Okay, this is the answers to the uh 20 homework problems. They're not homework problems. These are just for practice. You don't have to turn anything in. I sent you the exercises yesterday, so hopefully you all have that uh word document. You're going to need to open it for this class. So, take a minute. Let me know when you're ready to jump back in. It'll be in your inboxes. It was sent to you yesterday. You guys able to find it? I can give you another minute. It's should be there in your email. Were you talking about the email you sent letting us know that class was going to be online? No. There's just an email with an attachment. The chapter 5 practice. Yep. Chapter 5 practice is the name. Yes, I have it. I sent it Wednesday at 11:06 a.m. I have it. Okay, good. You got it, too. Let me get back to my team screen. Yeah, I found it. I'm just trying to go back and forth on my phone right now. How about the rest of you, Mike? You got it? Yeah. Robin. Okay. Yes, sir. So, um, okay. I understand, uh, Clara, your phone's going to die. We are recording this class, so you'll be able to have the whole thing. And I should add that up until the time of the final, I'm happy to meet with you all one-on-one. We can use Teams and go over any questions you might still have about the final exam. So, don't hesitate to shoot me an email if you want to set up a time on Teams to meet me and review. Happy to do it. Only time that might be a little tough is Saturday because I got to go to Athens for my daughter's birthday party. All right, so let me just double check and make absolutely sure I'm recording. Okay, we are recording for sure. So now we're going to start. I'm going to share out a document with you. And here's what I'd like us to do. Um, now these are the answers to the homework, but don't look at them while we do the homework because um then you won't have a chance to work through it yourself and see if you understand it. So, please don't use them. But I did want to show you uh this thing at the top here. You see it says not both S&T and not either SRT. In my own mind, it seemed like there was a little confusion about these things last time we met. So I wanted to just do a quick lesson on that. So um does anyone remember when is the statement S and T true? Let's think it through. If I tell you I am sitting and I am wearing glasses, would that be true? It's not true. It's not true. So any statement of the form a and b is only true if a is true and b is true. Right? So that's just how the dot works. Now, if you bear that in mind, if we want to say not both S and T like we have right here, you take the statement S and T and you put it in parenthesis and then you put a tilda outside of the parenthesis to say the statement in the parenthesis is false. So now let's look at the statement in the parenthesis. Remember, you always do what's in the parenthesis first, right? So line one, S is true and T is true. So what's the truth value written under the dot? Y'all should be able to see it. What letter is written under the dot T? That's right. So that says that when S is true and T is true, the statement S and T is a true statement. Right. The next line down, S is true and T is false. So what's the statement? It's false. The next line down, S is false, T is true. So the statement is going to be again false because they're not both true, right? The final one, S is false, T is false. The whole statement is false. So that's what's going on under that dot. You have the truth values of S and T written under the letter S and under the letter T. And then you have the dot. That's the truth value of the statement S and T in each case. Now, this isn't something you're going to be tested on, but it's just going to be helpful when it comes to translation. So, so far, can you make sense of the letters that are within the parentheses and the ones under them? Don't worry about the tilda. Now, line one says when they're both true, it's true. Right? In all the other lines, s and t are not both true. So, it's a false statement. It's like I'm saying I'm sitting and I'm wearing glasses. That's false, right? Or I am flying on a kite and wearing glasses. That's also false. The only time S and T is true is when they're both true. So that's what's going on with the little dot here. Now what's going on with the tilda? Well, when S and T is a true statement, like in line one, then the denial of the statement is going to be what? False. If I say I am sitting and holding a soda can, can you see this in the camera? It's actually a water can. and I actually am and you come up to me and say that's a false statement. You're wrong. So that's why under the tilda we have the letter F. In every other case, if you look back under the dot, the statement S&T is a false statement in line two, line three, and line four. So that means the denial of that statement which is under the tilda is true in all those cases. So what's under the tilda is always going to be the opposite of what's under the dot in this sentence. So that's what we say that's what we mean when we say not both S and T. And then you see I have equals not S or not T. Those mean the same thing logically. Not S or not T means either S is not true or T is not true or both S and T are not true. So that means the same thing as not both S and T equals not S or not T. And then we go to not either. I did the same thing here. You say S or T. That's the sentence. So the or is going to be um a tilda except I didn't write it evenly. The S or T should be pushed over a little bit. So, um, you look at again, we have the truth values. Oh, no, I did it. I said S or T. My T just looks like a Y. Sorry. So, I put not S or T in parenthesis. So, if we look under S, we see true, true, false, false. If we look under T, we see true, false, true, false. That's every possible truth value. So, when they're both true, what goes under the wedge? Is it a true statement or a false statement? True. True statement. Correct. And when one of them's true, it's also still a true statement because or in logic means this or this or both. Remember we talked about the inclusive or. So the only time S or T is a false statement is going to be when S is false and T is false. So under the tilda last line that's not S or Y is a true statement because S or Y was a false statement. So it's true to say that it's false. In the other cases, not S or Y is a false statement because in case 1, two, and three, S or Y is true. So saying S or Y is false would be a false statement. And then after the equal sign, S or Y equals the same thing as not S. Sorry, S or T equals the same thing as not S and not T. So just like at top we have not both S and T. That's the same thing as not S or not T. And next we have not either S or T. That's the same thing as not S and not T. They're just saying they're both false. Not either S or T means S is false and T is false. That's why it's only a true statement on the last line where S and T are both false. So that seemed to be an area of some challenge. And we're going to now go over this in class and see how it plays out. And hopefully it'll make more sense to you. I have a hunch it will. So now um let's go ahead and open the document and I'm going to share it with you. Sorry, let me stop sharing and let me go to share again. And now let me go to here. Here. All right. This is your practice exam, not exam, practice on what you'll be doing for your exam. We're just going to work through these. And I would like to do it in a way that goes in the order that I see people on the teams screen. So that would mean if I go look over at teams, we're going to go in this order. Jordan, Alexis, Matthew, Robin, Sena, Isaiah. Whoops. After Isaiah, we're gonna have Mike. And now, where's my team's thing again? And then, um, looks like Jordan and Rodney and Moises. And then we'll loop back again until we finish the exercises. So, who was up first? What did I say? I said, "Jordan." Jordan, you ready to go? Jordan. As in me, female? Yeah. As in you? Yes. Okay. Number one. Um, okay. I'll be very honest. I do not know. Well, it's not that hard. It's pretty easy. Um, what's the proposition? [Music] Um, I'm assuming it is not. Does not. Not. It's not. So, not is a denial of a proposition. What's the proposition that's being denied? Um, that Cartier does not make cheap watches. Nope. What's being denied is that Cartiier makes cheap watches. Yes. So it's not C. Okay, that's it. Makes sense now? Yeah. So C, always treat a negation as uh separate from the proposition, right? That's the tilda and it gets stuck next to the proposition. So what we have here is the proposition Cardier makes cheap watches and then that proposition we're going to call C. And then we're denying that proposition. It is not the case that Cardier makes cheap watches till the C. Makes sense. We're okay with that. Okay, great. So, anybody else in the group have a question about that one? Okay, I will take that as a resounding no. and go on to Alexis. Could you please do number two for us? Um, Arizona has a national park, but Nebraska does not, right? Um, would be there's two, right? There's two propositions and there's a denial of a proposition as well. So the denial is always going to mean there's a tilda around it. It' be a dot a tilda and an N. Exactly correct. A is Arizona has a national park. The word but is always going to be a dot. That's what you said, right? Mhm. Uhhuh. And then Nebraska has a state park would be N. And then we have the denial. It does not have a state park. So, not in. Is everyone good with that? You think these are a little easier than the last ones we did? I think they're a little easier. This is more like what you're going to see on the test. Okay. So, next is going to be Matthew. Matthew, could you do number three for us? either Stanford or two lane has an ar architect architecture school and then it's going to be um S and then like the Vshape and then T. Exactly correct. S or T. Either S or T. All right. Very good. Uh, Robin, you're up. Both Harvard and Baylor. Yeah. Just a question on number three. That either always trips me up. Do you do anything with the either? Um, it's just a way of saying this is true or that's true or they're both true. Okay. So, okay. Again, this is where logic doesn't exactly match English, right? Because sometimes if I say either or, it's one or the other, but not both, right? Either you're alive or you're dead. I mean, you can't have those both be true at the same time. But in logic, it's used in the inclusive sense. So, it's like either we're going to go to a movie or have dinner. Well, why not do both? So, it's always either this or this or both. When you see either or in propositional logic, okay. Um, number four, both Harvard and Baylor have medical schools. Um, H.B. Exactly correct. Uh, Isaiah, could you please do number five for us? So, Isaiah, I can't actually see you when um I'm going between windows like this. Are you there? And can you hear me? He's trying to say something to you, Professor. Okay. So, maybe there's something wrong with the microphone. In that case, I Oh, is that you? Okay. How about since you're having some technical difficulties, uh would you mind if we just skip ahead and then we can circle back to you later on today? Um Mike, could you do number five? Um, if Chanel has a rosewood fragrance, then so does Lavend. Is that the one you're talking about? Yes, sir. All right. So, I would do um it's a if, so it's going to have a horseshoe. Okay. Uh, the first proposition is Chanel. Second one is Lavin. So, I would guess C horseshoe L. And you'd be perfectly correct. Good job. Moving on. Uh, who's who's JS? JS. Yeah, like I said, I have a blocked screen right now, so it's harder for me to see, but I have JS after Mike. JS, are you there? Jaylen Stevens. Jay. Jaylen, is that you? Could you please do number six for us? Number six? Yes, sir. Chanel has a rose freight. That's the same thing. It's not the same. Oh, never if. Never mind. The if's in a different place. So, rule of thumb, and I know Mike and I went over this last time, whatever follows the word if goes on which side of the horseshoe. It goes on the right side. Nope. Goes on the left. Oh, because the horseshoe is if then, right? So the last one uh that Mike did was if Chanel has a rosewood fragrance. So that's why C was on the left hand side of the horseshoe. But sometimes in ordinary English we don't put things in the right order like that. Right? So what if I say I will go out to dinner if you ask me. What I'm really saying is if you ask me then I will go. So whatever follows the word if is always on the left hand side of the horseshoe. So then what are you gonna get for this now when you translate it out? Okay. If so, okay. So it's on the left side. So if Chanel has rosewood fragrance. No, no. What? What follows the if always comes first? What follows the if? Always comes first. Oh, if. Okay. If Lavin um has a rosewood fragrance, right? So does if Yes. So then so does Chanel. Perfect. And so how's that going to look with letters and symbols? Um I always have a hard problem with that. So let's look at the propositions. Chanel has a rosewood fragrance. Can we call that C? Yes. and land vin has a rosewood fragrance. We can call that l. Okay. And when we have if then statements, the operator we want is the horseshoe. Looks like a horseshoe laid on its side, right? So that's what you're always going to use when you see if then. So you want L horseshoe C. L horseshoe C. Okay, which means if Levan has a rosewood fragrance, then Chanel has a rosewood fragrance. Okie dokie. Good job. Uh, what's next? Rodney, are you ready to go now? Yeah, go. Number seven. I'm doing well. Yourself? Uh, number seven. Uh, Maren Dow writes inclusive editorials if and only if Paul Kugman does. So, here's a quick hint that you probably already remember. Every time you see if and only if, what's the symbol? Uh, the three lines. That's it. We call it triple bar. So, how's it going to look? Uh, M triple bar. There you go. Hats off to you. I will cheer you with my bubbly water. Okay, next up we have uh Moises. Moises, you ready to go? Yeah. All right, number eight, please. Um, this one may be a bit tricky, but we'll see. wins best actress only if Martin scores scores scores wins best scorsi w director correct okay uh R go on you're doing fine it's Oh, it's M horseshoe R. Uh, nope. Here's the trick. This is why I said this one's a little tricky. Um, so we know that if is followed by um if whatever follows if goes on the left side of the horseshoe, right? Yeah. So, we all know that. But whatever follows only if goes to the right side. So, what if I told a girl, I will go on a date with you only if you ask me out. What we're saying is one can't happen without the other. That's what only if means. So, we call that a necessary condition. It's that without which something else can't ever happen. So, if I tell this person, I will only go on a date I will go on a date with you only if you ask me out. I mean, if you don't ask me out, I'm not going. So, you're going to put what? R horseshoe M. Okay, so quick review. Whatever follows if goes on the left. Whatever follows only if goes on the right. And in logic we call those sufficient and necessary conditions. A necessary condition is something without which something else can't happen. So the necessary condition goes on the right. I'm not going to go out if I'm not asked out. The sufficient condition means this is enough to entail that something happened. So if I'm going out then you asked me out. So a sufficient condition can make a necessary condition be true. A necessary condition is required for a sufficient condition. Let's take breathing. That's a better example. Um if I'm breathing then I'm alive. That means that breathing is a sufficient condition for being alive. It's enough to make it happen. Do we understand that? Can I be dead while I'm breathing? Someone help me out. No. No. So, if I'm breathing, I'm alive. Um, and if I'm alive, I'm breathing. That's when you're going to use a triple bar. A triple bar is A then B and B then A. So, breathing and being alive. If I'm breathing, I'm alive. Breathing is sufficient to guarantee I'm alive. And being alive is necessary for me to be breathing. But I can change it around and say if I am alive then I'm breathing because being alive is sufficient to ensure I'm breathing because I'm not going to breathe if I'm not alive and breathing is necessary for me to stay alive. That's when you use the triple bar. You have what you call necessary and sufficient conditions. So back to this one. Um, it's saying Reese won't win unless uh Reese will win, excuse me, only if Martin wins. So, Martin winning is a necessary condition. Okay, questions about that or shall we move on? What would What would the answer be? Oh, Moises, tell us again. Wait, what? What's the answer? Oh, uh, our horseshoe N. Right, there's your answer. Because whatever follows only if goes on the right side of the horseshoe. Okay, I think that's everybody in the class has had a chance. So, that puts us back at the top, which means that we're with Jordan again. Jordan, can you give us number nine? Okay, so Armani will launch a leather collection given that Gucci rejects skinny models. So I said G horseshoe a right. So given that is another word for if. So g horseshoe a. All right. I'm just pausing long enough for any questions also to open up my screen again. Uh Alexis, how you doing? You ready to go? Number 10. The Colts I say winning most of their games implies that Payton Manning is a great quarterback. Um, I'd put C Horseshoe P, right? Um, implies that and if mean the same thing, right? So the the horseshoe just means A implies B. If A then B. So that's the correct answer. Yes. Uh need to shrink something down here. Okay. So we got that's number 10. And number 11 is going to Matthew. Bill Gates. What do you think we got going? Bill Gates is Bill Gates. Bill Gates does not support malaria research unless Warren Buffett does. Um, does he remember the hint we had for this on Tuesday? Oh. Whenever you see the word unless what's the symbol that always goes with Oh, is it it's um it's like it's either or. Yes, it's or. Yes, exactly. Okay. So, it's going to be B v W. Exactly. So, you you just file these things away. If you see and or but that's a dot. If you see if then, that's a horseshoe. If you have if followed by something, that thing goes on the left of the horseshoe. If you have only if followed by something, that thing goes on the right of the horseshoe. So, and or but is a dot. And what did you say after that? Uh, I said, what did I say? Oh, um, I guess or or either or or unless those are all wedges. And if then is always a horseshoe. And the word following if always goes on the left of the horseshoe. What follows only if always goes on the right of the horseshoe. And did I tell you unless is always a wedge also? Yes. Okay. And um if and only if is always the triple bar. So you just kind of file these pieces of information away and they will do you well. So, um, yeah, that that's right. Um, so, uh, Matthew, what do we have here for this one? Number 11. I thought I thought it was Oh, you got it right already. Yeah. Yeah, you did. You said it. Sorry, I've been teaching all day long. I'm a little dizzy. No, that's good. Number 12 is going to go to Jaylen. No, to Robin. Robin, will you tackle number 12? Yes. Yes. Mercedes will introduce a hybrid model only if Lexus and BMW do. So L do B horseshoe M. I'm sorry. L.B. Horseshoe M. Uh, we're talking number 12. Yes. So what? Say that one more time, please. L for Lexus, dot for an B. Hang on, hang on. Um, this is number 12. So, so you don't want L first because L um is not if, it's only if. So that goes on the right hand side only if of a horseshoe. M. So it would be M horseshoe L.B. Right. All right. And where do you need parenthesis? Around the L and the V. Yep. Because they go together. So only if these two things happen at the same time or they both happen. That's the only time Mercedes is going to introduce a hybrid model. So uh yeah, M horseshoe open parenthesis L.B. close parenthesis. Got it. And that brings us to Senica. Could you take a shot at number lucky 13? Sneaky, are you there? All right. Sneaker said her phone was dying. So, I'm guess going to assume that it has now passed on to the great beyond. And we will bypass Sena. And Isaiah, can you speak now? Or is your mic working? Okay, I'll take that as a no. Uh Mike, how about you? Can you do number 13? Mariah Carey sings pop and either Elton John sings rock or Diana Cross sings jazz. Um I see three propositions. Y go uh M do E and then the wedge and D. Yep. And do you see where to put the parentheses? Yeah, I would put the parentheses with the M and the E. Uh you don't want to do that because if you look at Elton John and Dana Crawl, that's an eitheror. And when we say either or, we want we're lumping the things together. So, put the parentheses beside E and D. Uh, yes. Because they have either or. Okay. All right. So, he could have said the sentence could have run, Mariah Carey sings pop or Elton John sings rock or Diana Crawl sings jazz. And I wouldn't know what to do with that. I would say it was a a confusing sentence because you don't know what goes with what and there's no commas, right? So, we're relying on either or to tell us to group those two things together. Okay. So, when you see a either or that's what you want, parenthesis. Yep. So, is there a reason why we didn't use parentheses in number three in either Stanford or to lane as an architecture? Because there's only two propositions, so you don't need parenthesis. Oh, okay. So, it's only when you got three. Yeah. When you got three, you got to sort them into groups. Cool. So, M E I mean M dot parentheses EV D. Yep. Okay, great. Uh Rodney, can you do number 15? Okay. 15 or 14? Which one? Um, oh, I got they use Mariah Carey twice, don't they? Okay, number 14. Sorry. 14. Either Mariah Carey sings pop and Elton John sings rock or Diana Cross sings jazz. Uh, M. E. Uh, wedge and D. Okay. Uh, M. E. Okay. And where do we want our parenthesis? around the M and the E, right? Okay, that was number 14. And again, um oops, hang on, I lost it. Yeah. So again, the language of the sentence leads us to construct it that way. We're saying that this condition has to be fulfilled. Okay. Number 15. That's going to go to Moises. Uh, not both Jaguar or wait, not both Jaguar and Porsche make motorcycles. Not both. We talked about this at the beginning of class. I don't know if you'd logged in by then. I don't think so. So, does anyone remember what we talked about for not both and the tilda? Yep. And then parentheses. Oh, tilda. And then J. P. And J.P. P are in parenthesis. Yes. So what tilda parenthesis J.P says is the statement J.P is a false statement. Meaning they can't both be true. So not both Jaguar and Porsche make motorcycles means it can't be true that they both make motorcycles. So if you say it is true, you're just wrong. That's what not both and does logically speaking. Oh, it's tilda parenthesis J.P. Yes, sir. And number 16. Uh, we're back to Jordan. Okay. So, um, both Jaguar and Porsche do not make motorcycles. So, would it be parentheses J. P in parenthesis in the tilda. No. And here's why. No. Okay. When you say number 15, oh yes, not both Jaguar and Porsche make motorcycles, you're saying it's not true that they both make motorcycles. But when you say both Jaguar and Porsche do not make motorcycles, you're saying it's true that neither one makes motorcycles. So no parenthesis. It's just going to be not J dot not P. Okay? Because you're making the claim that um you're making the claim that it's not true. J is not true and P is not true. And the other one above was they both aren't true, but one might be. Makes sense. Okay. Number 17. Uh, who's here? Alexis. Either Nokia or Seikko make cell phones. Mhm. Um, would that be N Wedge S? Yes, that's it. Nice and sweet and simple. Okay. Um, so now we're at Matthew again. Can we go back to 17? What's the answer? Uh, Jordan, do you want to say or Alexis, do you want to say again? Yeah, it's um N Wedge S. Oh, okay. And now, Matthew, we have one of those not either against Which means the proposition F orM if we do Ferrari makes sport car makes economy cars or Maserati makes economy cars is a false proposition. Right? So how do we put that into logic form? Well, you just add parentheses and then put the tilda. What is it? The tilda. the ta and then it still would be uh f v and then m but in parenthesis. Correct. Exactly right. Okay, Robin. Uh could you do number 19? Um neither Ferrari nor Maserati makes economy cars. Neither is the tilda. Yeah. Tilda if nor is nor like when you say neither nor, right? You're saying that they're both false. If I say, you know, I am I am neither flying nor am I the king of Egypt, right? means it's false that I'm flying and it's false that I'm the king of Egypt. So tilda par open parentheses f you're right trying to figure out the nor part though nor is always going to be a wedge. Okay. So tilda open parentheses f wedge m that's it close parenthesis which is another way of saying it is not the case that f or m is a true proposition number 20. Uh who's that? Let me get it back over here. Uh, Mike, can you finish us off with number 20? Uh, either Ferrari or Maserati does not make economy cars. So, I F Wedge M. You're almost there. You wanted to get the um do not part in. Yeah. So that's a tilda. Where would the tilda go before? What you have here is there's not going to be any parenthesis. You're saying either it's not the case that Ferrari makes economy cars or it is not the case that Maserati makes economy cars. So either or is not the same as neither nor. When you say neither nor like in number 19, you're saying that the proposition F or M is always false. Neither F is true nor M is true. When you say just or, then you're just saying either F is false or M is false. Um or they might both be false. So till the F wedge M. Yeah. Tilda M. Oh, both of them get a tilda. Mhm. Because we're denying both of them. Either Ferrari or Maserati does not make economy cars. So, it's not the case Ferrari does it, and it's not the case Maserati does it. Hang on just a sec. My wife needs him. I'm back. Okay, so those are the um practiced exercises. I'm going to send you a few things as soon as class is over. I'm going to send you the answers to the practice exercises so you can look at them on your own. And I'm also going to send you a video of today's class. Uh, is there anything else you guys want to go over while we're still together? Our our agenda has been fulfilled, but we do still have time left. So, is exam five the final exam? Just trying to make sure. Yes, it's just like any other regular exam. It'll have probably 15 questions on it. Okay. And is there going to be a time limit? Yeah. Uh I don't remember exactly how long it is, but it's over an hour. Um it's usually two hours for exams, but I'd have to check that. Okay, that's cool. Or for final exams, I mean, I think it's usually two hours. Okay. Is there anything else we can discuss? All right, guys. So, please do remember if you want to talk about any of this stuff before the final, feel free to give me an email and we'll set up a team's conversation to go over it together. Okay. All right. All right. So, I will see you all next week and uh take care and have a good weekend. Bye bye. Have a good one, Professor. Have a good one. Bye.