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QUESTION 6An upper Doundset S mustbe greater than each element cfsgreater thanor equa tc eacn ement of 5be a rational numoerbe lesstnan or equaltc any otner Upper ...

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QUESTION 6An upper Doundset S mustbe greater than each element cfsgreater thanor equa tc eacn ement of 5be a rational numoerbe lesstnan or equaltc any otner Upper bound on $

QUESTION 6 An upper Dound set S must be greater than each element cfs greater thanor equa tc eacn ement of 5 be a rational numoer be lesstnan or equaltc any otner Upper bound on $



Answers

Show by example (a) that the least upper bound of a set of rational numbers need not be rational. (b) that the least upper bound of a set of irrational numbers need not be irrational.

This question. We are given a house, a diagram off this pasha ordering, and we are asked the following questions about this graph. So let's go with them one by one. First, is that a maximal element? There are, and two of them, In fact, we have l and M. So these are just element that has nothing above it. So in described, we have him in l minimal element. So conversely, in a man that has nothing below it, so it will be there will be a B c. All right. Greatest element, no least element non. These are different from Massimo and minimal in that. By being a greatest, they must be compatible with everyone, at least as well. So greatest Ellis Element must be compatible with any anything in a positive. But we don't have such element. Forks Sambo. MM is not compatible with out, right? And why so what's up? So even though am dominate everything else it can I don't mean it out. And so on. Next up about off this set ABC. So ABC is here up about years, in fact, set off. L can him so. So each one of them is considered upper bow because they are compatible with on tree of them. You have to track the line a bit, but you will see that that's the case. And the least among these will be the least up about, which is okay. All right. Last to question. Lower bowel off if G and ish. So let's see. This is F. This is the This is ish. You see that there's no lower bow for these three together because there's no element that ah compatible with all three of them and are lowered and him, right, Evan Demon Help see Ishmael Health be. But but they are not compatible. So we have none. And since there is no low about, that can be no greatest lower bow. So no as well. And here you have it. We have answer or the questions. Thank you.

This question we are asked to deter my wish of the following Posit is well ordered. So where are other said is such that any non Indy subset has the least element, which in this case mean there's no other element that's not itself. Wish it's a predecessor of it, so it relates to that element in front. So I should say that the path why X is in the relation right being a predecessor me that it related to eggs on the left, right? This and it is less element. There's nothing else before it. That is the meaning of it. Let's go with this one by one. So for those obvious one, I've you just skip. So this is were ordered. Why you should know that said, Plus the positive vintages and the usual less than or equal to this is where order already. And this is just a subset off that is a positive in major, but start at 10 to 8 House they were ordered probably like otherwise it will break the property off. Set one. It is It's not where order okay, and so you can try to prove it. The sketch off the proof would be something like you. Suppose is not Where are you? When you suppose that then they must exist myself said, calling a subset of ISS such that that is no least element. Right? And you go on from there and you're gonna come to ah, contradiction soon. And I relive it that bad. Um, is not too high because we know that s itself has the least element. Captain, you can go be load out. Okay. Next. This one is No, This one is not. We're order. Why? Well, the nation off. Rational numbers make it so that if he takes upset A as the whole thing, excluding zero. So we take zero out, but leave everything else intact. This thing they said has no least element. Why will foot if he If he supposed that is at least eight women, then it must be awful for me. It will be right with a and B both not not equal zero because because we take zero out can be zero. Now, when we have this, you can You can see that that exists. It will be plus one wishes strictly less than it will be. So is not it will be, but is less than it will be. And so it why always the Assam Shanda, that it will be easily element. And so by contradict Chinese, it means that we can help these elements. Okay, so next one, this is it said all positive, rational with denominator not exceeding three. Okay, now this is It looks similar to the previous case, but it is We're order. Why? Let's just say that First Leader said yes. It's so stoddard somewhere started one over a tree and and goes on. And one of our tree is at least element off the whole, the whole set. And any subset we have less element because we have this. What should I call you? This? We have this discreet nous. Because of these, we construct any element with restaurant there. No minute there. So let me explain this. When we have when we have any element off original number, it will be the way to decrease to create another. Another element. See, Andy, that is less than the 1st 1 We can do two things we can either reduce. Hey, all we can increase be or both, right? We can do one of them or both of them. And one over a tree here is clearly delis. Element off s because we can increase three. We're law. We can decrease or reduced one further because we can only positive we can go to zero. And with this in the same way any any subset day off is we can prove that it has less element by again supports. It doesn't have so support is not where order support. It doesn't have these elements. We're gonna have some some element in didn't set a and we had keep reducing it right. But we can we can reduce it indefinitely. We're gonna hit and in point at some point. And that's gonna lead to contradiction. That's the structure off the proof like differs First cases. And so I will leave it at that. It is. We'll order this. Well, gonna get too long. This okay? This one is a bit unusual in the sense that you rally. See, this greater than or equal to. This is essentially re worse off off the less dinner or equal to. So when we say please element. Now when we sell, this element is actually translate to great this element in our I should that get there, You know, uh, you sure sin? Oh, I should say, according to the less than or equal to. So the least element here would mean the greatest element in straight instead. And the said off negative integer yah How great That's element, which is minus one right, which act as a least element under this under this relations and in the same way that we prove are the so is essentially re worse off everything off these cases that we have positive int ages and we have less than or equal to We re word both sigh and and relations off the set. And I do just convince you that OK, it is Where are there any? Any substance, huh? Negative vintages. Wu Hao. The greatest element. Because if it doesn't have we we can keep increasing, like the element inside us. It right in the usual sense. And when we keep increasing, it is gonna hit minus one. Eventually, you can't go further than minus one. And so we're gonna have a contradiction. All right? And that is it. So this in conclusion, this one is well ordered. Okay. Sorry for long video. It's a lot to explain. And that is it. Thank you.

In this video, we are arcs. Which of the following post set is We'll order. So where are other said is to set that any non empties upset often. Well, how least element which in this context mean at least in a cot, then to the relation off the post it right. It depends on what relation we're talking about, but in general it means element that has nothing preceding it. So that is, there's no said in a such that said these strictly less than this. All right, you say it said, is less than or equal to accident said must be X itself. Okay, so let's go over this one by one. First. This is we order why you could use the fact that in positive integer itself, he's well ordered on there this less than oh, you call to relation or you can prove it yourself. I'm gonna show one Prue that we can use, like four other cases as well. So I'm gonna do it once and in on the cases. You can use the same argument. Suppose is not so. Suppose it's not be order Then that access at least ones upset. We call it a such a depth. It has no least element, right? And but it is not empty. So? So that is ex India. We know that there is something in there and eggs can be. Listen, limn. And so what? What does that mean? You calling X one who start a sequins? This this implied it. That is another element. Call x two, which is different from X one in a such step x two Precede X one Right? I use less done because we know that they're not equal on dhe. And this can go on because it takes two again can be less elements that mean that is something called a tree less than it and is go on and on. And this sequins this sequence off in well is strictly decreasing, right? It's It's strictly decreasing by at least one, so x two is at least X one minus one. But it can be like much more nowhere we don't know, but this sequence defy like this will eventually. Hey, 10. Right. So this subset sail, uh, like proof that we assume exes who go on and on and TV hit 10. And when that happened, when that happened, we will have a contradiction. Why? Because 10. Now Axe asked the least element because in the entire pulls it There's nothing preceding 10 anymore, right when we hit it. When When we show that 10 is in this subset, we get a contradiction because 10 will act asked the least element for you. But we assume that it doesn't exist, all right. And so the acumen from the start that that we supposed that is not the order is wrong. So it is, in fact, were ordered. This is the proof by contradiction every other cases in this question that Israel order can be approving this way. So I will skip the others now because the video will be too long. Okay, Next. This one is not were ordered. Why consider this subset? So we just take the whole thing and exclude zero out of it. This has no least element. Why you, I suppose, by conservation again support that is Lisa. Women call it will be then a over B plus one, which is strictly less than it will be. We'll also be interstate right, because it will be here bought A and B is not Ciro because we take zero, then it will be plus one exists in a as well, and it's strictly less than it will be. So it's a contradiction. No matter what original we come up with, you're gonna be able to create another one that is less than that number. So it cannot have any leads element. So it's not the order next. So now is a rational again. But we have restriction on the nominee. It cannot exceed three Quiet. And is this a positive images? This one is actually we're under. Why consider this fact? That is like the entire thing has leased element won over three. Right? And this gonna goes on forever. So any any subset Theo is must has less element. Otherwise, we can do the following process. Soapy. Anything in a right is not empty exists. We can do one off. The two things either reduce Ah, by one. Ah, not by one Sorry, reduced a or increase be my one off the two things off to thing at once to create this is the only way to create See over the such that it is less than it will be strictly less than right from From the assumption. Like in the first cases, we assume that it has no least element. So the only way to create see and see what the would be to do one off the two thing or both, right. But this view who have limits, so do you know me? They can go up beyond tree, and the number above can go down below one. So it gonna stop at one over a tree, like in the first case. He's gonna if we keep doing this eventually we hit one of our tree and we'll break the argument so we can prove this with contradiction. And it is We're order. Okay, Next last one. So this is negative. Integer right. So is stop at minus one. But this one has a different relation. Relation here is greater than or equal to think of it as a re worse off the less than or equal to that, be usually encounter in this subject. What? What this means is that now lease element becomes or is equal into great this element in our usual sense. So I should put a blanket on this in the less than a week or two cents, right? We can think of it as, like the probably will become any subset. How? The greatest element according to how would you show less than or equal to? When we do that, we're going to see that. Oh, the entire city has the greatest element, which is one sorry, minus one. And so, any subset Eve, we assume that is not where order we're gonna keep increasing again, right? The the human body Thanks. I going to keep increasing, and eventually we're gonna hit. We're gonna hit minus one, and we can go on. So the sm, Shanda, this it has no where this element will be false, and that is how we prove it. So this is we'll order. All right? You don't really need to convert this to the greatest women, but you're gonna have to operate on a different kind of relation, and you're gonna be confused sometimes. So this is one way to do it. And that is it for this video. Sorry for the long one. Thank you.

For giving the hunt the instagram of definitely instagram. No zero comma one oh X plus one times X squared those two X squared the X. If you recall this substitution for definitely instagram states that you have a function of F O G of X. Time G problem will be equal to G problem A G F A. The lower and gov the awful in this. So you have the function F. Of you see you here we can tell you we can make our U. To be equal to X. Where for us to our and the eu these two works for us to yeah. Now the substitute our values. So we have to take out take out the one half. So you have the integral of zero Lower limit and one upper limit of to work plus two. Um X squared was two x over times two the X. This becomes one half you said to see you so you two the two becomes one third, 1/3 you to the third. So you have nine my pack. So to solve our definite integral We get nine half


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