if-c-in-mathbbr-and-s-is-a-set-define-c-sc-mathbfx-mathbfx-in-s-let-s-be-a-convex-set-and-suppose-c0

Question

If $c \in \mathbb{R}$ and $S$ is a set, define $c S=\{c \mathbf{x} : \mathbf{x} \in S\} .$ Let $S$
be a convex set and suppose $c>0$ and $d>0 .$ Prove that $c S+d S=(c+d) S .$

Chris Trentman · Accepted Answer

so given a conduct set and to positive numbers, C and D were asked to show that CS plus D s going to be equal to C plus D s. So if we in other words, stretch the elements of a convex set by sea and stretch the elements of a complex set ID and then add thes two new sets together we get a set, which is three elements of s stretched by C plus D. So we&#x27;ll let s be convex set. Let&#x27;s see, be greater than zero and d be greater than zero. Now let X be an element of CS plus ts and we confined S one and s two in s such that X is equal to see X one are sorry CS one plus D s to then it follows. The X is also equal to C plus d times. See over C plus d. We can divide by C plus Decency plus D is greater than zero since both c and D or greater than zero s one plus de over C plus D s to now because S is a convex set. It follows that see, oversee plus D s one plus D over C plus D. As to linear combination of elements from s also lies an s. So it follows that X is an element of C plus D s. So we&#x27;ve proven one direction. That is, that C s plus D s is a subset of C plus D s. Now to prove the other direction Similar exercise. So we&#x27;re going toe Let X be in the set C plus D s. This means we confined. A vector s in the set s such that X is equal to C plus D. Yes, and of course, this is equal to CS plus D s. After distributing the scaler, we could say, even distributing the vector across the scale Er&#x27;s And this is clearly an element of C S plus D s. So X is an element of CS plus ts and therefore it follows that C plus D S is a subset of CS plus D s. Combining this statement and the previous statement we get that CS plus D s equals C plus D s, which is what we wanted to prove

Chris Trentman · Answer

rest to prove that a point in a context set is an extreme point of the set if and only if the set minus that point is still convex. So we&#x27;re going to let the be an element of s where s is Convex set. And now let&#x27;s prove the yeah, it direction first. So suppose that V is an extreme point of this And let t b the set of all vectors X in s such that X is not equal to our extreme vector. The now if x and why are in t them it follows that the line segment by X y is an s. Since Esa&#x27;s convicts now the is an extreme point of s so follows the V does not lie on the line segment y z. If it did, it would have to be one of the endpoints x or y, which it&#x27;s not not lying line saying the next why and therefore falls X Y is a proper subset of tea. So it follows that t is convex is what we wanted to prove now to prove the other direction Who&#x27;s that he is not in extreme points were actually going to prove the contra positive, then this means that we confined two vectors X and Wyness such that the lies in line segment X, Y and V is not equal to X and V is not equal to why has to find the interior of the line segment. So it follows that from the previous paragraph x and Y R N t. But the line segment X Y is not a subset of tea because V allies on the line segment India&#x27;s not in T so follows. T is not convicts. We have proven what we wanted to prove.

Chris Trentman · Answer

were asked to prove that the sum of conduct sets is again a convex set. So let A and B he convict&#x27;s now let the vectors x and y B in the set some a plus B. This means that we confined in a in C in A and A B and a D in the set B Such that X is equal to a plus B and why is equal to C plus D. Now let t lie between zero and one Them we have that the point on the line segment x y It&#x27;s going to have the form W, which is tee times X plus one minus t times why you could do the other way around. This is the same as tee times a plus B plus one minus t times C plus d. Of course, this is equal to grouping together vectors from the same set. So we have t a plus one minus t see plus t B plus one minus t the now because a and beer convex it follows that ta plus one minus t See, this is an element of the line segment between A and C. This lies in a and we have that T B plus one minus TD. Since B is convicts and this is an element of the line segment between B and D. This also lies in B, so it follows that w lies in a plus. B. Therefore, it follows that a plus B is convicts, since it contains any points online segment between and between X and Y.

Gideon Idumah · Answer

so two shoulders the sides s which is a gold toe x In our end such that f of X is in C. To show that this set is a convict sense convicts subsets off our aim would peak sue points See, let&#x27;s call them X one x two in this under. We sure that&#x27;s the line segments the line segments between Damn, Isn&#x27;t this no leads X one be in this By definition, it&#x27;s implies deaths f off X one isn&#x27;t seen Now this implies dunce f x one de quarto. Why one for why one in c all rights? Because we say f excellent isn&#x27;t seen. So I miss fo x one is altering Why one for which why one has got to see this would imply that X one accord to f involve us off what I want. You just take the inverse of what side? Similarly, let x to be element of Bess. This implies that f of x two by definition, beaten See, which implies that f of x two. Because why too for a white soon in C. This implies that X Toole accord to f involve us off. Why? I want so to show that the line segments between them it&#x27;s between s we must show We must show That&#x27;s one minus C x one lost The ex too must be in s for every zero lesson opposed to see less than a course one. So let&#x27;s show these now off one minus c x one lost Seen X to disappear one Miners seem our x one is f ive us of y one loss seem f universe off white too. This is nothing Bad&#x27;s when you multiply it. Woman listed by these are FBI of us off Why Juan minus C f f s of y one lost seen f inverse off. Why, too, which of course toe f any of us or why one miners? Because f is Nina f impossible genius So I can pull this inside. We&#x27;ll see why one loss as one of us off. See why, Sue? So this is just equal to f ive us because this sort of putting things f is linear. Ft of us is also Minya. So that&#x27;s why I can do all these operations. I can pull out every bus I have what I want minus tse. Why one lost see Why soon? So this is recalled to f Reverse off one minute. Just see why Want plus c? Why soon Now, just pull these down. What obvious? That things is a board to bees? No, Because of these, you call that See his converts. So let&#x27;s go back. That&#x27;s so since C is convicts the points one miners see why one lost See white soon Most be unseen all rights Because this is the convicts combination. Because why one on White? You isn&#x27;t seen now from this three questions, Tom and Nicole is a question star. No, from star. What would our duds one minus c? Why? Want lost? See White? So I&#x27;m moving this f in verse to this side. So this is equal to f off this. So that&#x27;s is this will be equal soon. F off one minus c. That&#x27;s one lost X zoom. Now we said this isn&#x27;t seen, so this most b c so we&#x27;ll show that f off one minus c x one Must be in C by definition. Oh s If this isn&#x27;t see by definition of s, this would imply that one minus X one lost. See X two Must be s on. This implies that this is a conference

Angelo Rendina · Answer

So we&#x27;re told that the sets A, B and C for such that the carnality away is lower, equaled in the car originality off B and the cartoon ality of bees lower equal in the car the night of C. So here, karate of a lower than chronology of being means that there exist, um, a map f that goes from a to be, which is injected. And similarly there is a map G because from B to C, which is injected. But now we can consider is the map well age, which goes from a into sea and is defined as hey goes into B via ehf and then be causing to see Reggie and so h oven element X. He&#x27;s defined as G off f of X, but now H is injected because it&#x27;s the composition off to inject it functions. Which means that well, we have an injection from a to C. Their means that the cardinal it your way is lowered equal them the card inanity off, see, as we wanted

Gideon Idumah · Answer

so we have some show of force. Question. We&#x27;ll see that the A part is true because this is a symptom of US definition given on pains. 4 57 The be pot is also true. This is you can see Sarah AIDS on page 4 50 seats. The Sea Potts is also true because this is exactly this sister insane. That&#x27;s his car, it&#x27;s your dog returns.

Problem

Find an example to show that the convexity of $S$…

Problem 19 Medium Difficulty

If $c \in \mathbb{R}$ and $S$ is a set, define $c S=\{c \mathbf{x} : \mathbf{x} \in S\} .$ Let $S$
be a convex set and suppose $c>0$ and $d>0 .$ Prove that $c S+d S=(c+d) S .$

Answer

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