What is going on guys welcome back to the channel critical overlord here just want to do a video today talking about the purge More specifically the forever purge which is set to be released this july it was supposed to come out last summer in july But of course, you know things changed as they did with many films that were supposed to release last year They got shifted into 2021 and beyond due to the ongoing pandemic Uh, the purge has been a franchise that's been around since 2013 now I remember going to see the original film with ethan hawk in theaters I thought it was one of the more I thought the first film definitely was a very much grounded home invasion film that was exploring this idea Of what it would be like if we had this type of event going on every year where For a short period of time all crime would be legal to financially and economically assist the the climate in in america because apparently it's very beneficial from what the data is Showing in this universe and this has all been brought on by the new founding fathers It was their idea and we know that the purge series from the most part takes place in the 2020s Uh, the last film that came out was the first purge a prequel to the events that came before the other three movies Now the forever purge is set to take place after the purge election year which came out in 2016 subsequently Ironically the same year that we all went to go out and vote for either uh, 45 or hillary, but We had a new image released today Not today, but recently of the of the film is nothing special here nothing special nothing mind blowing It's a purge movie. It's something chaotic total film released this It looks like it's typical purge activities going on in this image as you see And given everything that happened last year and things that were going on still this year as we just saw recently in news The recent events that happened at the capital Uh, nothing in this image is shocking. I don't think anyone's shocked by this. It looks very much like a purge movie We've basically been living a purge movie for the past year, but go nuts. I guess I'm not necessarily the biggest fan of this franchise, but I appreciate the concept the first movie Yes was very very much a grounded a grounded way to handle the concept and what I think a lot of people Loved about the future sequels is how they branched out and did things that was an issue With that first movie instead of just being grounded in this house There's a lot more exploration But I still feel like they're kind of squandering this concept because everything's just murder murder murder murder murder It felt like they could do a lot of different angles with this concept and they're not doing it so News recently came out about what's gonna happen with the franchise going forward and just a brief synopsis about this So this is just from bloody disgusting Sources recently told thr that the film takes place That this film takes the purge out of its usual Metropolitan setting while still dealing with class and race issues in the way of the previous films have And we've also been told that this is to be the final movie in the franchise Now details about the film those synopsis is saying that the the movie will take place after the events of election year And will center on adela and one who finds solace at the texas ranch Who finds solace at the texas ranch? Having failed fled a drug cartel in mexico Things go awry when a group of outsiders decided to keep purging Beyond the allotted time when people can break any and all laws So apparently in this movie what we'll have here is just a purge going on after it's allotted time And we'll be centering on two two new characters who we have not yet met adela and one Now the film's cast looks to include josh lucas Leavin rambin cassidy freeman will patent who is so far the only recognizable name that i see Anna de anadella regara and veronica falcon the only name i recognize is will patent Uh, that's the only name i recognize in this cast again. Ethan hawk was in the original film The second and the third film. They had this one central character that returned in both films I can't think of his name at the moment, but He was in the second and the third film. He was in anarchy and he was in election year the My favorite personally out of this franchise so far has been the purge election year I was a real big fan of this girl here the candy the candy bar girl The way she was just she she was probably the most insane character They they presented in this franchise over a candy bar a candy bar She was ready to go in on this store clerk over a candy bar. I just was I was just enamored by her character. I thought that's my honestly my favorite character from the franchise How insane she was over a candy bar but As you heard this is supposed to be the final entry in this franchise. I'm honestly quite glad I don't think that's going to be the case. I think ultimately what'll happen is they'll keep making more They'll just take a brief So they'll take a brief hiatus for the time being and then they'll return to it The reason I think that is because look at paranormal activity paranormal activity The ghost dimension was supposed to be The final entry a very lackluster final entry it was but now we're going to get another one We don't know really what's going to happen yet there, but I'm just saying look at what happened with paranormal activity Um, you know a lot of series do they say they're going to be the final chapter? We know screen 5 is coming but screen 3 We know scream was originally supposed to be a trilogy You know things always change with time things do change So like for the time being yeah, we'll rock we didn't say this is going to be the final film in the franchise Because that's what their intentions are but in your intentions could be different a few years from now especially if Another group of creative minds get together and they have something they want to offer to this franchise because like I said For the most part I felt like this purge concept has been squandered the Creator of it james demonica. I believe his name is he's even he's the one stating He's always wanted to end this franchise recently He stated that recently in an interview whether last year or this year or the year prior He has ideas already about ending the franchise. So the fact that they want to end it here Um, depending on how it plays out I'll that's when I'll say whether it was a good ending or not I felt that the purge election year was a solid end I thought the purge was just fine as it was as a trilogy Because you know if you're not gonna branch out and explore this concept in other ways without just showing us murder and chaos like that Then stop making these movies because like I said There's so many things that other people would be doing on purge night But I'm just seeing they're more interested in exploring Murder murder murder murder and I feel like a concept like the purge where all crime including murder is legal Um, that's something that could be explored in a variety of different genres Like the purge doesn't just have to stick to this hardcore horror It could be exploring all these other things because like like they say themselves all crimes Including murder. They're just including murder in the concept with these films. So they're not exploring all crimes They're just they're just exploring Including murder aspect. That's why I feel like the purge has been a squandered concept for the most part with these movies So with the purge 5 whatever they do. I'm just glad it's gonna end. Hopefully it ends on a high note I can't wait to see what happens Can't wait to see how things play out with these people that are clearly just gonna keep on purging beyond the allotted time They're supposed to be purging. Let me know what you guys think about all this down the comment section below What do you think about that image? If you haven't already make sure you subscribe turn on post notification at numbers of video in the description I have links on my social media accounts on facebook twitter and instagram. Of course, you can mention me there, of course So let me know if there's any movies news or reviews you would like me to cover in the future with all that in mind guys I will see you in the next video


This is stock number 9J227A, we are here at Summit Automotive in Fond du Lac, Wisconsin, your new and used Jeep headquarters. Today we are checking out this super clean 2019 Jeep Cherokee, limited with the high altitude package. This vehicle has the two liter turbocharged four cylinder motor, granite crystal is the color, and from this HD video you will be able to tell that this Jeep is extremely clean and like new all the way around, inside and out. We shoot all of our videos in 1080p, so if you have HD capabilities on your computer tablet or a smartphone device, turn them on right now because it's like you're right here looking at the vehicle with me, and definitely your best way to check out the vehicle before seeing it in person. Bridgestone, Duler, a lens of tires, these are 23550R19s, they have just about all the tread buff, only 4,000 some miles on this vehicle, no denser dings on the driver's side front fender, has the LED fog lamps, the LED headlamps, and the LED running lights, no dense dings, scuffs or scrapes on that front bumper, and the hood is absolutely perfect, no denser dings on that. Pastner side rim, very clean as well, no scuffs or scrapes on that, and as you go down this side of the vehicle you can see just how clean the body is, how reflective and mirror-like that paint is. We take these HD videos, so if you are far away, or even if you're close by and just cannot make the trip down you can still see the vehicle. Hear the vehicle and have confidence in what you're looking at before you even get here, so that when you get here there's absolutely no surprises. Back rim, excellent condition as well, and the back tires have just as much tried as the front. New thing on the 2019s is the capitalist fuel fill, it's a really nice feature if you've never used it. As we come to the back of the vehicle and see it has the rear parking sensors, the dual rear exhaust, LED tail lamps, you got the granite crystal Jeep lettering, this is a power lift gate, just press a button it'll go up like so. Back storage area is very very clean, no rips or tears, it does have a factory subwoofer, and underneath here we have a full-size spare and jack tools. That goes down like so, and those seats do go down nice and flat.  And as you go down this side of the vehicle just as clean as that passenger side, no dents or dings on that rear quarter, and this back rim for full disclosure, no scuffs or scrapes on that.   There are no rips or tears on those seats, has driver's side lumbar, power driver's seat, factory floor mats throughout, auto headlamps, memory driver's seat, and power windows, power locks, and power mirrors. As you hop inside the vehicle here, you can see that this one has 4,497 miles, you get the outside temperature display, as well as the range to empty and your digital speedometer. This one has your heated leather-wrapped steering wheel, cruise controls on the right, and information center and Bluetooth controls on the left. I have a little bit of a problem with my focus on my camera here. There we go. It has the 8.4 touchscreen radio with the 4C radio, which means that you get the AM, FM, Serious XM radio capabilities, as well as Android Auto and Apple CarPlay capabilities. This one does have the heated seats and heated steering wheel, as well as the backup camera right there. There's your AM, FM, and Serious XM radio capabilities. This one also has the factory navigation system, which is a nice feature. Tool climate control right here has the select terrain, auto, snow, sport, sand, and mud, active drive 1 system, the 9-speed automatic transmission, AUGSA and a USB jack. And the passenger seat is in nice shape, as well. No major rips or tears on that. And that seat is heated, too. Headliner is very nice and clean. You get your map lights up there, as well as your home link buttons for your garage door security systems and lighting systems. And finally, there's an SOS and assist button in the mirror, which are features of the vehicle that you never have to pay a subscription for. We'll take a quick look at the back seats, and then we'll check out under the hood.  No rips or tears back here. Does have a latch child safety system. 115 volt, 150 watt plug-in, USBs and USB-C jacks back here, as well. And these seats do fall down for extra storage, like so. Take a quick look under the hood. I would personally like to thank you for checking out the video today, hopefully from this HD video. You'll have been able to tell just how clean this Jeep is all the way around, inside and out. Under the hood, we have the 2-liter turbocharged 4-cylinder motor. Engine bay is very clean, runs very smooth. 270 horsepower. This Jeep has been fully safety-danspected by our service shop, has a fresh oil and filter change. All the fluids have been checked and topped off. Jeep has been gone through mechanically 100% and is 100% ready to go to see more pictures of this Jeep or one of our other 450 new-used cars, trucks, SUVs, minivans, Wranglers. You name it, we got to go to our website, www.summitauto.com. Full pictures and descriptions of every single vehicle from our lot, from two locations all at summitauto.com. If you'd like to check out more HD videos, you can go to youtube.com slash summitauto. Remember to like, subscribe and share on this video and all the videos that you see there. In fact, in a second, you will see a link to subscribe to our YouTube channel on your left. A link to more used inventory videos like this one on your right if you've not been to our website on the bottom link to this vehicle on our website. Click those, check us out and we really look forward to helping with this super clean like new 2019 Jeep Cherokee limited high altitude. Thanks again.


Minnesota. This is where we slept overnight. I just went across the street now, grabbed myself a coffee and I'm ready for the day. So today I'm gonna unload this lumber that we loaded yesterday. It's under the tarp. We're gonna take the tarps off, get in their lumber and then we're gonna drag our empty trailer down to Shakopee, Minnesota. We have a load there to pick up. That's part of Minneapolis once again. We're gonna pick up our load there. We have a loading appointment at 4 p.m. We should make it no problem. And then we're gonna start heading home. So we're not gonna make it home tonight yet, but we'll be home tomorrow. Beautiful day again for trucking. Just about coming up to Brainerd here already. Gotta quickly get unloaded and then rush on down to Minneapolis. Once again an empty trailer. Hell's a nice quick one. Now it's time to go down to Shakopee, Minnesota. I love delivering here. Perceiver is such a nice guy. Usually they would be going home around the time about when I got here. I called ahead and it was nice enough to stay a little bit longer to get me unloaded. And now that I have an empty trailer, I can go put new stuff on it and make some money going home. I thought I'd be going home empty. And I was like, ah, I guess I gotta get home for the weekend. Gotta get old blue polish. I guess I'm gonna have to go home empty. No. The load gods came through for me and they found me some freight to put on my trailer. Now I can get paid to go home the way it should be. I should have plenty of time, but you do also have to add extra time for traffic in Minneapolis. Since my appointment's at 4 o'clock and I gotta be there around that time, I can't be too early. Means I'll be driving right into afternoon rush hour. Hard to guess how much extra time you need traffic is behaving. And if there's any fender benders or accidents, I can really slow things down. It's not a Friday, so at least you don't have the cottage crowd, you know, plugging up the roads, going to their cottages and cabins. Not as much anyway. Just the lucky ones who get to go take off a day early. Mankato last weekend at home. Mankato. Some of you guys live out here. I wonder how close I am to here right now. What a nice day though. In 500 meters, get to the left on, US 169 Mankato. US 169 south, Mankato. Just stay straight. That's all you gotta tell me. I'm six kilometers away now. My measurements are all messed up. I use both. I use American measurements and Canadian measurements like any good Canadian. Outside temperature is always in Celsius. Inside temperature could either be Fahrenheit or Celsius depending on the household. Ours is Celsius. Most Canadians are Fahrenheit, I believe. Cooking temperature on your oven always in Fahrenheit. Temperature of, you know, oil temperature or water temperature of a motor, like in my truck here, Fahrenheit. But if you're talking about outside temperature, Celsius. And you're talking about weights. Weights on a truck? Kilograms. But weights, if you weigh yourself on a scale, pounds. We're a big mess. We should have just stuck with the miles and the old metrics. I mean, we changed over to the metric in one of the 60s. The last Trudeau guy did that. He changed us over to metric. We used to be just like the US. Miles and hour, feet, inches, ounces, pounds. And he changed us over. Because part of the Canadian identity is to separate ourselves from the American identity. Because we're so similar, probably we're the exact same people. But our, it seems like Canada's whole goal is just to prove to the world that we're different in some way. We always got to be different. We're not American. We're Canadian. I'm sort of like that too. I mean, yeah, we're different. We have different histories, but we all come to the same place. I mean, we all came from over the Atlantic. We came here. We built two new countries. And we did a pretty good job, I'd say. They're going pretty well. I mean, they need a little maintenance. I'll tell you that much. We haven't been maintaining them very well, but they're good countries. And distance is in kilometers as well. As a Canadian distance is always in kilometers. But if you're talking generally, then it's in miles. And as a truck driver, I get paid, I get paid percentage of the load. But if you're getting paid for distance driven in Canada, you're paid by distance. You're paid by the mile, even as a Canadian driver. So it's, it's all over the place. I can't explain it. I don't know what's going on. I've got my new load behind me and we're in St. Cloud, Minnesota. They got the cheapest juice again. Got about a little over a third of tanks left. Let's swing in here, grab some fuel. 364 per gallon US, as opposed to $4 a gallon up in the Grand Fort, or 396 or something. As opposed to way more in Canada. I don't really like stopping here because it's always a gong show to get in there. And I can see it already. Line up onto the street. Yep. Trucks lined up onto the street. Well, this is fun. I parked in a way that I'm not blocking the road, but this guy in front of me is blocking the entire road. So just a heads up if you do that. It's all very nice. Oh, the car's just squeaking by him. Yikes. Absolute chaos. Very small a lot to begin with. And for some reason, I've noticed at this specific place that guys will stay parked in the pumps for a little bit longer than they maybe should. But that happens everywhere, though, too, right? I mean, maybe I'm just, because I've been stopping here more often. Maybe I noticed that. It's mostly, I guess, the size of the lot. We've got this guy going out backwards against the arrow. There's an arrow right on the ground right there. Check this out. There's an arrow on the ground right there. You see it? Arrow to go that way. And he's going that way behind us. And the guy in front of me isn't fueling.  So even if he's done, he can't go. And we're lined up out onto the street. So it's a little bit of a hit and miss. Sometimes you come here and it's not bad.  That might be for the same reason I'm here. It's cheapest fuel on I-94, west of Minneapolis. It is what it is. Hopefully it won't take too long. The North Star Truck Wash. It's at the stay mark on the north side of Fargo off I-29. I-29 is literally right behind me over there. I'm going to be honest, I'm trying something new today. I've never been to North Star. I usually go to the Blue Beacon over on the other side of town at the Petrol. But the guy who does my polishing requested that I get the truck acid washed before I bring it to him to make sure it's all nice and clean. Makes it a lot easier on his side. And I said, that's no problem. No problem at all. Whatever I can do to make your job easier, right? So he recommended North Star because apparently they do a much better job than Blue Beacon. So it's my first time here and let's see how good they do. I don't know what their prices are, but they should be pretty competitive to the Blue Beacon, right? I mean, that would be their main competitor. We'll see. They're open 24 hours except for Fridays and Saturdays when they close from 12 a.m. to 8 a.m. I guess just to clean the place out or something. We're here at 10 p.m. Let's see what they can do. I always love getting truckwired. It's like going to the spa for me too. It makes me feel good. Looks like they're letting the guy in front of me in. Moving up. That means we're next.  You want to see the finished product? I think they did a pretty good job. I'm actually very impressed. I will be coming here instead of Blue Beacon from now on. So we got the whole truck acid washed and that's to prepare it for polishing tomorrow. That is clean. Clean, clean, clean. Look at those steps. Imagine how good they're going to look after they're polished and the tanks. So we got all the dirt and grime off them. Using the wheels in the back that's way better already. I'm playing a bit of a risky game here. I'm gonna drive up because I got some hours left on my clock. I'm gonna drive up a little closer to home in the dark here. Usually in the summertime this would cover my truck in bugs, my clean truck. But I think it's early enough in the season that there's not too many bugs out. There are some but not too many. A bit of a risky game here. So hopefully I don't get my truck covered in bugs. If I do I mean I can just rinse it off tomorrow morning before I bring it in for the polish. But I want to get closer to home because I don't want to have such a long drive tomorrow. This paperwork is not cleared yet so we're pulling into the gas tracks at Pembina North Dakota right before the Canadian border. And we'll sleep here and hopefully we'll be cleared by morning. No I'm gonna turn right. The entrance to the right. No I'm gonna go to sleep. How about that? I'm still in the U.S. I'm still here and it's morning time. My border paperwork uh something got messed up or something something something's not happened I don't know and I couldn't get cleared last night yet so I had to wait here at the border in Pembina. Well that's okay whatever it's morning time we're cleared now at least I'm told we are. I'm just gonna double check. Well that's double check. There's usually like a par's tracker that most brokers have. As long as you know who the broker is you can check to see if your par's has passed. That's my customs clearance. I don't know what par's stands for. P-A-R-S. Eh. Probably. Okay come on. Where's the uh shipment tracking? Why is it not giving me the option? Oh it's hot in here. Let's open some windows. It got it suddenly got hot very fast. Okay well my shipment tracker's not working right now. That's inconvenient. Maybe I just need my glasses on. Glasses on so I can see the world in HD. I can see just fine right now that I can read all the signs around here. I'm fine. It's everything's got a little bit of a fuzziness. A little bit. I can still read everything just fine but then you put these bad boys on. HD 4K. Oh there it is. See that's all I needed to see my glasses. That's weird. Okay it worked just fine. It popped up. Okay let's see. Am I clear? Declaration accepted awaiting arrival of goods. Nice. Good. We're set to go. So thanks for joining me today. I'm going to continue this tomorrow. Tune in then. Don't forget to subscribe. Hit that like button and we'll hang out some more then. Take care.


I am with the Maris family and Rich will start with you. Did you ever think that this record would be broken? Uh, I didn't have my doubts, but you know, it's been a long time since somebody's come close to really doing this in a so-called manners, but it's nice to see somebody come along like Aaron Judd to give it a shot. Roger, what's it been like for the family to watch him pursue 61? Well, it's been exciting for us.  And to be here the other night, we've been here this our second night here in the first night. We had that crazy ninth inning, judge it as 60 or so. I mean, that was all really, really cool for us to witness and we're just happy to be here taking it all in and seeing what he can do this weekend and hopefully hit against the Red Sox just like dad did, you know. What are the emotions as you're taking it all in? Well, you know, I don't know how they feel, but for me, I just, you know, I try to be in the moment. I try to get too far ahead of yourself. I mean, I think we realize the record is going to be broke and just matter of win. And I just hope for the Yankees fans and for us, it all happens right here in New York so we can actually, you know, get the feel what happens here in New York with something big like this happens. Kevin, what do you think your dad would think about all this? I think he'd be excited to see it, you know, being a Yankee and the being the type of player and judges, you know, the quality guy on and off the field that he is. In the teammate that he is, is special and dad would be excited to see it. Sandra, as you're watching, are you hoping that he breaks the record? Are you holding out hope that, you know what, there's still a lot of games left this season, but maybe it won't happen? Well, it's bittersweet, you know. I mean, we being in Yankee Stadium and playing the Red Sox, which is who dad hit his 61st against. So it would be kind of a special time to tie it all in, how it has been with judge and 99 two nines and, you know, 61 and it's been 61 years and it's just a lot of similarities and we're just excited. So I bet it's going to happen next couple of days. It seems like it all does kind of fit a little bit when you bring up the numbers, you bring up the dates and all of that. So just each of you, when Aaron judge steps to the plate, what are you doing? What comes to mind?  Honestly, I think my heart stops for a moment because I don't know really, you know, what to think, you know, I'm excited for him. He's had a phenomenal year and, you know, I think, I think dad would be happy to pass the baton to him. I really do. Yeah, yeah, what are the bats like for you? I'm just a baseball guy. I don't get too wound up either way, but just excited to see the challenge. Will it be meaningful for you to be here if it happens against the Red Sox? Meaningful for me? Oh, yeah, if it happens against the Red Sox, I think it'll be very exciting, you know, because obviously being big rivals that they are and, you know, since dad did it against the Red Sox, I think it'll be pretty cool finish to the year. Thank you all for your time and enjoy the night. Send it back to you.


Hi and welcome to the session. Let us discuss the following question. Question says, form the differential equation representing the family of curves given by x minus a whole square plus 2y square is equal to a square where a is in arbitrarily constant. Let us now start with the solution. We have to find the differential equation representing the family of curves x minus a whole square plus 2y square is equal to a square. Or we can say we have to find the differential equation whose solution is x minus a whole square plus 2y square is equal to a square. Now we will consider this equation. Now we know x minus a whole square is equal to x square plus a square minus 2ax. We will write this plus 2y square as it is and here we will write is equal to a square. Applying this formula to this bracket we get these three terms. Now a square and a square will get cancelled and we get x square plus 2y square is equal to 2ax dividing both the sides of this equation by x we get x square plus 2y square upon x is equal to 2a. Now differentiating both the sides of this equation with respect to x we get x multiplied by 2x plus 4y multiplied by dy upon dx minus x square plus 2y square multiplied by 1 upon x square is equal to 0. We can find derivative of this term by applying quotient rule also derivative of 2a is equal to 0. Clearly we can see 2a is a constant and derivative of a constant term is 0. Now multiplying both the sides of this expression by x square we get x multiplied by 2x plus 4y dy upon dx minus x square plus 2y square is equal to 0. Now this further implies 2x square plus 4xy dy upon dx minus x square minus 2y square is equal to 0. Multiplying x by this bracket we get these two terms and multiplying this negative sign with this bracket we get minus x square minus 2y square and here we can write is equal to 0 as it is. Now we know 2x square minus x square is equal to x square only. So we can write this further implies 4xy dy upon dx plus x square minus 2y square is equal to 0. Now adding 2y square minus x square on both the sides of this equation we get 4xy dy upon dx is equal to 2y square minus x square. Now dividing both the sides of this equation by 4xy we get dy upon dx is equal to 2y square minus x square upon 4xy. Now we know dy upon dx is denoted by y dash so we can write y dash is equal to 2y square minus x square upon 4xy. So this is our required differential equation. This completes the session. Take care and have a nice day.


idealized sampling means to replace a continuous signal by a train of pulses, sharp pulses if you would like to call it that. Let us catch what we are saying. What we are saying is if you have this continuous signal and if you sample it here and here and then here and here and here, what you are doing is just ultimately considering this signal. So, let me call this X s of t and let me call the original signal X t. Now, what is the purpose of reconstruction? The purpose of reconstruction is to get back X t from X s t simple and where is the problem? Now, we know the problem. The problem is not that you do not have an X t which you can get back, you have just too many of them. If X t, even if X t were one sine wave, this process of sampling has created an infinite number of sine waves which could have generated those samples. So, the problem is not that you have no X t. The problem is you have just too many X t s which could have all resulted in the same set of samples. Now, the difficulty is to identify which is the correct X t and that is where we need to see when we will be in a position to do so. You see a simple approach to identifying the correct X t is to go back to what we did on the frequency axis. On the frequency axis, what sampling does? Sampling creates copies at a collection of frequencies. In fact, let me try and illustrate this to you with an example or at least with some intuitive explanation. So, you know we have done the exercise for one sine wave, but let us do it when you have a few sine waves. So, suppose we have a few sine waves, we have a 0.1 kilohertz sine wave, we have a 0.13 kilohertz sine wave and a 0.15 kilohertz sine wave. So, what is the copy that is created around 1 kilohertz when you sample at 1 kilohertz? Again, you know you have so what I am saying is you have these three and in fact, let us give their amplitude some values. So, we could say this has amplitude A 1 phase phi 1, this sine wave with amplitude A 2 phase phi 2 and this one has amplitude A 3 phase phi 3. Then if the sampling rate is 1 kilohertz or sampling frequency is 1 kilohertz, if you prefer to call it that, then you get a copy at around 1 kilohertz, you get 1.1, you get 1.3 and you get 1.5 and of course, the amplitudes are A 1, phi 1, A 2, phase is amplitude and phase I mean A 2, phi 2, A 3, phi 3 here and you also get copies behind. So, you get 0.9, 0.7, oh well, I am sorry, this should be 1.1, this should be 1.13. So, you know in fact, 1.15, this would be 0.9 and this would be 1 minus 0.13, that is 0.87 and this would be 1 minus 0.15 and that is 0.85. The amplitudes here would be A 1, A 2 and A 3 respectively, but the phases would be minus phi 1, minus phi 2, minus phi 3 and you can keep doing this exercise. You can do it around 2 kilohertz, you can do it around 3 kilohertz and so on, but the central idea is that you have this original what we call spectrum.  Spectrum is the decomposition of the original signal in terms of the sine waves that comprise it. You have this original spectrum. So, let me repeat, what is a spectrum? Essentially, a spectrum is a Fourier transform. What is the physical meaning of the spectrum? The spectrum is essentially the collection of sine waves specifying the frequencies, amplitudes and phases. Now, here we have a discrete set of frequencies. We have 0.1, 0.13, 0.15. We could have a continuum of frequencies. So, remember if the signal is a periodic, then we need to have a continuum of frequencies. You know, the frequency axis could be continuously occupied in a small segment. Then you have a band of frequencies occupied. In fact, you talk about the band of frequencies on which the signal lies. So, here for example, we have taken a case where there are only 3 frequencies, 3 sine waves, 0.1, 0.13, 0.15. So, the band of frequencies is between 0 and 0.15. But of course, you could have a continuous occupancy of the frequencies all the way from 0 to 0.15. So, we will say in either case that the band of frequencies occupied is between 0 and 0.15. The difference is here, we have a discrete set of frequencies occupied. You could have continuous axis occupied all the way from 0 to 0.15, whatever it is. What is the effect of sampling? The effect of sampling is that the original spectrum is copied as it is after 1 kilohertz, after 2 kilohertz, after 3 kilohertz and so on and it is copied in mirror image fashion. So, you know you can visualize this has been mirror imaged and in mirror imaging, you have reversed the phase as well. So, you know you have a copy of the original spectrum here and a mirror image of the spectrum here and this is repeated at every multiple of the sampling frequency. So, around 2 kilohertz, you have a copy of the spectrum after 2 kilohertz and you have a mirror image of the spectrum behind 2 kilohertz. In mirror imaging, the phase is also reversed. So, it is very easy. How do you reconstruct x t from x s t? If you look at the spectral domain, it is very easy. In the spectral domain, what we need to do is to retain this and throw away this, throw away not only this, all such copies. Let us write that down. It is a very simple idea. Reconstruct x t from x s t means retain the original spectrum and throw away the copies, all copies and mirror images in fact. Now, the answer is very simple. When can you do this? You can do it if the copies and the mirror images do not mix with the original spectrum. Let us write that down too. We can do this if neither the copies nor the mirror images pollute or overlap the original spectrum and again now we have a simple answer, which is the trouble making copy or mirror image? In fact, if you think about it, it is only the first one, which is the trouble maker and that too not the first mirror, not the first copy. It is the first mirror image, which is a trouble maker. That is the one which comes first as you move away from the original spectrum. So, let us write that down. The trouble maker is essentially the first mirror image. You see, again let us write this down on the frequency axis and this time let us use more general symbols. Let us not confine ourselves to this 1 kilohertz, 0.1, 0.11 and so on. Let us use 0 to f m. This is the maximum frequency or the band occupancy.  So, the first mirror image occupies the range between f s and f s plus f m. I am sorry, the first copy occupies the range between f s and f s plus f m and the first mirror image occupies this range. This is an f s minus f m. So, now we are very clear. As long as this point is after this point we are okay.  Now, this is a very simple principle. This is called the Nyquist principle of sampling. In fact, I dare say many of you probably know this principle. You know, this is called the Shannon Nyquist, Wittaker. There are so many names associated with this principle. In fact, many people have arrived at this conclusion by using different approaches. We have also used a simple approach to come to this conclusion, perhaps a little intuitive, maybe not that rigorous, though we can complete the same discussion with rigor. And I have left a part of you, part of that as a challenge for you. Whatever the case, I am saying that the principle is very simple. One simple way to get back x t from x s t is to take the lowest part of the frequency axis as containing the original spectrum and to ensure that all the copies and the mirror images are well spaced away from the original spectrum. They have not polluted the original spectrum. So, now this brings us to the situation when will there be a problem, when this does not happen. In other words, if that first mirror image happens to overlap on the original spectrum, there is trouble. And what is that trouble called? That trouble has a name in the literature. If this is not true, then we have aliasing to spend a couple of minutes on the term alias. Now, we use the word alias in common discussions, in common parlance quite frequently. There are two situations in which the word alias is used. One in the context of troublemakers and the other in the context of highly gifted people, writers, poets. So, you know that some writers or poets take on aliases. You know, I do not need to give you examples. There are several writers either in the English language or in the Indian languages who like to take on different pen names as they call them. They are called aliases. And then you also have many people who create trouble in society who take up aliases. Or even if they do not take up aliases, the authorities give them aliases or the authorities know them by their aliases. Now, the reason why I mention this is that the spirit behind this is also true in signal processing. Aliasing is not necessarily always bad. So, it can be done in very desirable situations or it can be done in very undesirable situations. In this first course, we shall treat aliasing as undesirable in a way, because we want a very simple methodology for reconstruction. We do not want to complicate our reconstruction process too much. We want the original spectrum to be unpolluted. Now, I will explain why the word aliases is appropriate. You see, take the frequency, let us say, take the frequency 0.7 kilohertz. Now, suppose you sampled at a rate of 1.4 kilohertz exactly. You know, if you had 0.7 kilohertz as the signal frequency or if you had a signal whose maximum frequency component went all the way up to 0.7 kilohertz, you should have sampled at more than 1.4 kilohertz. Suppose you happen to sample exactly at 1.4 kilohertz, where would the first mirror image come at 1.4 minus 0.7, that is at 0.7. So, the original signal and its first mirror image overlap at 0.7. So, what happens is that the first mirror image assumes a false identity. Actually, it is not a 0.7 kilohertz signal, but it pretends to be the 0.7 hertz signal, which the original signal is. So, in other words, you know, you have a false or another man posing to be a 0.7 kilohertz frequency and interfering with the original 0.7 kilohertz signal frequency. That is why we call it a raw signal. There is an imposter, there is an intruder there. Now, you can take this argument further. So, now suppose the original signal had components all the way up to 0.9 and you have sampled it only at 1.4. Now, what is going to happen? 1.4 minus 0.9. So, you have 0.5. Now, you have a component at 0.7 kilohertz, at 0.5 kilohertz coming from the original signal and you have a component at 0.5 kilohertz contributed by the first mirror image. Actually, that component at 0.5 kilohertz is a mirror image of the component at 0.9 kilohertz in the original. So, 1.4 minus 0.9, that is 0.5. So, you have an original 0.5 kilohertz component and you have a false component at 0.5 kilohertz coming actually as a mirror image of the 0.9 kilohertz component as a consequence of sampling at 1.4 kilohertz. So, you have an imposter and the original one sitting together at 0.5 kilohertz assuming a false identity and interfering with the original signal. We can understand it like that. Now, this is the bad part of, you know, we cannot reconstruct if we choose to use the simple approach of taking a region around 0 frequency to reconstruct. But I want to put a second challenge before you here. Let me explain the challenge to you, but I would not answer it too far here because it is a challenge after all. The second challenge is suppose I know that I have a band occupancy of the signal that means if I look at the sine waves in the signal, the frequencies where there are nonzero amplitudes are between some f 1 and f 2. Let us for the sake of example take f 1 equal to 3 kilohertz and f 2 equal to 5 kilohertz. So, what I mean is I have a band of only 2 kilohertz occupied actually, but if I go by the highest frequency occupied it is 5 kilohertz. So, of course I am guaranteed that if I take a sampling rate of more than 10 kilohertz here, I am of course able to reconstruct the original signal by retaining all the frequencies up to 5 kilohertz and throwing out the rest. That is perfectly possible.


Welcome back in the last lecture we discussed began with the semantics of predicate logic where we discussed about when do we say that a given well formed formula in the predicate logic is true when it is going to be false etc. In continuation with the last discussion of the last lecture we will be continuing and then we will be talking about some more examples so that we can get this idea in a better way. So we will try to talk about the semantics of the predicate logic in greater detail with some examples in this particular kind of lecture. So what is important for the semantics of the predicate logic is there are two things one is the domain it does not make any sense to talk about the truth value of a given predicate logic formula without respect to some kind of domain. So we need to fix a domain it can be natural number it can be real number it can be set up people say rivers etc and all we need to fix the domain and then you need to have an interpretation function I and that constitutes D and I constitutes what we call it as a model or a structure etc. So now in this particular kind of context we defined what we mean by we provided formal definition of a structure so essentially what it talks about is that in the predicate logic we have variables constants predicates and functional symbols each one when you assign some kind of some values to these things it has to find some kind of an entity in the domain where the predicates are mapped to 0 and 1 that means the property whether or not it holds or not is the one which we are going to see and then each individual constant should have a member in the domain D etc each function symbol finds another kind of energy functional symbol in the domain etc. So now how can we define truth the truth of a sentence Phi a given formula Phi in an L with respect to some kind of structure which consists of domain and the interpretation function structure a in which an element a belongs to a is named by a ground term of L and is defined by means of some induction like this for atomic sentences are T1 to Tn in a given structure a that is going to hold that means our T1 to Tn is going to be true with respect to a structure a if and only if if you have R to the power of a T1 rise to the power of a to T to the Tn rise to the power of a so that means the relation R a on a n assigned to R holds of the elements named by the terms T1 to Tn otherwise it is going to be false. So we will give some examples to talk more about the definition of truth with respect to the formulas in the predicate logic in the context of propositional logic we have seen what we mean by saying that a particular formula is true or false with respect to a structure a a model a for example not Phi is going to be true in a structure or model a obviously when it is not the case that Phi follows from a actually it should be written in this particular kind of sense so a does not belong to Phi if that is the case then not Phi is considered to be true in this particular kind of model a in the same way Phi R Psi is going to be true in a model a if either Phi is true in a model a and Psi is true in a model a that is a standard definition and the conjunction is going to be true when both conjuncts are true that is taken care by the third one and implication that is going to be true only when if you have premises true and the conclusion falls in the same way Phi if and only if Psi these are the things which are exactly same as the case of propositional logic so the additional things that we have in the case of predicate logic are some of the truth values with respect to quantifies what is extra in predicate logic are two more operators they are considered to be quantifies first is there exists some V Phi V that is true with respect to a structure a if that is going to be true at least for one ground term T that means if you substitute one ground term T then this there exists some V Phi of V holds then that particular kind of formula is true with respect to a structure a that means for some ground term T Phi of T that means X is V is substituted by T and then Phi T has to be true in a structure a if that is the case for at least one of the ground term T then obviously that is called as there exists some V Phi of V is true with respect to structure a and for all V Phi V is going to be true in a structure a if it happens for if Phi of T is going to be true for all the values of T whatever value that you are going to take into consideration for T as in all these cases Phi of T has to be true in that sense we call it as for all V Phi V is going to be true with respect to a model a we can talk about some other important logical properties such as satisfiability validity in this context. So a sentence Phi in a predicate logic first of all what is considered to be sentence in predicate logic we does not have any free variables then obviously it is called as a sentence otherwise it is going to be formula in the predicate logic so that sentence Phi of L the language of predicate is considered to be valid which is usually represented as models and Phi especially if it is true in all structures for L if that is going to be true in all kinds of structures whatever interpretation that you are in all interpretation that is going to be true it is considered to be topology so that is what we mean by validity truth in all structures is considered to be what we mean by validity and given a set of sentences Sigma which consists of P1 to Pn we say that P1 is considered to be logical consequence of Sigma if and only if P1 is true in every structure in which that particular kind of structure all the members of Sigma are also going to be true in all the members of Sigma are true P1 also has to be true if that is a case then we say that P1 is a logical consequence of Sigma and third important thing is that a set of sentences let us say P1 to Pn is going to be satisfiable if there is at least one structure a in which all the members of Sigma are true where P1 is also true in such a structure is also called as model of a given set of formula Sigma if Sigma has no model then obviously that means there is no interpretation in which X is true it means you are not able to find out at least one interpretation in which your X is true then it is called as unsatisfiable. Let us consider some examples let us take this into consideration FB connection of the following three formulas they are F1, F2, F3 the first one is represented as red as for all X there exists some Y or XY now our XY means here that X is less than Y suppose if you take one obviously it is less than two if you take the natural numbers into consideration if you take pick up X as one and Y as two and obviously one is less than two that is what we mean by RXY and then you have F which is a conjunction of all these things F1, F2, F3 now we are going to show that it is going to be satisfiable in the domain of natural numbers it might be false with respect to real numbers some other numbers etc but we are going to say that it is going to be satisfiable when you say that this conjunction of formulas are going to be satisfiable at least in one interpretation in which this particular kind of property F1 and F2, F3 is going to be true then it is called as satisfiable otherwise it is going to be unsatisfiable. So now F1 is the formula which is represented in the sense for all X there exists some kind of Y or XY in the context of natural numbers for all X whatever number that you have taken to consideration in the domain of natural numbers there always exists some kind of Y where that particular kind of X is always X is less than Y. So for example if you take into consideration one to be the particular kind of thing there always exists there exists some kind of Y2 which is less than sorry if you take X to be greater than 1 then all the elements all the numbers greater than 1 2, 3, 4, 5, 6 etc and all for all those numbers obviously one is less than those particular kind of numbers. The second one is there does not exist X such that our X relation is X is less than X and for all X for all Y for all Z the third one is stating that if X is less than Y and Y is less than Z then obviously X has to be less than Z it happens to all XYZ in that holds otherwise it is going to be false for example if you take three numbers 1, 2, 3 etc and all one is less than 2, 2 is less than 3 obviously one has to be less than 3 in the same way 2, 3, 4 if you take into consideration in order 2 is less than 3, 3 is less than 4 that means 2 obviously has to be less than 4 so now analysis is like this so now in at least one particular kind of case where it this property holds in particular then F1 is going to be true F2 is going to be true F3 is satisfiable so then each F1, F2, F3 are satisfiable then F is obviously considered to be satisfiable so the first one for any natural number X there is always a number Y number Y there is a number Y such that X is less than Y you always find some kind of arrangement like this for any number that you take into consideration X that always exists some kind of Y where X is less than Y natural numbers it consists of 1, 2, infinity and all for example if you take any number such as let us say 25 you take into consideration then always there exists some kind of number which is greater than that one which is less than another number set 26, 29, 30, 35 etc and all so at least one such kind of situation it happens so that is why for all X whatever number that you are taken into consideration in the natural numbers there always exists some kind of Y where X is always less than Y so that holds that satisfies. Now the second thing is there does not exist X or X so it is written as for all X it is not the case that R X is there is no number of X is less than itself which is obviously the case in terms of natural number suppose if you add 0 to it then things will change or minus if you add integers to it this may not hold for in the case of natural numbers if you take 234 anything into consideration that 2 cannot be less than its own number that is 2 it has to be equal to 2 it is definitely not less than 2 so that also holds and the third one for any numbers you take any natural numbers into consideration in some kind of order if X is less than Y this holds and Y is less than Z and then obviously X is obviously considered to be less than Z if you take 234 etc and all two is less than three three is less than four obviously two is less than four. So now let us consider some interesting formula that is stated as stated in this way for all X P X implies there exists X P X this formula is going to be valid in all non empty domains when you say that domain is non empty at least it has at least domain is domain has some kind of objects otherwise the domain is considered to be empty for example if you talk about set of people at least some kind of people have to be there in that domain otherwise if there are no people etc and all only animals non living beings etc and all the domain is considered to be empty. So if every element X has a property P then of course there is at least one X in it having that particular kind of property for example if you say that all human beings die in some day or other if at least one human being has that particular kind of property I mean everyone has to die in some day or other. So then it means by saying that some XYZ if you take it arbitrarily from the domain of people that they also have they also satisfy this particular kind of property P so obviously the formula seems to be certainly valid in case of non empty domain that means the domain consists of set of people in that if it happens for all the things for example if you take into consideration set of birds for example birds crows in particular if all crows are black most of the crows are black and all then if it holds for all the crows and all then you take any two or three birds into consideration which are considered to be crows which are also considered to be black obviously. So for all XPX if it holds then there exists some XPX also holds so this happens only with respect to a non empty domain but what happens if you take into consideration empty domain like like for example unicorns devils demons etc all empty domains it does not exist so in this case what happens is that for all XPX is going to be true for any choice of P because empty set is set of all the sets so in that sense for all XPX is going to be true of any choice of P but the consequent in this conditional that is there exists some XPX that is going to be false because that leads to the existence of X and so for all XPX does not we do not have any commitment that that particular X has to exist in the universe with respect to empty domain for all XPX is going to be true and with respect to empty domain there exists some XPX is going to be false. So for any interpretation I mean any structure that you are taking to consider which has domain and interpretation function etc where the antecedent is true here I mean the forward XPX is going to be true whereas the consequent is going to be false here there exists some XPX is false hence the given well-formed formula is going to be false and hence this formula is going to be invalid with respect to empty domain. So in general when we try to evaluate the well-formed formulas that means evaluating the truth conditions of given well-formed formula in a predicated logic we usually take into consideration that the domain is non-empty you can also take into consideration the empty domain then in that case only universal quantifiers the formulas which begin with universal quantifiers are going to be true and others the property px with universal quantifier is going to be true and existential quantifier there exists some XPX is going to be false. So these are some of the things which we need to talk about in the context of semantics of predicate logic. So now a formula Phi of a language L which consists of free variables v1 to vn is considered to be valid in a structure a for L which is represented as Phi models a Phi is a semantic consequence of a structure a if the universal closure of Phi so that is the sentences for all v1 to vn to Phi which you got it by putting for all vi in front of Phi for every free variable vi in that exist in Phi and that if it happens to be true in a for all vi that the formula is going to be true in that structure a then obviously Phi is true in that particular kind of form structure a so a formula Phi of L is considered to be valid if it is valid in every structure for L otherwise it is considered to be an invalid formula. So now let us consider some more examples considered a language which is specified by some kind of binary relation symbol or which relates to objects in some way it can be plus it can be greater than minus etc and all there are binary operations and we have some constant C0 C1 to Cn and we can talk about two possible structures in the context of the formal definition of validity a structure that we are given earlier now let us talk about a domain which consists of here D sometimes you write it as a D etc it consists of natural numbers and let R a be usually the relation or with respect to structure a that is usual relation we take into consideration less than and then there are some constants which find some kind of members in the domain you write it as C0 rise to the power of a that is when it is the case it is 0 and if you take 1 into consideration C1 it takes the value 1 and now in that context the sentence for all x there exists some y or x y says that in the context of natural members we are taken the domain domain as D and now we assigned some kind of values to the constants and we have a relation which is a function between this thing and now the sentence for all x there exists some y or x y states that states like this for every natural number there is a larger one that R x y stands for this thing for all x there exists some y means for every natural number that is for all x there exists some y means there is a larger one y so obviously that formula for all x there exists some y x y where x is less than that particular kind of for example if I take a number as 25 25 etc and all 41 there always there will be number 42 which is this 41 is always less than 42 you always come across with a number which is greater than 41 so if R a is considered to be usual relation greater than then this particular kind of sentence is going to be false for example if I take 1 and 2 into consideration 2 and 1 into consideration then for all x there exists some y that is there exists at least one y that is one which is less than greater than that one one is not greater than 2 so that is why this sentence is going to be false so depending upon how you define your function that R x y and the domain that matters to us so your interpretation also changes so now let us consider domain a to be rational numbers q to be q0 to qn and r a to be again it is taken as a relation less than and constants are represented in this sense we see the power of C0 a is 0 and C1 a is 1 you are taking into consideration to constants 0 and 1 now the sentence for all x for all y R x y implies R there exists some x R x z and R z y that is going to be true in this structure it says that usually rationals are dense however the same thing is going to be false with respect to this is going to be true with respect to q but the same thing is going to be false with respect to natural numbers so what essentially I am trying to say is that same formula is going to be true with respect to some kind of domain of natural numbers same thing when you take real numbers into consideration the same formula here in this case for all x for all y R x y implies so and so the formula is going to be false so now let us consider some more examples two formulas for all x px and there exists some x not px now let an interpretation be as follows you have a domain D which consists of two numbers usually natural numbers one and two and you have an assignment for P so whenever you have P P to the power of one is going to be T when it is to that formula is going to be false and now we need to show whether the following formulas are true under this particular kind of interpretation so now the first formula for all x px this kind of property px is going to be false when it takes a value to so it is not true for all the values of x so that is why for all x px is going to be false because px is not true with respect not true for both it is true for x is equal to one but definitely it is not true for x is equal to two because we said that P of two is false it is not true for all the things is true for only one particular kind of thing only there exists some x px holds rather than for all x px so now if you take the second thing into consideration there exists some x not px which is going to be true in this interpretation because not of P to obviously is going to be true in this particular kind of interpretation so if it is satisfied by at least one particular kind of interpretation then there exists some x not px is going to be satisfiable otherwise it is going to be unsatisfiable if it is true in all the interpretation it is unsatisfiable if it is false in all the interpretations it is considered to be unsatisfiable so here there exists some x not px is true in this particular kind of then at least one interpretation in which the formula is going to be true and that will serve over purpose so now let us consider important theorem which is which is stated in the sense is stated as follows let ? be an open formula of a predicate logic that means quantified free I mean it is at least free variables in that kind of thing it is a formula and we may view ? as a formula ? of prepositional logic by regarding every atomics of formula ? as the prepositional letter. So what is theorem essentially says is that you have some kind of tautologies in prepositional logic and if you substituted with some kind of instances when you have a substitution instance which are formulas in the predicate logic that are also going to be tautologies for example in this case p ? p is considered to be a tautology or px ? px so now you substituted it like this thing for all x px implies for all x px for example if you say that thing it is obviously going to be a tautology so in this case for all x px implies there exists some x px that is going to be go going to be true in the non-empty domain but definitely is going to be false that is what we have seen earlier that is this formula is going to be false so now if you have if you have a formula PC if something holds for some particular kind of entity then you can say that there is at least one kind of entity which has this particular kind of property if at least one chalk piece is white in color and you can say that there exists some chalk piece such that this chalk piece is white in color so the always holds so it is considered to be a tautology. So in the same way for all x we have we know that p implies p if and only if not not p is true then you replace it with px in this particular kind of formula an instance of preposition logic is a substitution instance of tautology in the preposition logic and that is also considered to be a tautology. So now let us consider some more examples so that we will understand this particular kind of things the semantics of the predicate logic in a better way. So let us consider one simple example and we will stop here. So the problem states like this what do the following formulas mean meaning of a formula means giving the truth conditions that is what we mean by that are they true are they true or false so now we are taking into consideration few examples simple examples so for a predicate logical formula to be true or false we need to have a domain first of all and then you need to have an interpretation right. So now this is the formula which we have x2 greater than 0 so now where the universe of discourse or a domain is like this is set of real numbers this universe of discourse is considered to be set of real numbers which are represented as this R so what are real numbers we have all these things natural numbers 1 to infinity and then we have whole numbers that is 0 1 to all the natural numbers together with 0 whole numbers and then we have race integers and it is just like minus 1 minus 2 minus 3 etc this is minus infinity and then plus infinity 1 2 3 etc and then you have rational numbers 1 by 2 2 by 3 etc and all so all these things are considered to be real numbers so now if you take this particular kind of formula into consideration with respect to real numbers now we want to see whether this particular kind of formula is going to be true or not so now if you take you know universe of discourse to be only natural numbers so now for example if you take natural numbers into consideration to take x into x as 1 then it says that 1 square is less than 0 obviously it is less than 0 so for natural numbers it seems to be the case that whatever value that you substitute for x is going to hold 2 square which is 4 which is obviously greater than or equivalent to 0 which is greater than 0 so now this particular kind of formula for all x the x square is greater than or equivalent to 0 for every real number x we have this particular kind of thing x square is greater than 0 is the case so that is why this is going to be T that means in all these situations even if you take into consideration minus 2 or minus 1 etc and all minus 1 whole square is equal to 1 obviously 1 is greater than 0 so this formula holds for the real numbers so hence that is that formula is going to be true so now let us consider another example for all x x square is greater than 0 you remove this this thing x square greater than 0 but here real numbers also consist of this whole numbers also example if you substitute 0 square then definitely 0 is not greater than 0 and all but 0 is greater than or equivalent to 0 so now if you take x square greater than 0 now if you take this into consideration and that is going to be false in at least one instance this formula is going to be false then this does not hold for all x x square is greater than 0 does not hold so that is why this formula is going to be false whereas this particular kind of formula going to hold because if you take 0 into consideration this formula is telling us that at least one x for all x for example if you take 0 into consideration 0 square is 0 only that is greater than or equivalent to 0 the second condition holds and all that is 0 is equivalent to 0 but in this case it is strictly stating that 0 is greater than 0 which is considered to be false so this formula does not hold in particular for the real numbers so now if you take another kind of formula so now let us consider the domain to be real numbers only that is the domain which is written in this sense so now if you take another example such as there exists some x square plus 1 is equivalent to 0 so now in this case so is there any real number which satisfies this particular kind of property for example if you take 1 2 3 etc and all natural numbers then suppose if you take 1 square plus 1 is equal to 2 is not equivalent to 0 it does not satisfy this particular kind of thing or you take 2 or anything into consideration any natural number that you are going to take into consideration is always it is not equivalent to 0 so now coming back to the whole numbers if you take 0 into consideration 0 square plus 1 which is obviously equivalent to 1 so there also it is not going to satisfy the whole numbers also I mean it is not true in any domain so now let us consider integers it consists of even negative numbers also suppose if you take minus 1 minus 2 whole square for example it is considered to be 4 4 plus 1 5 which is not equivalent to 0 even that also it will not hold and then this is the integers and even if you take into consideration rational numbers and this is not going to be equivalent to 0 and that means that this formula x square plus 1 is equal to 0 it does not hold in any structure and all so the formula which does not hold in any structure is considered to be contradiction so x square plus 1 is equal to 0 for example if we just talk about only x square plus 1 is equal to x square plus 1 is equal to 0 and usually we write it as square is equal to minus 1 and x is equal to something like pressure minus i x is a complex number and all which is different from the real numbers so there is no model or no structure which satisfies this particular kind of formula that means this formula has to be contradiction and now if you take into consideration some other examples such as there exists some x x square plus x minus 2 is equal to 0 whether this is going to hold in some cases are not is the one which you are trying to see whether if you take natural numbers into consideration if you substitute one for it what will happen 1 plus 1 2 and 2 minus 2 is equal to 0 that means it holds in at least in the case of natural numbers at least one instance this formula is going to be true then this is going to be the whole formula is going to be T so that means this formula is going to be true that is true it holds for all at least natural numbers then that particular kind of formula is obviously true. So in this lecture what we have seen is that we started with the semantics of predicate logic the definitions and then we have seen with some examples when a given formula is true and when a given formula is considered to be false the same formula is considered to be true of some kind of domain which is considered to be false in some other kind of domains. So in the next lecture what we will be talking about is some important decision procedure method which is called as which we have been using it in the case of the context of prepositional logic that is the semantic tableaux method and using semantic tableaux method will be dealing with some of the important logical properties such as when group of statements are satisfiable with us in the predicate logic when a given formula is considered to be a tautology when a given formula is contradiction etc all these things we will be talking about in the next class.


ATV, the newsroom for tech and telecom professionals, and JSA Radio, the voice for tech and telecom on iHeart Radio. I'm Jamie Scott-Kutaya and joining me here today at the New Jersey Fiber Exchange launch is Mr. Jeff Brown. He's in charge of product and marketing over at Windstream, Windstream wholesale, correct? So tell us a little bit about the latest and greatest news coming out of Windstream. Okay, well, Jamie, Windstream is a Fortune 500 company. We're one of the largest national providers, sixth largest fiber network in the U.S. We're now truly a nationwide provider of high bandwidth transport, carrier ethernet, and IP services with our new network expansion going all the way out to the West Coast. And we provided many services to multinational corporations, small businesses as well as consumers. And at the wholesale unit, caters exclusively to content providers, wire line wireless carriers, cable companies, other over-to-top providers, and digital media companies. And those OTT players today, they're really making a mark for sure in our bandwidth. So that's a great, great list of customers for you. Tell us a little bit about Windstream carrier solutions specifically. Yeah, so we focus on those verticals that we talked about within the last year. We broke up our sales teams to focus on providing customized solutions to those specific verticals. So for example, with these content companies, there are a number of web scale providers that we are providing both off the shelf and customized high bandwidth transport solutions for them to get their transatlantic traffic back to certain locations within the United States or to locations between major markets within the United States where we've been embarking on a major network expansion aggressively throughout the U.S. And at the end of this year and in a couple of weeks, you'll see some press releases announcing our network expansion going all the way to the Bay Area and down to southern work down to L.A. and all the way through the Phoenix area. My neck of the woods these days, loving that. So Jeff, tell me, how does the right data center connectivity enhance that 100 gig network like yours? Well, obviously it's capital intensive for us to spend money and we're in a capital constrained environment so we need to get the right return on our capital. So when we look at our data centers, and my team's responsible for determining the data centers that we want to add to our network and put them on net, it's very important that we get to aggregation points where there's a hub of a lot of traffic. Where the content providers want to be, where the subsea providers want to be, the foreign and the domestic carriers. So NJFX really represents a lot of that for us and it was a natural fit for us to partner with them. And what do you think of this facility that Gil and his team here at NJFX have built here today? Well, Gil and I started talking probably a year ago and frankly I was a bit skeptical that the facility would be in place. So it was an extremely aggressive build plan and it's extremely impressive just to see what's been accomplished in that timeframe and I think they have a great vision, good leadership team and we're ready to be a good partner of theirs. Just even looking at their video they have of the building being built sort of in fast forward. They had a camera on the site for a year, 11 months. It's mind blowing, just the amount of work, hard, hard work that went into building this. So really quite an accomplishment. Coming back to Windstream for our viewers who may not already know, how can they get in touch with you? Well, you can reach out to me personally on LinkedIn under Jeff Brown, 0409 at LinkedIn or you can reach us on our website carrier.windstreambusiness.com. We've recently rebranded from Windstream Carrier to Windstream Wholesale. So we will be changing that URL but for right now it's carrier.windstreambusiness.com and I'm also like I said available on LinkedIn and you can reach me there directly. Windstream Wholesale, thank you Jeff for joining us and thank you viewers for tuning in to JSA TV.