sets here. That one's way out of range. And then this one here is a little bit closer, but still, still well out of range, but still a good chance of getting back in by expiration. So just going to hold this one for now. SMH, uh, we've got this adjusted short strangle made back about 700 bucks since we rolled it, uh, just continue to hold that for some more theta decay. That's in May, uh, SPY I mentioned, WIN, another one of our bunkers. So we're up pretty decent in WIN.  Uh, but if we get a sharp move lower, you know, that's going to benefit that. So that's the, you know, the cool thing about these bunkers, and let me go to one of others here. I'll skip over XBI, go to XLI.  So instead of having the price on the, on the side, we have the percentage. So, you know, we hit a low, uh, we were, this was down 40% at one point. Now it's down 23. So that's like a 20% rally that we've seen. And the fact that we can see that kind of rally and we're down a couple hundred bucks on these bunkers, that's pretty good. Now, obviously we would love to see the downside to benefit this position, but the whole benefit of these bunkers is that if you do get that rip to the upside, you've got very little risk.  And so, uh, the other one XRT, we've got two different sets here. Let me, let me separate these, you know, so this one is down 24 bucks. And this one is down 190 bucks. You know, so even with these massive moves that we've seen, just rip your face off rallies, you know, in this case, from this point to this point, you know, that's about it again, about a 20% move in the stock. And to only be down a couple hundred bucks, that's the value of these things. You have a huge potential to the downside, but you don't get, you don't get killed on the upside. You know, some of these, uh, some of these verticals that we have, you know, they have more risk to the downside. Whereas these, these things, if this market does continue to rip higher, we have very little risk and we have all the potential to the downside. And so that's the benefit of those things. Um, and then, um, and then, you know, if you compare it to other kind of bearish or short Delta strategies, I already mentioned short verticals or short call verticals, long put verticals where you have risk in both directions. You know, the other thing is, you know, you could buy deep in the money puts, but again, on big rallies like this, you're going to be getting killed, uh, on the, on the P and L where these just really minimize your losses to the upside. So I think between all of our bunkers were down, you know, like five, six, 700 bucks. I haven't, I don't know exactly, but, um, but we have huge potential to the downside. And if we do get this, this, uh, drop off in the market again, like I think we will, uh, these things are going to be nice to have in place.  I like to bring the heat. I do not like to take the heat, but, uh, you know, I talked about being patient in one of the videos, uh, earlier. And this is part of it too, is when you're, you know, when you're in a, when you're in a strategy, again, you're never going to pick the tops. You're never going to pick the bottoms. That's not how trading works. And anybody tells you they can, uh, is, is full of it. And so you're going to have, and that's why we've just been kind of layering into these things, layer in a little by little on the upside, get some movement in our direction, but then it turns on us, just kind of layering in. And, you know, if this thing does roll over, we're going to benefit that from that, even though we're taking some heat right now. So stay a little bit patient. My friends, uh, I think the winter is still coming. Uh, okay. So I skipped over a couple of these XBI, we've got this adjusted short strangle here, uh, and this is, uh, this is inverted. So we've made back about 900 bucks since we, since we did our last roll, you know, we're at a point now where, you know, we're over 50% of max profit. So we'll let this sit a little bit longer. I mean, we've got 36 days to expiration. So we're not looking to roll out in time yet. So we'll just continue to let this theta decay. You know, if we look at our theta at this level, at the current price, you know, we've got 18 bucks a day today, and that'll accelerate as we get closer to expiration. So no, no hurry to do anything with this one yet. And then lastly, XLK, another long put vertical that's out of range. So we need some downside in that to benefit. So those are all the alerts. Those are all the positions. Everybody have a great long weekend. We'll check in with you Monday. See you then.


So did you hear about that hostage situation at the reform temple in Texas where rabbi and three congregants were taken hostage by a guy protesting about a Muslim in prison and wanting wanting a Muslim take you know taken out out of prison and so the rabbi was quite a lefty so Reform rabbi is about 99% are on the left Conservative quote-unquote can the conservative denomination in Judaism about 95% of Quite-unquote conservative rabbis are on the left Probably reform Jews 90% are on the left Conservative Jews probably 85% on the left. Anyway, this this reform rabbi is quite the leftist who's taken hostage in Texas He regards the Jewish state of Israel as an apartheid state And I actually don't have any problem with that I think that there is you know some accuracy to that that accusation, but He thought it was an apartheid state that against Islam. So he was He was quite the lefty and what I'm struck on Twitter is all these Jewish lefties are saying that armed gods at synagogue are not the answer.  So there are plenty of rabbis who don't want any armed guards at their synagogue and don't want any of their congregants You know carrying guns and so there are accusations about this particular rabbi that he was you know very much against having armed congregants or armed guards and This is just craziness All right, if the founders of the state of Israel the modern state of Israel being pacifist There'd be no modern state of Israel. So So On the alt right, there's a lot of conception about Jews that they're you know duplicitous and they're you know recommending You know one thing for non-Jews and the very opposite thing for Jews, but it's not generally true Right left-wing Jews promote left-wing policies for both Jews and for non-Jews for both the Jewish state and non-Jewish states and so Jews who believe in gun control, you know Jews who believe in pacifism They feel the same way for their synagogues and for the Jewish state and for Jewish communities It's not just something where they want to impose something entirely different on non-Jews that they want for their own groups So generally speaking right-wing Jews Pro-nationalists both for the Jewish state and for other states and left-wing Jews are anti nationalism and anti guns Whether it's for the Jewish state or for the non-Jewish state, but it's just amazing to me that on twitter There are all these Jewish reactions that having armed guards is not the answer To hostage situations that the gun control is the answer Now I think you can support both gun control And having armed guards at at certain institutions Right, you can support gun control and recognize that gun control is not going to be perfect.  So I am not a big supporter overall of gun control But you can support both gun control and having armed guards at synagogue that I just see this dominant reaction among many Jewish leftists on twitter In their reaction to this hostage situation in texas that Armed guards is not the answer. Well armed guards is the answer Armed guards are absolutely essential for any institution like Jewish institutions that might well be under attack


.In today's lecture, we will discuss Homsky hierarchy. Already, we have seen a few class of languages, regular languages, convex languages and recursive languages and then recursive and immorable languages. We have characterized all these class of languages grammatically and also using automata. We have found that regular language is a proper subset of convex free languages because every regular language can be generated by using some contract free grammar. But there is a language like say a to the power n b to the n and gatoran equal to 0, which is not regular that we have already proved, but this is a convex free language. Similarly, every convex free language is recursively immorable language because every convex free language can be generated by using some unstructured grammar and except by some Turing machine. But there is a language a to the power n b to the power n which is recursively immorable, but not convex free. So, that way regular language is properly contained within convex free language, convex free language is properly contained within recursively immorable language and then we will introduce some more class in between which is called context sensitive language and these four classes, these four classes of languages are said to be Honsky hierarchy is named after the famous language Honsky who proposed these languages as natural model for natural languages. So, this confinement we can show that regular language is properly contained within convex free language, convex free language is properly contained within recursively immorable language and we have introduced that regular right linear grammar generates regular language that means regular language generated by regular grammar and finite grammar accepts regular language and finite grammar and regular grammar these are equivalent that we have already shown. Similarly, two zone .tomata and convex free language, convex free grammar they are equivalent. So, p d a accepts convex language and a convex free grammar generates a convex language. Similarly, a Turing machine accepts recursively immorable language and similarly, a recursively language, immorable language is generated by an unrestricted grammar or structured grammar the same thing. Now, let us recall the definition of structured grammar.  That means production rules is written basically by this notation alpha goes to beta where alpha is from v star n v star and beta is from v star that means beta may be epsilon, but the left hand side alpha must contain at least one non-terminal symbol. Now, if left hand side of each production is a non-terminal simply then we say that the grammar is a c f g. So, this structured grammar is set with c f g if the left hand side is a single non-terminal symbol. In addition in the right hand side of each production rule is a terminal string flowed by at most one non-terminal symbol then a grammar is set to be a regular grammar. So, based on this we can have a classification of grammar. So, alpha goes to beta if there is no restriction other than the rule that we have already given is said to be a type 0 grammar. If we keep on imposing some restriction on both sides then we get some other kind of grammar if length of alpha left hand side is less than or equal to the length of the right hand side then we say that this grammar is a type 1 grammar. And if the left hand side is single non-terminal it is called type 2 grammar or context free grammar. If besides this if the right hand side beta is either a sequence of terminal symbols or a sequence of terminal symbols followed by a non-terminal then we say that is type 3 grammar. So, already we know that the type 3 grammar generates regular language, type 2 grammar generates context free language, type 0 grammar generates recursively and invariable language, but we did not mention so far this type 1 grammar. We see that the type 1 grammar is nothing but it is called context sensitive grammar and it generates context sensitive language. So, we will first introduce this context sensitive language we will characterize this context sensitive language and then we will show that context sensitive language is properly contained within this recursively and invariable language in fact recursively language and there is at least one context language which is not context free. So, this shows the proper contentment of CSL within recursively language and CFL is within context sensitive language. Just consider an example of context sensitive language say this is a language n equal to equal number of ways followed by number of ways followed by same number of bs followed by same numbers of c's where n is greater than equal to 1 that means n equal to a to the n b to the n c to the n n greater than equal to 1. Now, we can generate this grammar using generate this language using the following production rules is a set of production rules where the first one says s goes to a b c s is a start symbol or a x b c x is a non-terminal x goes to a x b capital C capital C non-terminal or a b capital C then c b goes to b c and c c goes to c c. Now, we can show that this grammar can generate this language we know that this language is not a CFL, but this production rules set of production rules which follows the restriction imposed by this context free language in no production rule the left hand side length of the left hand side is more than the length of the right hand side it is at most equal at this point c b equal goes to b c, but in other cases it is less than the length of the right hand side. So, it is basically the type 1 grammar which is a context sensitive grammar it is context sensitive grammar. Now, therefore this set of non-terminal is s x c and set of terminal symbol is a b c. So, it is quite clear that this is a context sensitive grammar c h g just have a look at how can drive the strings or the form a to the n b to the n c to the n for some n greater than equal to 1. So, if n equal to 1 we simply generate use a first rule s goes to a b c to get a b c if n equal to we first use the second rule for s that means s goes to a x b c and then this x will be terminated by one of this we can term it by one of this if we keep on continuing with terming using this one will keep on generating more and more as b as n b n c and whenever we use this one we stop here generating any more as and once we use this production as goes to a b c then we keep on shifting the b which is there towards the right hand side will keep on shifting towards the left hand side. So, that all b's appear immediately after the group of as and eventually all capital c's will go towards the end extreme end and eventually all capital c's using this kind of rule capital c goes to c capital c c goes to c c small c c we can use or we can have we can convert all capital c's to small c's. So, that is what we have done for n equal to a x b c. So, this x is replaced by a x a b capital c now using the rule c b goes to b c we have shifted this b towards the left hand side by one symbol and we have got this string and eventually capital c c goes to small c c we have used the last step to get the c's similarly if you keep on continuing to use the production x goes to a x b c then we will have this kind of derivation as derives a x b c then replace this x by a x b c keep on continuing to replace this x by a x b c then after number of steps we will have this kind of string a d power k minus 1 a b c b c to the power k minus 2 then b c and eventually shifting this b in the extreme right towards the left hand side by using the rule c b goes to b c b capital c number of times we can shift all the b's towards the left hand side getting a power k b power k and then k numbers of k minus 1 numbers of capital c and small one small c at the extreme end and eventually using the rule capital c c small c goes to small c small c k numbers of steps k numbers of times we will eventually get the string a d power k b d power k c d power k so this grammar generates the language a d n b d n c d n so this is an example of context free grammar we will introduce a kind of automata called linear bounded automata and we will show that the linear bounded automata accepts context free languages now a linear bounded automaton is a non deterministic Turing machine satisfying the following conditions the first one is that it is input alphabet includes two special symbols one is left angular bracket and the other is right angular bracket the left and right these are the left and right end markers respectively and the l b a has no moves no more moves left from the left end marker and or right from the right towards the right end marker nor it may print another symbol over left end marker or right end marker that means left end marker and right end marker cannot be erased or can be replaced by any other symbol and head read head cannot move left towards the left end marker and right towards the right end marker now consider a linear bounded a linear bounded automata automaton m where q is a set of states input alphabet transition moves and q 0 is the star state say it is an l b a linear bounded automaton the language accepted by this linear bounded automaton m the set of all strings w over sigma other than what which does not contain left end marker right and marker or the head symbol such that it starts with this configuration initially that means it starts with a star symbol q 0 the input w is placed between the left and marker and right marker and the head read read head is reading the right end marker initially eventually from this if it moves to this configuration where it enters a state through the whole thing state and in that case that the con may be anything may have any content within this left angular bracket and right angular bracket alphabet any strings over string of terminals or any non-terminals then it is reading some symbol a since it has eventually entered the whole state for this string will be w will be accepted will now show that the l b a linear bounded automaton automata and conduct sensitive grammars are equivalent and this equivalence is shown as follows given a csgg conduct sensitive grammar g derision l b a linear bounded automaton a such that lg equal to l m similarly given an linear bounded automaton m derision csgg such that lg equal to l m minus epsilon so here since l b a might accept epsilon but since the csg the left hand side of the grammar is less than length of left hand side of grammar is less than or equal to the right hand side so therefore this conduct sensitive grammar cannot generate epsilon so epsilon cannot be generated by a conduct sensitive grammar so therefore it will be equivalent in that sense that it generates any string except by the linear bounded automaton except for the string epsilon now will first show that if there is a csg there is an equivalent linear bounded automaton m accepting the same language generated by the grammar g now clearly g is a structured grammar a special case of a structured grammar now will construct a Turing machine m equivalent to g in exactly the manner that was discussed earlier except that earlier that in the in the context of showing that a Turing machine and structured grammar are equivalent already have shown that Turing machines and structured grammars are equivalent in that context whatever construction we used will use the same construction except that instead of multi-tap consider it in a multi-track in such a case suppose w belongs to l of g then there is a derivation like this as derives w1 w1 derives w2 eventually w n minus 1 derives wn which is equal to w and in every step since we are using the production rules of a conduct sensitive grammar therefore the length of s must be less than equal to length of w1 because of the restriction that we have for conduct sensitive grammar and the length of w1 greater than length of w1 minus 1 and length of w1 is basically w now give the input w to the linear bounded automaton in the form w within this left n marker and the right n marker since each sensational form wi which is generated over various steps of m is not of length greater than w greater than the length of the w m does not use any steps beyond the n marker so this is for sure because in every step the length of w or wi is not greater than the length of the final string that will be generated so therefore in no step the Turing machine will use any cell towards the left side of the n marker or towards the left side of the left n marker or right side of the right n marker therefore m behaves as a linear bounded automaton as is at therefore m is an LBA which accepts the language generated by the C H G G let us now see the other direction that means if the original LBA m accepting some language so L m there has to be an equivalence C H G G except accepting the same generating the same language and that is your L G that means L m equal to L G consider that the linear bounded automaton is m with the elements q sigma delta and q naught we write sigma naught to be sigma minus these two symbols that means sigma naught does not contain the two n markers now construct grammar which is a required contraction grammar from this LBA we will see that T is a set of terminal symbols of sigma whatever terminal symbols is there in sigma we will have in T and n is a set of non terminals we have two special two non terminals s and a besides that we will have we use some composite symbols to represent the non terminals the composite symbols are of the form a alpha is a pair basically within square bracket a alpha where a is a terminal symbol and alpha is from this set C so it may be x for x in sigma 0 or it may contain angular bracket left angular bracket then x or x left angular bracket or it may contain both the angular bracket and x x may be any symbol from sigma naught or it may contain some state symbol q q belongs to capital Q and it may contain both state symbol and angular bracket along with this x so C may be anyone from this set alpha may be anyone from this set C and P consist of the set of production will consist of the following productions we will now see what are production rules to be used so the first production rule says that as goes to capital A and then this composite symbol a a q naught a comma a q naught or as goes to the single symbol non terminal is a comma within bracket within bracket a q naught here a is a terminal symbol and this a for a can generate this kind of strings a a and a left bracket a in fact using this one and two we can generate the input string here once we generate a string using as and a the first component if you collect the first component from each of the composite symbols that will give us the exactly input string and the second component basically represents the tap component and while deriving during derivation this grammar will simply simulate the moves of the Turing machine and linear one automaton m suppose that the automaton contains a move like this delta q x equal to P y instead q if it reaches a symbol x then it goes to state P and y y may be a print or y may be left left move or r y may be a right move if y is a print from some some symbol from input alphabet it prints then what is done if currently suppose a Turing machine or linear one automaton was reading symbol x it was in state q reading symbol x that is what given by this move then it simply changes that from q to P and x is sends to y that means it prints a new symbol y and here z 1 and z 2 may be any symbol may be epsilon or it may be left angular bracket or a right angular bracket so therefore z 1 and z 2 will remain same now if suppose the Turing machine or a linear one automaton in state q reading x goes to state P and it moves towards the right hand side here is moves towards the right hand side that means y equal to r in such a case suppose this is a last this difference this is a last symbol last non-terminal in such a case this angular bracket will appear at the end so therefore it will simply reading q reading Turing symbol x at state q it will simply go the right side of the symbol x that means it is now reading the right hand marker P otherwise if it is not the last symbol then there will be some other symbol towards right hand side so it is in state q read symbol x then it will simply move over the symbol x that means it will read the next symbol therefore it will go to read the symbol b so from a z q x next symbol is b b z dash where z dash may be any right angular bracket or epsilon it may be empty also so the state symbol q will move towards the right side to indicate that it is now reading the next symbol b so it is P b z dash z this may be empty and here this z may be the left angular bracket or it may be empty as well similarly you can see the move when it goes towards the left side these two are exactly similar to the previous two and a special case when it is reading the left hand marker it simply goes towards the right side it moves towards the right side so this is right side and it may sense a state to p so a q left angular bracket a z it is reading the left hand marker so there is no move towards the left side it has to move towards the right side so it can now read the symbol a similarly we can write the rule for when when the lba is reading the right hand marker in such a case of course there will be no right move so it has to go towards the left side to read the symbol which is there towards the left of the right hand marker and finally when the lba enters in a state which is a halting state is h is a halting state then simply we reduce this composite symbol to a where a is a symbol that is there to write the right side and then from then onward we can reduce this combination some composite symbol and the terminal symbol to a b so a comma alpha b will be now a b taking the first terminal symbol and these are terminal symbol similarly b a alpha will convert to b a so we see that each of the production is follows the rules of a contraction rules of contraction of grammar that means type one grammar so this grammar that constructed from the move towards the reading machine is exactly is a contract sensitive grammar now it is a written verification as in the case of the reading machines that this grammar g generates the language except by the linear model automaton except for epsilon suppose w is a string which is a 1 a 2 up to a n except by the automaton m so using rules 1 and 2 we can simply generate in a few steps square bracket a 1 comma left angular bracket a 1 then square bracket a 2 a 2 square bracket a 3 a 3 finally square bracket a n comma a n q naught left angular bracket you see that if you collect the first symbols from every composite symbol it constitutes the string a 1 a 2 up to a n which is the input string a 1 to a n and then we can use the other symbols following the moves of the grammar we can use other rules of the grammar eventually to generate the string eventually this can be converted or transformed to this string so we can transform this string from s we will get the generate this string eventually we can transform it to this string provided that the reading machine m accepts this particular string that means we can have we have some moves to accept this string by the linear model automaton we will last this by an example say this is a linear model automaton which generates or accepts all strings containing an a any string containing a will be accepted by this linear model automaton which is shown in the table the moves of the linear model automaton is shown in the table so q 0 q 1 q 2 are the states of the linear model automaton a b are terminal symbols and left angular bracket and right angular bracket are two special symbols that we have as left unmarker and right unmarker here in this capital a shows some arbitrary moves you can choose anything there because that situation does not arise since q 0 in the star state it always read the left right angular bracket so in such a case will enter in a state q 1 and go run left side because the string will be given like this a 1 a 2 up to say a n within this and will be reading this left unmarker at state q 0 so therefore first we have to move towards the left side that means we will move towards this side and enter instead q 1 now we will be reading the symbol a n that is the first move and once we are there in the state q 1 will keep on skipping the symbols until we see an a once we see an a if at q 1 we see an a we enter in the halting state and we move towards the right side or left side does not matter but if we see any other symbol b other symbol b then we are in the same state q 1 but moves towards the left side we keep keep on skipping the symbols but if we see the left angular bracket then we enter in a state q 2 and move towards the right and in q 2 again if we see an a we enter in an arbitrary state because that situation does not arise otherwise we would have entered in a halting state earlier in the state q 1 and if instead q 2 we see a b you enter in a state q 1 and move towards the left side so therefore at that point it will keep on oscillating so just consider a move of the Turing machine it starts say input string is a b whether the Turing machine did with an lba whether it accepts a b or not so q naught a b is that to n markers currently reading this this indicates the currently reading symbol right angular bracket so in the next move following this one q 0 right angular bracket goes to q 1 l so it will enter in state q 1 and it will move towards the left hand side head move towards the left hand side so it is now reading b so on q 1 b q 1 b it will again skip and move towards the left hand side the head move towards the left hand side state remains m now q 1 a q 1 a it will enter in a halting state it will enter in a halting state it will move towards the right hand side so since now it has now enter in a halting state this string a b is accepted by the linear bounded automaton now let us see whether this c h g which is constructed using the rules of the grammar generates the string a b or not so the if we apply the first rule so as derives a b b q naught for every for all a belonging to T we have this kind of rule as derives a as goes to a small a q naught is a rule we can apply and then this a goes to a a this rule can be used in the next step to terminate the non-terminal a so once we have this we see that the first symbol a b represent the input string a b and the others the second part can be used to similar the type of the l b a now this q 0 right in a bracket in this case it will simply move towards the left hand side following our rules of the move of the Turing machine q 0 right in a bracket it will move towards the left hand side so we will use the rules of the grammar corresponding rules of the grammar says b q 0 and then q 1 b in this step since q 1 b we have this rule q 1 b it goes towards the left side so therefore and it remains the same state q 1 so in this case so since there is no way to go here in the left side so this q 1 will move towards the left this symbol towards the left side composite symbol towards the left side so to have to read the symbol a that means it will be now left hand bracket q 1 a and similarly now q 1 a since we have the move q 1 a q 1 a is halting state and r so therefore the corresponding rule in the grammar csg will be according to construction so q 1 a will go to the right side and will read now symbol b and enter in the halting state so this will this will be a new situation so since it has entered in the halting state so this now will reduce to b according to a rule this rule that we have so according to a rule this rule on the context free connection with grammar so whenever we have halting state it will reduce to the symbol that is there towards a which is a left component so therefore in this case so this will reduce to b so there is what we have used over here and now b allow this now will reduce to a b so this a b so therefore s derives a b so this how this corresponding grammar generates the strings so therefore we have shown that context sensitive language context grammar and linear bound automaton automata are equivalent and the language generated by context sensitive grammar is said to be context sensitive language now we will show that context sensitive language is properly contained within recursive language first we will show that every context sensitive language is recursive let l be a context sensitive language generated by the grammar context sensitive grammar g will construct an algorithm a which accepts l that is given any string w from sigma star a accepts w if and only if w belongs to the language l so it goes like this given w belongs to sigma star the algorithm performs the following steps it constructs a graph whose vertices are the strings from n union sigma star whose length is less than or equal to w length is less than equal to w so here the vertices represents the strings over n union sigma star but the length which must be less than or equal to the length of the string w now for vertices alpha and beta are vertices of the graph we know that alpha there is beta in the grammar g if and only if there is an arc from alpha to beta so it is quite a g or you can see that alpha there is beta if only if there is an arc there is an arc from alpha to beta now note that the parts in the graph represents derivations in g therefore w belongs to l of g if and only if there is a part in the graph from vertex s to the vertex for w so if you can somehow find out a part from the vertex representing s to the vertex representing w then we know that s derives w now a simulation method which finds a part from s to w if exist and there are algorithms we can do it easily to find out a part from one vertex to another vertex in a graph and it reports s if there exists any such part to indicate that as derives w that means w is x generated by the grammar g otherwise it says that no it is exit so therefore we know that there exist such an algorithm and hence every csl can be shown to be recursive now we will show that there is a recursive language that is not context condensative to show the proper contentment we will show that there is a recursive language that is not condensative to show this we will consider a binary encoding binary encodings of all csg's context condensative grammars we will just consider some binary encoding suppose this g in angular bracket represents a binary encoding of a csg g now we define the language l like this l is a set of all binary encodings of all the csg's g in angular bracket binary encodings of all the csg's such as g is a csg and this encoding is not accepted or not generated by the grammar g now this l of course it belongs to 0 1 star because this is a binary encoding so here g must be a csg and this is the binary encoding and this is not accepted by or generated by the grammar g that is this encoding does not belong to lg now we will show that l is recursive this language l is recursive but it is not a context sensitive language to show that l is recursive if any input is given say any w belongs to 0 1 star is given we will say where are w defines a csg if it does not defines a csg obviously w does not belong to l that is quite clear now if w defines a csg then using the algorithm just we have described above given a previous result we decide whether or not w belongs to l of g so that we can always do by constructing graph and then looking for a path if there exist so therefore we can show that l is recursive because we can decide whether or not w belongs to lg now we show that l is not a context sensitive language so assume for contradiction that l is a context sensitive language then there exist a context sensitive grammar so g dash such that l g dash equal to l there must be some context sensitive grammar now we pause equation where are the encoding of that grammar g dash belongs to l there is a question that we ask suppose that this encoding belongs to l then according to a definition that this encoding does not belong to l of g dash so therefore this encoding does not belong to l so this is a contradiction that we have got similarly if we assume that this encoding belongs to l encoding of this grammar that we have assumed belongs to l then since g dash is a csg then encoding of this belongs to l of g that is encoding of this belongs to l again we have arrived at a contradiction by this argument so therefore our original assumption that l is a csl must be wrong so therefore such a grammar context sensitive grammar generating l does not exist so therefore it is not context sensitive language hence l is a there exist l is a l is a l is a recursive what is not context sensitive so hence we have the following strict containment between the between the language classes the class of context sensitive language is strictly contained in the class of recursive languages and hence we have got this hierarchy in fact we have some other classes like say first they have got this containment say regular language is properly contained within context free language context sensitive language is properly contained and cfl is properly contained in the context language and csl is properly contained within la disputa in the bother language in fact we have some other classes for example say DCFL determine condition language so reguar language is properly contained within you drink on language thatých let also we have taken earlier tQ determined context language proppers of some other context speech English context learning which purpose of some other context language and context language purpose . . . . . . . . . .


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