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Exercise 3: Define the following propositions: j: Sally got the job. Sally was late for her interview Sally updated her resume.Express each pair of sentences using ...

Question

Exercise 3: Define the following propositions: j: Sally got the job. Sally was late for her interview Sally updated her resume.Express each pair of sentences using logical expression_ Then prove whether the two expressions are logically equivalent: If Sally did not get the job, then she was late for interview or did not update her resume_ If Sally updated her resume and did not get the job, then she was late for her interview:

Exercise 3: Define the following propositions: j: Sally got the job. Sally was late for her interview Sally updated her resume. Express each pair of sentences using logical expression_ Then prove whether the two expressions are logically equivalent: If Sally did not get the job, then she was late for interview or did not update her resume_ If Sally updated her resume and did not get the job, then she was late for her interview:



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) Everyone in your class has a cellular phone.
b) Somebody in your class has seen a foreign movie.
c) There is a person in your class who cannot swim.
d) All students in your class can solve quadratic equations.
e) Some student in your class does not want to be rich.

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

And so the objective is to go for a path. The objective is to translate the statement. Someone in your class can speak hindi into logical expression. Since the domain consists of the students in your class, they're going to let them sea of X to the proposition of function. Ex student Yeah. In your class. Yeah. Mhm. So now you're going to write a statement in logical expressionless. There's there's some X where H of X and there is this some X where C. Of X, conjunction H of X. So here H of X is um X can speak in hindi. So let me just write you two over here. So each of X means um X. Yeah, can speak hindi. Yeah. Okay. And I said earlier that sea of X be the purpose proposition of function where excess and the students in your class. So this is for the part 8 to for part B. Mhm. So for part B. So B your objective is to translate the statement everyone in your classes friends into logical expression. Since the domain consists of a student in your class, you're going to let's see of X be the proposition of function where X. Is the students in your class, as I said earlier to see of X. Is the proposition of version and X here is mhm. Is the student. Mhm Yeah, students in your class. So we are going to write the logical expressionless for all X for all X of F of X and for all X. Yeah of C of X weapon into F of X. So here F of X is X. Is friendly. So F of X over here in this access friendly. So this is how to write that. This statement. Everyone in your class is friendly in logical expression. So there's no one should I see. But so the C paths the objective is to translate the statement. There is a person in your class who was not born in California in theological expression. So here again see access their proposition information. So we are going to write the logical expressionless. There's this so for C fat there exists the assist um X relegation of B of X and they exist some X. Oh I see of X. And conjunction medication be of eggs here. B of X. S. X was born in California. So this is the logical expression. So for the price as well your objective is to translate the statement students in your class has been in the movie into logical expression. Sea of X is of course the oppositional function. So you're going to write this looking car this into a logical expression. So it's going to be, there is just some X. What am of X? And there exists some X for C of X. Yeah, conjunction M of X. So in this situation as well M F M of X. Yes X has been in a movie. So for parts E. Your objective is to translate the statement. No students in your class has taken a cost and logic for government In logic pro government into logical expression. So a sea of X is the proposition of function. So we are going to write this into political expression as four oh expo for every X minus L. Of X. And for all X. Sea of X, not into minus L. Of X. Where L. Of X is X, has taken a course in Logic Program for government.

Upsy daisy. All right. We are asked to show that each of these propositions are tautology is by using the fact that uh these conditional statements can only be false when the conclusion is false and hypothesis is true. So for a I'm sorry before I jump into that, I'm gonna explain a little bit why it is enough to show to show that fresh the statements showing that if you make the conclusion false, it will make the hypothesis false as well. So in the event where this happens, it means that you can't make the statement. It's in its entirety false because it is a implication or Yeah, yeah, it's an implication. You may go by other names but I used that one. All right. Let's start with a So we have to make a conclusion false. So therefore we have to have I'm sorry. We have to have P equal false zero. However, this here's an end command and P also arrives there is peace false. The end command is false. Therefore the hypothesis false. Therefore we can't make this statement false. Their force and pathology. Okay, so being we have here a conclusion that we have an or command and only way to make an organ and false is to have both of its arguments false. Therefore P must be the same as Q. Which must be false. However, since P is false. Therefore the hypothesis false, you know, I repeat myself and therefore you can't make this this induction false. Therefore it's a dermatology. Okay, so here we have a nested uh implication. I'm not sure if I'm using the right word for that anyways Mhm. The only way to make the conclusion false is by making the sub conclusion false. So Q. Is false and making the hypothesis true. So P has to be true. However, that leads us to to a contradiction here because to make hypothesis true, he must be false. But we already require P to be true. Therefore we have the conviction for this not being anthropology number deep again we have this nested thing or I can make this false is to make Q true. That's right cue false and P true just like before and the left hand side is an and symbol which means both of these have to be true but one of them is already false. Therefore like policies must be false. So therefore this is a tautology. Uh again we have this the whole thing except the inner the sub implication implication. It's in the hypothesis this time. So to make a conclusion false, P must be false. Uh for the hypothesis to be true inside inside the negation must be false. And the only way to make this false is to have Q equal false and P equal true. But here again we have a contradiction so therefore we cannot make a statement false. Okay, continuing this pattern here we have H. N. This tells us that Q. Yeah must be true to make the conclusion false. All right. And the hypothesis can only be true if the inside of the allegation is false and to make the inside indication false, Q. Must be false, and P must be true. However, again here we have a contradiction Alright to reiterate what we were doing is we're using the nature of this operator here implication that tells us that the only way to make this false, the statement false by having my hypothesis true, the conclusion false, and it literally isn't possible for any of these. And if you can't make a statement false with any assignment of the variables, it is tight tautology. Yeah.

Okay, so we want to show that each of these conditional statement is a pathology. That is, that it's always true, regardless of what the truth values are for each of its variable. And we want to use the fact that a conditional statement is false only when the entity that is true but conclusion this fall. Therefore, we want to show that if the student is true, then there's no way for the conclusion to be falls. So for a we have not P and he or Q as our antecedent. And we want this to be true. Well, if we want not Pete to be true, then it must be that p this fall soapy p this fall. And if P is falls, thank you must be true. Then you must be true and cue. It's also our conclusion, which is true, therefore, a pathology now for being we have. If peas and Q and excuse and arms and P then then if P then are well, this whole thing is our entre student, and we want it to be true. So we want we want p p, then Q and A. If Cuban are to be true well both since both of them are conditional statements. They are always true, except for when the activity that is true and the conclusion is full. Well, let's begin with P p. Thank you. We have that. If P is true, then Cube must be true by the conditional rule. And if Q is true, then our must be true. Similarly, because of this on DDE, it's p s true and are true. Then we have that The statement is p even our is true and this is our conclusion. So we have that he then our is true. So this is true now if he is false, If Pius falls and we have two options either Q is true. O. R. Q is full. Well, if Q is true, then our must be true because accused the antecedent of Cuban are So then our is true then Pete. Then we have that The conclusion is P then our is always true. Therefore, this whole statement is always true. If the conclusion is true, then it doesn't really matter what the answer is. He didn't. Now it's cute fall we have that we have now two options. Our fault O r are is true. Well, if our soul, then if our souls then if p even our it's true both p and R r fault because we say that we're here, that he was whole and if our therefore the conclusion is true. So this is a tautology because it doesn't really matter what the antecedent is anymore. And if our is true, then if people are it's true. Since the entity and P is false and the conclusion are sure there for this is still a cytology. Now for three, we have that we have our antecedent. Is he on def even Q. Thank you. Our decedent Is this Well, we need Pete to be true for the entity that to be true And if he is true Thank you must also be true for the for the decedent to be true So we have that you must be true Therefore But we also have the queue with our conclusion and we know that q is true Therefore this is true and this is a pathology now for d We want peor que to be true And if people are to be true and if Cuban are to be true. So we want p or Q to be true. And if he is true, we have two options. Either P s true or Q is true or both are true. Well, if P s true, then we have that are is true because of this conditional right here. If Pienaar and we want this whole thing to be true that we want this to be true, then our must be true. But then right here are our conclusion and it's true. Therefore, this whole thing is true. Now if Q it's true, then we have that similarly our must be true because we have this conditional statement right here which is this Cuban are and we have that the embassy and this is true then the conclusion must also be true for this whole thing to be true. So then we have that are true. So yeah, and our is our conclusion again. So we have that are true. Therefore, regardless of what the values of the variables are, we have that are iss always true. Therefore, this is a tautology


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