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

Q: (a) Define the T1-space.

A: Define the T1 - space

Q: be a closed subset of space Xa

A:

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: Prave that every Ramadule is a homomarphic image of free R-madule.

A:

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

A:

Q: Prove that every well order is a linear order,

A: A set  is said to be well-ordered if it has a total order and has the property that every non-empty…

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…

A:

Q: Find the set of ordered pairs (T0, Yo) for which Picard's Theorem guarantees that the IVP y ry %3D y…

A: Given,            y'=xyx+y, y(x0)=y0

Q: Show that R is not compact.

A:

Q: Show whether the set {(2, –1, 3), (0, 3, 1), (- 1,1, -2)} in the space R³ is linearly independent.

A: Definition. Let V be a vector space. Vectors v1, v2, . . . ,vk ∈ V are  called linearly dependent if…

Q: Given a set L sR", Following are true statements regarding conv L(convex hull of L):

A: In this question, we have to choose the correct option for the convex hull. the set L⊂Rn By using…

Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

A:

Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: Suppose H is an inner |(x,y)| ≤ 11x11-1111 product space. Show Ужуєн.

A: Given: H is an inner product space. To prove: x,y≤x·y, ∀x,y∈H

Q: Find oxthogonal uncdary diagonalizing an or matri'y to lMowing tor One 2 (a) (b) 3 -2 -2

A: A square matrix A is called as orthogonal matrix if AxAT=ATxA=I where AT transpose of matrix A. and…

Q: Generate the valid inequalities needed to describe the convex hull of the following sets S. • S:=…

A:

Q: Generate the valid inequalities needed to describe the convex hull of the following sets  S = {x,y}…

A: The set S is S=x,y:x+y>b,x>0 integer,y>0. Let a=x1,y1∈S and b=x2,y2∈S. Since a=x1,y1∈S,…

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: Sketch a model atleast 2 axioms of Pappus's Geometry and identify those axioms

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Q: Prove that the convex hull of a bounded set is bounded.

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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: Find the null space for A. 1 2 -2 1 0 1 3 A = null(A) = span

A:

Q: Determine whether or not T = (6,-3) belongs to the null space of A' if A 1

A: the answer is

Q: Find the null space for A. 1 0 -2 A = 0 1 null(A) = span

A:

Q: Let ā = (- 4, - 5) and b (- 5, 0). Find the projection of b onto ä. Round your answer to at least 3…

A:

Q: Show that a subset X of a matroid is a hyperplane if and only if X is a maximal non-spanning set.

A: As you question specifies only to show X is a non-spanning set and there is no other data provided ,…

Q: If A and B are nonempty bounded subsets of R with A C B, prove that sup A < sup B.

A:

Q: Let à = (1, 1, 4) and b = (1, 2, - 5). of b onto a. - Find the projection proj- 6 = b =< -

A: For the given vectors the projection of b on a is given by

Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

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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: Sketch the graph of the convex hull of S. In R, S is the set of points where y =x and x 20 A. В. D.…

A: The Convex hull or convex envelope or convex closure of a shape is the smallest convex set that…

Q: Show that and are not strictly convex.

A: To show that l1 and l∞ are not strictly convex.

Q: Suppose U and v are conformally equivalent. Prove that if U is simply connected, then so is V.…

A: Given that U and V are conformally equivalent that is there exist a mapping f from U to V such that…

Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

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

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Q: Let R = {(0,1), (0,2), (1,1), (1,3), (2,2), (3,0)} Find R', the transitive closure of R

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Q: - Consider noncoplanar points A, B, C, and D. Using three points at a time (such as A, B, and C),…

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Q: Prave that every R-madule is a homomarphic image of free R-madule.

A:

Q: Describe in your own words the convex hull of a set of points in S in the plane. Show that the…

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Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Show That R is Banach Sface.

A: To Show -

Q: If A and B are distinct points in 3-space then |AB| = -|BA||

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Q: Generate the valid inequalities needed to describe the convex hull of the following sets S. • S:=…

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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: 3. Find a minimally invariant set in (C, R) that contains the circle [2 -(1+21)| = 4.

A: It is given that z-1+2i=4. Note that, z=x+iy. Substitute z=x+iy in z-1+2i=4:…

Q: Show that X is Hausdorff if and only the diagonal A = {(x,x) | x € X} is closed in X x X.

A:

Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.