Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

Q: Determine whether the set in P_2 is linearly independent.    S={x^2 , +1, 2 - 1, 2}

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Q: Show that if A is such that m (A)> 0, then there exists a bounded subset BCA such that m'(B) > 0.

A: By properties of Measurable set,         If then  .

Q: (a) Define the T1-space.

A: Define the T1 - space

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

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Q: Consider the collection of all points of the form – 2w] 3w w Express this collection as the span of…

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Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

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Q: Assume that A and B are nonempty, bounded above, and satisfy B CACR. Show that sup B < sup A.

A: Introduction: The supremum and infimum properties are the very important properties of any subset of…

Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

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Q: be a closed subset of space Xa

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Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

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Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: Show that if A is closed in Y and Y is closed in X, then A is closed in X.

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Q: Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a…

A: We start the proof by assuming there are no such maximum element in C.

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

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Q: Let A be any set with the discrete topology. Show that A is normal.

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Q: Prove that intersection of any class of convex subsets of N is convex?

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Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

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Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

Q: Give an example of a compact set A and a closed set B in R2 such that (conv A) N (conv B) = Ø but A…

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Q: If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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Q: Exercise 6.65. Prove that if E₁ CR and E₂ C Rm are convex, then E = E₁ × E2 is a convex subset of…

A: Given :  E1⊂ℝn and E2⊂ℝm be any two convex sets. To show :  E = E1×E2 is a convex subset of ℝn+m…

Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: Each nonempty can be represented as the convex hull of its extreme points. --------

A: Each of the nonempty compact convex set can be represented as the convex hull of its extreme point.

Q: Find an example to show that the convexity of S is necessary in Exercise 19. Reference Exercise 19:…

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Q: Is p continuous? Is p an open map?

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Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: If there exists open ball B (a, r) such that B (a, r) nG=Ø then a is not limit point of G

A: To check whether there exists open ball B(a, r) such that B(a, r) ∩ G= ∅, then whether a is not…

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

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Q: Find the transitive closure of R if M. is

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Q: Find the transitive closure of R if MR is

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Q: Suppose the collection = {B₁ (x): xe (0,1)}. Is B an open cover of the interval [0, 1) CR? Give an…

A: Given: The collection B=B12x : x∈[0, 1). To determine: Whether B is an open covering of the interval…

Q: Which characteristic/s implies(y) that a set S of real numbers is closed? S has no accumulation…

A: To choose correct option from given question.

Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least…

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Q: (ii) Find a connected subset of R² which is not convex.

A: Note: Hi! Thank you for your question. As per the guideline we can solve one question at a time in…

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: The set {V₁, V₂, V3} is independent. Use the Gram-Schmidt process to produce an orthogo set with the…

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Q: Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…

A: X and Y are two discrete spaces then X and Y   The topology is  discrete then for x∈X  is an open…

Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: Let A be dense subset of (X, t)

A: Given: Let A be dense subset of x,Τ and let B be non empty open subset of X. To find: Fill in the…

Q: Let the non-empty set S = {x - yv3:x,y ER}, show that S is closed'under multiplication then find the…

A: We have to show that the given set S=x-y3:x,y∈R is closed under multiplication and also find the…

Q: Show that both ℝ and ℝ2 may be covered by countably many open balls

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Q: Sketch the graph of the convex hull of S. In R, let S =. :0sx<3

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Q: Let A be a subset of R and let c€ R. Explain what it means to say that c is a cluster point of A.

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Q: Let D be the closed region in

A: Given,                    x2-2xy+y2+x+y=0   and      x+y+2=0