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Translate the following sentences iuto first-order logic. (Give oue trauslation for each:) a) Every student passed the elass (b) There were nO studlents in the clas...

Question

Translate the following sentences iuto first-order logic. (Give oue trauslation for each:) a) Every student passed the elass (b) There were nO studlents in the class Suppose (b) is true, according to your translation. Is also true? Explain YOur answer .

Translate the following sentences iuto first-order logic. (Give oue trauslation for each:) a) Every student passed the elass (b) There were nO studlents in the class Suppose (b) is true, according to your translation. Is also true? Explain YOur answer .



Answers

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people.
a) Someone in your class can speak Hindi.
b) Everyone in your class is friendly.
c) There is a person in your class who was not born in California.
d) A student in your class has been in a movie.
e) No student in your class has taken a course in logic programming.

For sure. So we are going to elect L. X to the ego too. The proposition of furnishing which is um X is in your class. So A. Of X. Is equal to access in your class which is the proposition of function. So for the a part. Uh huh. I'm getting that message into a proposition we are going to get for all X. Or for every X we have B. Of X. And for all X open a racket we have a of X. No it was only two. Yes B. Of X. And we close this racket. So in this case we have B. Of X. To be hard to to be of X. Is mm X. Has a cellular phone. So this is how to write or translate the message into a proposition over here. So far we pat when we translate it we are going to get they exist some X. Where I. M. A. X. And there is this some X. Uh huh. Where A. Of X. Conjunction enough X. So in this scenario two we have M. Of X. Yes eggs has seen good has seen a foreign movie. So for our C. Path we have so when we translate that message into this into a proposition we are going to get there is there's some X. Where the negation of end of X. And there exists some X. Yeah for a. Of X. From junction the negation of and of X. So in this scenario do we have any of X. Is S. Can swim for a deep part. Um When we translate it we are going to get for all X. P. Off X. And for all X We have eight of X. I've been into P of X. So P of X. Here is as can solve a quadratic equations as can solve okay, quadratic equation. And for the last part which is the E. But so e when we translate it, we are going to get for all X. The negation of kill of X. And for all X. Sorry there's there's some eggs. So for parts E we have there is this some X. The negation of Q. Of X. And there's there's some X. We have a of X conjunction. The negation of Kiev X. Why kill off eggs? I want as you go to excellent or excellent to be rich.

In this problem, we're going to be focusing on quantify IRS where were given a statement, and we have to find essentially the domain of that statement using our knowledge of mathematical quantify IRS now quantify IRS are very important. And they're going to come up, um, and be significant when you get into introductory proofs and proves later on. But for right now, we're just getting used to the concept of quantify IRS and how to use them and the difference between them. So, please, while you're going through this video, please look back at the problem to remind yourselves of the original statement that we were given. I think that's going to help. Um, if you're new to quantify IRS and learning it for the first time. Okay, so for part A, we're going to let the proposition all function B y of X, and we're going to say X is the school. Then we're going to say we have this additional function. Um, X is that X has visited Uzbekistan. And so the domain of this using these conditions is that there exists y X, and you have X. So what does that mean? There exists someone in the school. Who has visited whose Pakistan now suppose that a Person X has visited a country? Why, which will denote be of X comma y? So now when we take all of this into account, we can say that the domain of the original statement is that there exists why FX and a V of X comma use Pakistan. Now let's move on to part B. We're going to follow essentially the same procedure. Let's consider functions see of X and P of X. So the domain of just these functions at the moment would be for all X. We have X. Pardon me, Why X maps to see f X and p of X. So suppose that there is a Person X who studied subject why and will denote this by s comment. Pardon me s of X comma y. So the domain of this remember, we're considering the grade and calculus is that for all X? This would hold Y X maps, toe s of X comma calculus and s common X C plus plus. Now on to part C, we're going to follow the same procedure, so we're gonna let be of X and M of X B functions bead notes, bicycle m denotes motorcycle. So the domain of this would be that for all X. We have this thing to be true y x maps to the negation of B F, X and innovex. Now remember, negation means we is essentially not so This is not be of X and mm of X. Now suppose that Person X has an item. Why, Given that we have in this essentially notation we have O of X comma y. So now we have to find the domain of our original statement For all X. This would be true. Y X maps to the negation of zero. Pardon me. Oh, of X com A bicycle and o x com A motorcycle Alice who want to deep We're gonna let h of x b that X is happy. The domain of this would be that there exists an X such that y x and not h of X. So this is saying that there exists a person such that they are not happy. So suppose a Person X is in the mental state e of x comma y. So the domain of this taking all of these conditions into account is that there exists an X such that y X and not e of x com happy. And now, finally, we're gonna say, let t of X for party. Sorry for tea of X. Be such that X is born in the 20th century. So the domain for this statement alone would be that for all X. We have y x maps to t of X so that for all people, they would be born in the 20th century. So suppose a Person X is born in Wyeth century. Maybe not the 20th century will to note that by b of x comma y. So the domain of this original statement would be for all X such that y x maps to be of X comma 20. So I hope that this problem helped you understand or maybe even introduced you to the notion of quantify IRS and how we can use them. Particularly we have to define. All of the, um, conditions were going to be quantifying. And then we could use things such as, for all, for every such that Teoh essentially take a statement in English terms and move it into a mathematical statement

You know this problem we want to translate each of the given logical expressions, each of the statements into logical expressions. Now let's let P vivax B that a perfect person is perfect, I'm gonna take your votes, be that the person is your friend. And so it's find all this out here. Someone A Yeah we want no one is perfect. And so that means for all acts, we have not P events for all X. None of them are perfect. Only we want not everyone is perfect, which means there exists an X. Such that not P events. That leaves open the possibility that there could be a perfect person and see we have all your friends are perfect. All your friends are perfect. Speak for all X. If they are a friends would be Q of X. That would imply that they are perfect. Mhm. On the we have at least one of your friends is perfect and so their existing X. Such that they are perfect and they are your friend. Yeah, there is one person who is both. Mhm. E everyone is your friend and is perfect. And so this means for all X. They are all your friend and they are all perfect means we want both of those. And that upside that character means an Mhm. Now last. We want that everybody is your friend or someone is not perfect and so it can be for all X. I'm sorry not everyone is your friend. So that would be there exists an X. Such that not Q of X. Okay that means that there is someone who is not your friend or someone is not perfect or turns to someone who is not. So here we have. Yeah.

Hey, it's Claire so enumerate here. So we're gonna translate the statements into three different ways. So for party, it says a student in your class has lived in Vietnam. So we're gonna let a F X means excess in your school and p f X means excess lived in Vietnam and p of x comma. Y means X has lived and why? So we're gonna let the first I mean be the people in your school in the second domain be all the people in the world. So three ways to write it is using the existential quantification of p of x, the axe essential quantification f x and P f x on the existential quantification Can't square so enumerate here. So we're gonna translate these statements into three different ways. So for party, it says, a student in your class has lived in Vietnam. So we're gonna let a F X means excess in your school and p of X means excess lived in Vietnam NP FX and P of X comma Vietnam. I'll just put via for sure short he of X comma y means X has lived and why? So we're gonna let the first domain be the people in your school in the second domain, be all the people in the world. So three ways to write it is using the existential quantification of P of X, the axe essential quantification f x and P f x on the existential quantification For part B, we have there is a student in your school who cannot speak. Indeed. So we're gonna let be of xB X is in your class que of x means that ex btk Lindy. Thank you. Have X comma y means x weeks. Why? So let's let the first domain be the people in your class in the second domain be all the people in the world. So we're gonna use existential quantification, not Q of X, which means that ex speaks Hindi existential quantification be of X. That means that X is in your class and not speak. That's effects and p of X comma Vietnam. I'll just put via for sure, short for part B. We have There is a student in your school who cannot speak. Indeed. So we're gonna let be of X B access in your class, que of x means that ex btk Lindy. Thank you have X comma y in the instinctual quantification P of X and not a queue of X comma didn t for Parton's X weeks. Why? So let's let the first domain be the people in your class in the second domain be all the people in the world. So we're gonna use the existential quantification, not Q of X, which means that ex speaks Hindi existential quantification be of X. That means that X is in your class and not speak. That's a C. We have a student in your school, knows Jabba Prologue and C plus plus. So we're gonna make a of ex mean exes in your school s of ex mean ex knows Java tea of X means ass nose prologue and you have X means x no c plus plus unless make s of X comma. Y mean ex knows why. So let's let the first domain be the people in your school and the second domain be all the people in the world. So one way we could write it is using the substantial quantification mass of X and he of X. You have X Henschel quantification p of X and not a queue of X comma didn t for a part. See, we have a student in your school knows Jabba Prologue and C plus plus So we're gonna make a of ex mean exes in your school s of ex mean x. Another way is a f x using the existential quantification e f x means X is in your school and SFX and t a X and you have x on another reader. It it iss using any of bucks and s of X comma knows Java tea of X means ass nose prologue and you have X means x no c plus plus unless make s of X comma. Y mean ex knows why. So let's let the first domain be the people in your school and the second domain be all the people in the world. So one way we could write it is using the substantial quantification mass of X and he of X. Then you have X Java and s of X comma pro law and s uh, X comma c++ Another way is a f x using the existential quantification e f x means X is in your school. And as of x and t a X and you have X on another reader. It it iss using E of X and s of ex common for a party we have. Everyone in your class enjoys Thai food, so we're gonna let the first domain be the people in your class and the second domain be all the people in the world. So a of X means excess in your class v of X means X enjoys Thai food and V of X comma wines. Ex enjoys Why so this is the universal quantification which is unlike the other previous examples that we did P of X, the universal quantification and java and s of X comma pro law and s uh, X comma c++ for a party. We have everyone in your class in jokes. Is this if then we have X, which means that ex enjoys Thai food, the universal quantification of a of X, then the, uh, X comma Thai food. Then we have always Thai food, so we're gonna let the first domain be the people in your class and the second domain be all the people in the world. So a f X means excess in your class V of X means X enjoys Thai food and V of X comma linings. Ex enjoys. Why? So this is the universal quantification which is unlike the other previous examples that we did p of x, the universal quantification. You have a key, which means that someone in your class does not play hockey. So the day of X B exits in your class w of X means x plays hockey and w of x comma y means x, please. Why? So we're gonna let the first timing be the people in your class and the second domain be all the people in the world. So I'm gonna use the existential quantification, not w of x e m x, and not dumb. You have X Is this if then we have X, which means that ex enjoys Thai food, the universal quantification of a of X then the, uh X comma Thai food. Then we have E, which means that someone in your class does not play hockey. Finally, we have that substantial quantification am ex, not end, not W of X comma hockey


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