give-an-example-of-a-closed-subset-s-of-mathbbr2-such-that-conv-s-is-not-closed

Name: give an example of a closed subset s of mathbbr2 such that conv s is not closed
Uploaded: 2020-09-11
Duration: 1 min 54 s
Channel: Numerade
Description: Give an example of a closed subset $S$ of $\mathbb{R}^{2}$ such that conv $S$ is not closed.

Question

Give an example of a closed subset $S$ of $\mathbb{R}^{2}$ such that conv $S$ is not closed.

Chris Trentman · Accepted Answer

to find an example of a closed convex set in our two such that its profile is not empty. But the convex hole of the profile is not the same as the set. So perhaps the easiest example I can think of such a closed conduct set s is going to be a ray in our two. So we have an end point and then from that a half line, well, question is what is the profile of this set profile? P only contains the end point. So the point where the ray originates from so it follows that P is not empty. But convex hole of P is going to be equal to P since P contains only single endpoint, which, as we know, is not the same as s.

Chris Trentman · Answer

were asked to find an example of a bounded conduct set s in our to such that the profile of set is non empty. But the convex hole of the profile is not the same as the set. So this is different from a previous problem in the previous problem. Exercise 10. We&#x27;re looking for an example of a compact contact set with this property, but now we&#x27;re looking for a bounded convex set. So an easy example is to consider a square. We&#x27;re only two of the sides are included in the set, So we&#x27;re going to let s be a square with just two adjacent edges. In other words, it&#x27;s going to look something like this. The dotted lines indicate that this boundary is not included. Now we see that the context hole Well, Katie profile is non empty because mm profile P includes the three. Yeah. Mm hmm. Corners shall guess I&#x27;ll put in quotes because technically, two of them aren&#x27;t really corners. But the three corners of this square, So the profile looks something like this is the only extreme points. So you have these three points for the profile, and we have that the convex hole of P is a triangle, so forming the convex hole which are drawing green, you simply draw a triangle between these three points and it&#x27;s clear at the convicts. Whole of P is not the same as the set s.

Jonathan Turovsky · Answer

right. So in this question, were given a false statement. So, uh, we&#x27;re getting the same in that next plus y squared is X squared. Uh, plus one. That&#x27;s all missing fools. Remember, you&#x27;re staples be square. That equals a squared plus two a being. So if we were to give a counter example to say that it is indeed false, you can see that X equals to present Why eagles? For so we could say, If we do follow that of false equation, you can see two plus four squared of us to square was forced. You know, if we solve the left side, that gives us six square equals four plus four squared, which is 16 and six squared is 36 had four plus 16 is 20. We know that it&#x27;s not true, so

Gideon Idumah · Answer

not to prove these. We know that into sexual being will always be a substance off a right because you&#x27;re intersections. Something would be it&#x27;s always you&#x27;re intercepting a woods be. It&#x27;s always contended day. So this implies that from exercise eight scene we have That&#x27;s the convex all off a intersection. B must be a subset of the comics all off, eh? Similarly, we know that intersection being will be a subset of Be so from exercise 18. Just doing the same thing over game we have that the convicts all off a intersection b will be, is so set off the convents all off B. Now look at these three questions. The left answer on something Sam is because this is Indy&#x27;s Andi. This is also indeed so. This implies that the convicts all off a intersection, being be a subset or contained in the convicts. All off a intersection. The convents all off. You can just put this now for the deep parts. Wanna showed us equality, Need notes. Oh, so let&#x27;s give an example. So take or consider four line segments warn out. Sue out 34 dots. Mick is square, so want to use an example of his square? No. Um, let&#x27;s A which is a cold and one comma l three on b which is out. So come on. L four be sets dogs, countries, a poor city lines. But that&#x27;s what means something like these. So this is everyone out soon off for So you have everyone l three outs for the opposite lines. This would imply that the convicts all off a intercision the record that this is a So this is a gun, then this is B. Now the contracts all off a intersection being would just be the convicts, all because it into sexual being. He&#x27;s just Did I go now on this side of the square off the square? I discussed this point this point this point This points So the convicts all off eight ounces of people be the convicts all off the boxes. That&#x27;s is this four points. This four points into Section B. Nazis. So Dobby, the convicts all off a into section will be that sees. Did the argot knows on sides off the square off the square. That is a dog going on the side of the square. No, we&#x27;re us. The convicts hold off into section convicts. All off being is the old square. So this is that all square where us D&#x27;s is decides? No, just the bodice. Is that say This side&#x27;s I&#x27;m Doug going out. So the dog owners on decides where us This one is the old square us old. So we see that equality doesn&#x27;t recorded, See doesn&#x27;t hold one is the old squared whereas one is the convicts all off the vortices

Abigail Martyr · Answer

in this problem were given a set. We&#x27;re a vehicle. Do a B C. The Ohio contains the elements A, B and C, and we need to come up with an example of a relation Are such that this metric Luger&#x27;s of the reflective closure of the transitive closure of our, um that is not transitive. So keep in mind, um there are mortar. One possible answers to this question. So what I&#x27;m doing here is just one possible you may be doing and get a difference. And also be correct. So, um, I will select value are on the relation arm going, said that equal to what are equal, the element be okay. And the reason why I chose something of that form because at the end, we know we do not want it to be transitive. So when we did the transitive closure, what we find is that it is equal into our itself. So here in the stones thing be a me a. Okay, now we&#x27;re going to do the reflexive closure. Oh, our transitive closure and what that means. Bassem, listen, what we know about being reflective then for each element of the said a we should be a a BB and sits on so forth so well to reflect the closure bits. Then I would have to end the point. Yeah, I would have to also add BB and now it also have to add So this would be to reflects the closure of the transitive closure of our Now we need to do the symmetric closure from the previous of the previous. Yeah, and based on the definition or symmetric, what that means is in the previous step everywhere that I have, uh let&#x27;s see, because I have a b a Then that would be no. It also have to have a baby. And here because I have see a that implied that I should also have a c still our symmetric Cordier. Some of the previous thing I would give us the discovery and coordinate a in a b Chasteen be in being being three a and the don&#x27;t know that with complete engine the same metric closure of the reflective closure of the transitive closure. Let&#x27;s test to see if this is transiting. They&#x27;re not here. We have a warrant B A and B c Do we have on athe Sorry. Yeah, So we have B A and hey, Speedy. That should imply, based on our definition here, that should imply that we also have the point. Meet the okay. And if we examine, I will last that there is no point. You see? So this would imply that disclosure is not forensically. I don&#x27;t think the relations are that I started out with the sudden spider conditions setting the problem. No. Here at he had being okay.

Problem

Give an example of a compact set $A$ and a closed…

Problem 27 Easy Difficulty

Give an example of a closed subset $S$ of $\mathbb{R}^{2}$ such that conv $S$ is not closed.

Answer

Then $S$ is closed, and conv $S$ is open, because conv $S$ is the upper half-plane.

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David C. Lay, Steven R. Lay, Judi J. McDonald

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Then $S$ is closed, and conv $S$ is open, because conv $S$ is the upper half-plane.

Linear Algebra and Its Applications

Catherine R.

Caleb E.

Michael J.

Joseph L.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Catherine R.

Caleb E.

Michael J.

Joseph L.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications