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Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.")
a) Some old dogs can learn new tricks.
b) No rabbit knows calculus.
c) Every bird can fly.
d) There is no dog that can talk.
e) There is no one in this class who knows French and Russian.

Okay, So before we have that all dogs Nazis. Okay, if we're supposed to rewrite this, let's flip you, Becks. Mean excess, please. And then we can rewrite this. As for ox, he have explodes. So for all dogs, uh, dogs have please gate. Now, let's not get this. We negate this. We get that in this end, are for all Genji's there existed next negation of p of X. And this means that exist some X to the existing dogs that shuts that dog does not have. Please. Now, for part B, we have that there is a horse that can add. Okay, no, let's let p of x mean That ex came ad and we said there is a doctor, Of course, that can add. So there is Is there exists an ex such that p of x old. If you negate this, what do we get if we pull in our nation or sickness becomes for all the negation of p of X holds, this means that for all exes, which is for all horses, the negation of P of X code. So for our horses, horses cannot add hit enough. See, we have That's every cool looking climb. It's like p of x b X kim climb Okay, so we can put well, the statement says every call, affection climb. So we have We're all ex p of x sold the bigness, yes, that exist some exits that the negation of p of episodes so exists There is a quality that's cannot claim his innovation party. We have that no more speaking French. So let's let p of x be can't be French. Excellent. Speaks French. So for all exes, for all monkeys. Uh, you can speak French. You cannot know monkeys can speak French. The negation of cancer crunches. Uh, all monkeys cannot speak French. And Alexa Gate this name it yet that there exists a mex. It's that p of explodes suitor exists. Some monkeys upset. It can speak French. Now we have exist. Pig second swim and catch fish. So there exists, um ex such that can swim and it can catch fish so it's but catch fishes. See you, Becks. And as of xB against one hoping to get this we get for all ex segregation of SFX and and comes or an indication of cfx. So this means that for all Thank you. Um, we either cannot swim or we cannot catch fish

Hello. Welcome to this lesson in this lesson. US person. The that the the largest, as negations of themselves. So here we have for some Z for all y for all X for the project. Katie of X, Y and Z. Yes, President. US delegation. So here. Still we will not touch the domain. All of them would be in the same domain by here. The statement or the project. Kate would become negative of war. He used to be okay. All right, so And if the negation follows the Project eight Okay, so All right, that's the negation. Oh, the next one is local of this for for certain X for certain y for the project p of excellent y and for certain for all x for all y For certain educate care of x and y so here because he had and and we want to negate it would negate board project Kate so that the and would become negative of what used to be. Okay. So, for example, if we're looking for the height, that is a plus X plus y and X plus two y now would be having the opposite of both of them individually. Okay, so here becomes for some X for some y of the project eight p of x and y negation. We negate the per educate and for us for all y we negate the per educates. Mm. Okay, so that is just for B. Let's go to the sea. Bad. Okay, so for all for some X for some. Why? For the project, Kate. Tale of x and y, by implication do in the gates. Mhm. They kill. Okay, So that the K would now by implicate the negative of itself. Okay, He used to By implicating yourself. Now, by implicates, they are, by implication of their negatives. Let me put it that way. Okay? Yeah. So the by implication of the negatives this by implicates it's negative. All right, so it means it has been negated. Let's go to the final one. Okay. Where we have the all so for why? For some X for some Z, both the the 90 gets the pre Lucchetti of X, y and Z or 90 gets the care for X and y Okay, so this is the negation of that. Okay, so thanks your time. That's the end of the lesson.

Conditions for a We have the negation of their existence. Why? Just accept P. Oh, come on. Why? We're asked to rewrite of these statements that allegations appear only within etiquette. Okay, let's pull in a nation that for all wine girl creation of P of X sound alike. Fuck me. We have indication of right this the Y p of X. Come on. Like that for an indication, descend into the distance for all. Why? Delegation of P of it. Come away. Hey, we have biggest ordination of There's just some wine substance, you of wine and for all ex delegation of P of X, come away. Okay. Could be pulling the nation. We get that existence and for all why navigation of Q of one And if we get the gate or end turns into an or an investigation of for all becomes just tonight and imitation of the nation of P of X y. Eyes peeled, Come away side are Hey, not for okay. We have irrigation of the wine for why? Or or like a warning. Okay, so let's get this fortification. All right. This becomes for all why negation of are in this company. Oh, negation of far. But why are or becomes an end? This tracing this and the negation of us collect. Come on. Right now for E. We have the negation of E. Why, right. Is this the such a key? External? Why coming, Pete? Or they're on the you have X cover. Why? Completely? He pulled his interrogation yet existed in for two for all. Why? Just turned into there. Thanks. This statement for all this is the negation now negation of t X color wire coming or or becomes an end This treason before on just preparing to exist something, uh, and then delegation of U. S delegation of you kind of white coming.

Hey, it's clear. So a new range. So for part eight, we're gonna express the negation of the statement. So we're given I'll just put the negation sign in front t of X. Why? I see we're gonna use the dim organs law for qualifiers three times and you end up getting existential, not tea of X y Z for part B. We're gonna use dum organs law. So we start off with thinning negation. Uh, this statement p of x y or universal q of X y used to Morgan's law to simplify X Y and not universal q X y. And we used to Morgan slow for qualifiers. I should put the equivalent simple Universal not he of X y and Texas 10 Show Universal not of Q. Next. Why for part c. We're doing the negation of the given statement p of x y and existential. Next Waas. See, we're gonna use the Morgans law for qualifiers and we got not p of X, my and existential X y Z, and we used to Morgan's law or not exist on show of X. Why see and finally muse to Morgan's law for qualifiers, not P of X y or universal? No, our of Why, that's why I see then for party. We need the negation of the statement p of x lie, then Q of X y we're gonna use to Morgan's law for qualifiers, then Q of X y finally muse logical equivalents. This is a universal symbol p of X y and not Q. Of X Y.