Consider the following mappings on RXR: * d1(x,y) = ex-y, d2(x,y) = |2x -y and d3(x,y) = √√x - yl O Both d1 and d2 are metrics on R. O Only d2 is a metric on R. O Both d2 and d3 are metrics on R. O None of the choices.

Q: A and Bare open and bounded.

A: For second one we can take any two open interval one of whose right end point ar same of the others…

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Q: Thm. Let v be an inner product Space then and HyE V жує ||≤|12||· 1191/ further equility is hold iff…

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Q: Thm. Let v be an V be an inner product Space V and then жує | 13 11211. 11111 further equility is…

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Q: Show that for all values of θ and φ, the point (a sinφcosθ, a sinφsinθ, a cosφ) lies on the sphere…

A: Show that for all values of θ and φ, the point (a sinφcosθ, a sinφsinθ, a cosφ) lies on the sphere…

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 A and B be two connected subset

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Problem 9 (6 points). For each of the following functions d: R2 x R² → [0, 00), say whether d is a…

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Q: Q8. Assume that (R²,T) is the usual Euclidean topological space. D = {(x, y) E R², y = x} C R².…

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Q: (a) Give an example of a metric on R2. (b) Is the set of all points (x, y) in R² such that y = x²…

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Q: A. Consider R3 with the metric d((X1, X2, X3), (V1, Y2. Y3) = |X1 - Yıl + |x2 - y2l + [X3 - Y3l. i.…

A: This problem is related with metric on ℝ3. Given that                   dx1,x2,x3,y1,y2,y3 = x1-y1+…

Q: 2. Let d1, d2 be two metrics on X + 6. Show that: d(r, y) = d1(x, y) + d2(x, y).

A: I have proved all four conditions of metric for d

Q: Example B Let M=IR², and let x = (x₁, x₂), y = (y₁, y₂). Then d₂ = √√(x₁ - y₁)² + (x₂-Y₂1²1 is the…

A: Let X be a vector space. A real valued function d:X×X→R is a metric if it satisfies For all…

Q: Decide which of the following are metrics on X = R2. For each, we let x = (x1, x2) and y = (y1, y2)…

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Q: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

Q: (2) Draw the following geometric objects in the SPACE. Consider the follow- ing function w = 4x + 2y…

A: The given function is, w=4x+2y+5z. To Draw: The following, (a) w=4x+2y+5z=20 w=4x+2y+5z=0…

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: 16 + 3 the following matrices festSIAM #71 47 purन

A: Topic = Matrix

Q: Thm. Let V be an inner product Space and ждек then | 1 ≤ 11all. 1171/ further equility is hold iff…

A: Using the properties of inner product And  aā=|a|2 , for any complex scalar  ||cx||=|c| ||x||, for…

Q: Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

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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: 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…

Q: (a) (b) X = S y = t = 0 2 = x - 3 4 = −2+2 (c) x + y = 1 (d) x = 2y + 1 3 -3z + 1 4 || = 2

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Q: ✓ 3.11 Find M+. 292 (a) M = { (c) M = { (e) M = { { (₁) |x=0} () (f) M={y|-x+3y+z=0} (g) M={y | x =…

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Q: 1. Let X R². For any (x1, y1), (2, y2) E R², define d₂((x1, y₁), (x2, y2)) := Show that d₂ is a…

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Q: c spaces. Show that X x Y and Y

A: The sup-metric d1 on X×Y is defined as d1x1, y1, x2, y2=max dx1, x2, ey1, y2. Similarly, the…

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Q: Let f (z) = 3O = 0 A = 10and B= 10 + 10iet C1 be the contour composed of the line segments OAand…

A: Solution  Given  A=10   ,  B=10+10i

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Q: Prove that the d1 (“Manhattan”) metric on R2 is really a metric

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Q: 6. Let A and B be two connected subsets of X and An B + 4. Show that AUB is connected. In…

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Q: Let A = (5− √24,0), B = (0,5− √24), and C = (0,0) be points in the Poincar ́e Disk. The hyperbolic…

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Q: Homework 2. Let d be a metric on X. Determine all constants k such that the following is a metric on…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…