Let d(x, y) = be a metric on R* then the open ball of center 2 and radius is: O ]1/2,1[ O None of the choices O 14/9,4/7[ O [4/9,4/7[

Q: 1. Suppose that (X, dx) and (Y, dy) are metric spaces and f: X →Y is a function. For a, b e X,…

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Q: ,A, B,C be 4 points in R², no three of which colinear. Fix A' e VA, B' e VB and C' e VC h that AB ||…

A: In the following question, concept of parallel line is applied. Parallel Line Parallel lines are…

Q: Let d be metric on X, and de f ined di on X as di = max (d (r, ).0}, then dis semi metric but not…

A: Given d be an metric on X and d1 is defined on X by  d1=maxdx,y,0                     (i) To verify…

Q: 2. (a) Let ds² = v du² − du dv + 2 dv². (a) Find the largest subset U of R² for which this…

A: (a) Given metric:ds2=v du2−du dv+2 dv2Comparing g11=v , g12=g21=−12 , g22=2Given metric will be…

Q: Find d(A) in the context of R with the standard metric when i) A = (0, 2) %3D ii) A= the set of…

A: d denotes standard metric on R which is  d(x, y) = |x-y|  , for all x, y in R.  For any non empty…

Q: 3- In a complete metric space (X, d), If A C X such that, A = A then A is complete as a sub space of…

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Q: non-empty set and d:X x X R de fined by d(x,y)=31 x- 41_ 2+|xl+lyl or disprove that d is a metric on…

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Q: Let d be a metric on X + 0. Show that the function e defined by d(a, b) 1+ d(a, b) e(a, b) = %3D…

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Q: Evaluate (z* + 1)dx around the circle [z]=1 A) 0 B) 2in C2 Evaluate fx8 (2x - 1)dx A) 1 B) 0 C) -1…

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Q: Consider the following mappings on RXR: * d1(1, y) = e²=y, d2 = |2r - 2y). and d3 = V12r – yl. %3D…

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Q: Let X=R2 and let d be the Euclidean metric.  Define d3:  R2 x R2 to R by d3(x, y) = min{1, d(x, y)}…

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Q: Q. Let (X, B(R), µ) be a Borel measure space. If AnB=¢, BnC = 0, u(AnC) = 0. Determine µ(AUBUC).

A: Measure theory

Q: Q2: In usual topological space (R.T), answer the following: 4- Is [0,1] is arcwise in R? Why?

A: Given that ℝ, Tu is usual topological space. To show: 0,1 is arcwise in ℝ.

Q: Consider the following mappings on RXR: d1(x,y) = e*-Y, d2(x,y) = |2x –yl and d3(x,y) V=yl O Both d1…

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Q: Q3: (a) In the metric space (S, d), Vx,y ES. If A = where S = [0, 1]; d(x, y) = |x-yl; A = [0,2),…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

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Q: 2. (i) On R², consider the metric d(x, y) = 2x1-9₁ +4₂-9₂- In this metric, sketch the closed ball B₁…

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Q: let AABC be a triangle S². Let the dual point C'ES² of C be defined by the following three…

A: Given: △ABC is a triangle S2 (the two-dimensional sphere). The dual point C'∈S2 of C is described by…

Q: d set): Let (M ,d) be a (SUT)' = S' U T'.

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Q: I need help for number 5. There are some parts in this section that I don't understand. Please give…

A: Problem 5.To prove that the given closed ball is closed.

Q: Let (X, B(R), ft) be a Borel measure space. ANC= 6 BOC =0 p(AnB) = 0. Determine ju(AUBUC) %3D

A: Since sets are disjoint so measure will be the sum

Q: Let F = (z,2x, y) and C is the straight segment from (0,0,0, to (1,2,2). Then F• dř = %3D a) 7/2 b)…

A: Parametrize the line and then integration.

Q: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

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Q: d, d': xxx-R be two metrices on x that Satisfy following for every nEX and >0 7>0 st. Bai (x, S) C…

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Q: Determine whether d is defined on R. Is it metric on R? d(x, y) = 13x - yl Also: if metric, show…

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Q: Please Lgent let VizLi = L3) and w=L4o=2) belongs to spaln (rloV2} %3D termine whether w

A: yes this belongs to span of (V1,V2)

Q: Let F = (z, 2x, y) and C is the straight segment from (0,0,0) to (1,2,3). Then ,F dř = %3D %3D

A: Given,                                   F→(x, y, z)=z, 2x, yand C is the straight segment from (0,…

Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given : As per the question we are given the following bivariate function : x2 + xy + 3y2 = 3 And a…

Q: QI/ a) Suppose that f: [0, co) y is a metric space. Show that d(x, y) = fCo(x, y)) is a metric on X.…

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Q: 3. Let d₂(x, y) = |arctan(x) — arctan(y)| for x, y € R. (a) Show that d₂ is a metric on R. (b) Are…

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Q: P2(x, y) = (E | x; – Yi [°)} i=1 s metric forms a metric space on ( ?

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Q: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

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Q: 3. Find the closure and interior of the set A = {(x, y): xy = 1} as a subset of R? (equipped with…

A: Closure and interior

Q: Find the matrix A, relative to the standard basis {e1, e2, e3, e4}, of t

A: Given, W be the subspace of ℝ4 spanned by the vectors                                    1-11-1 and…

Q: Exercise. Prove that in any metric space (X, p), a closed ball {r E X : p(a, x) <r} is closed.

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Q: Let X = R and x, y E R. Admit for all x and y the metric d(x, y) = |arctan x arctan y|. | Compare…

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Q: Consider the following mappings on RXR: * d1(x, y) = e2z-y, d2 = |2x – 2y|, and d3 = VI2x – y|. %3D…

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Q: Let d be a trivial metric on a non-empty set X and d₁: XXX → R such that x = y d₁(x,y) = { 2 if x…

A: Given:d1:X×X→R such that  d1x, y=0if x=y2if x≠y∀x , y∈X To Show: d and d1 are equivalent.

Q: 5. Let X = R? and define d, : R? ×R² →R by d (x,,y,),(x,,y2)) = max {x, – x,|-\y, -y}} . 13 a)…

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Q: Task 6 (Sobolev spaces) Let pe (1, 00), f e W((a, b)). Show that af = 0 f= const a.e. in (a, b).

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Q: tind Yaduis ot curuature of the Curue O r= 2t i t (1+); - 2t k at (2,1,-2) 13 Brofi+tietk at the…

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Q: For X = (x₁,...,xn), Y = (y₁, ..., Yn) € Rn, define n d(X,Y) := Σ |xi - vi| i=1 (a). Show that d is…

A: Given that, are two vectors.The defined function is: To check the d is matrix space on , have to…

Q: d1(x,y) = e*-y, d2(x,y)= |2x – yl and d3(x,y) = /]x – yl %3D %3D O Both d1 and d2 are metrics on R.…

A: We have,d1x,y=ex-yd2x,y=2x-yd3(x,y)=x-yWe will check whether they satisfy the properties of metrics…