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Task 4. For each of the statements (a), (b), and (c) below (a) If the teacher is absent; then some students do not complete their homework: (6) All the students com...

Question

Task 4. For each of the statements (a), (b), and (c) below (a) If the teacher is absent; then some students do not complete their homework: (6) All the students completed their homework and the teacher is present: (c) Some of the students did not complete their homework or the teacher is absent: Do the following tasks 1) Introduce appropriate propositions to write statement as compound propositions 2) Write statement as compound proposition using logical operations 3) Negate the compound proposi

Task 4. For each of the statements (a), (b), and (c) below (a) If the teacher is absent; then some students do not complete their homework: (6) All the students completed their homework and the teacher is present: (c) Some of the students did not complete their homework or the teacher is absent: Do the following tasks 1) Introduce appropriate propositions to write statement as compound propositions 2) Write statement as compound proposition using logical operations 3) Negate the compound proposition 4) Write out the negated compound proposition in English



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.

Okay, so we wanna rip right all of these compound propositions into equivalent compound propositions that do not involve conditional. And we do want to do this as an example theory, which tells us that it's even. Q. This logically equivalent to not pee or Q. So for a we have that not being than not cute this logically equivalent to not not pee or not cute by example. Three. And that's equivalent to pee or not killed because we have a double negation. Herbie, we have he or kill Len, not see. And this is logical equivalent, not Pete more kill or not Pee, which by the Morgans is equivalent to not pain and not Q or not pee. And by absorption law, we get that this is actually just not be foresee. We have. If p even not killed, then if pee, then you, which is logically equivalent to Not if Pease in not q cooler, not peace. Thank you. My example. Syrian. Similarly, by some free we have that this is logically equivalent to not not pay or not. Q. Well, we're not not pee or Q, and this is equivalent to by the Morgans smocks and double negation right here. We have the news equivalent to pay and cue O r he or Q because it can be a communication, which is a quick, willing to P and Q or P more kill. And if we observe right here, we can use absorption law, which tells us that this is logically equivalent to he. Four q. And we have rewritten every compound proposition into a form that involves no conditional.

We're trying to make a true statement with the negation. Navigation is when we take the opposite of something. So let's start with 22 statements A and B. We know a is true. Now we'll try to make B also true. So we have here New York City is in Utah. The negation of B Would this be and I see is not in Utah. Well, obviously this is true. So now let's put this together to make an overall true statement. For example, if we use a conjunction the and form both must be true. So we're gonna combine a which is true it on on its own with not be so. Our new statement is Boston is in Massachusetts. Excellent. And New York City is not in Utah also. Good. So we have here a true statement with the negation.

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.


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