2. Letde(x, y) = |x — y|, d₁(x, y) =for x, y E R. Show that(a) d₁ is a metric on R;(b) de and d₁ are not equivalent.|x - y|1 + x - y

Answered: Image /qna-images/answer/9ccb1871-aab2-4ca9-9916-53223a5b03c8.jpg

Answered: Let de(x, y) := |x — y|, d₁(x,y) := for…

Let de(x, y) := |x — y|, d₁(x,y) := for x,y E R. Show that (a) d₁ is a metric on R; (b) dẼ and d₁ are not equivalent. |x - y| 1 + x − y|

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: 3. Use implicit differentiation to find əz - for ?х x²+2y²+5z²=2.

Q: 2. Write the formula for tangent plane of a function f(x,y) at a point x = a.y=b. How does this…

A: Conisder a function f(x,y).We need to find the tangent plane at x=a,y=b.Let z=f(x,y) and…

Q: Let (R, d) be usual metric space, then Ris ot separable True False

Q: Evaluate fr (y-x-3x²i)dz where C is the straight line from z = = 0 to z = 1 + i. 2+i

Q: 1. Using Euler's equation, show that the shortest distance between two points (r1, yn, z1) and (r2,…

A: Given, the points are (x1, y1, z1) and (x2, y2, z2).

Q: Determine the intersection if any between the line L₁: [x, y, z] = [3, 2, − 5] + r[0, 1, 4] and the…

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: Let f function from (R,d1) to (R,d2) defined as d(x) = x, where d, is the usual metric and d2 is the…

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…

Q: Let 1, y E R². Show that p:=y is the closest point on the line Span(y) to z.

A: Given, x,y ∈ R2 To show: p=x.yy.yy is the closest point on the line span y to x

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)}…

Q: Consider an isometry f : E3 → E3 given by f(x, y, z) = (30 − x, z + 22, y). Find a line L in E3 such…

A: Given , f(x,y,z) = (30-x , z+22 , y)

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

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

Q: The pt. (x,y) is at a distance 5 from (0,-3)

Q: x-2 z+4 • Let L:*²=+3 = 2²+ and M: 2x - 3y +z = 6 are line and 3 2 5 plane and let P be the…

A: We have to find the equation of line

Q: Evaluate (x + y)ds where C is the straight-line segment x = 4t, y = (12-4t), z = 0 from (0,12,0) to…

A: Let where C is straight line segment from .We know thatwhere

Q: (b) Let (M, d) be a metric space. Define a function e: MXM-R by e(x, y) = min {1,d(x, y)}; vx,y E M.…

Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

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.…

Q: Let R = (c, 1,-1) where c denotes a real number. Let ä denote the plane x + y – z = 2. Find the…

A: Given Information - The distance from the point R = (c,1,-1) to the plane x+y-z = 2 is 2. We have…

Q: 2. Let (X, d) be a metric space and y E X. (i) Prove that the closed ball B[y,2] is a closed set in…

Q: Let (x, g) be a metric space. Is the function d (x, y) = k.g (x, y) (k € R) defined from XxX to R a…

A: Let X,g be a metric space . Given that function X×X to R defined as dx,y=k·gx,y where k∈R A metric…

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: Evaluate the line mtegral over the curve S(-x + y +z) dx + (x+y - z) dy + (x-y +z) dz where the line…

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

Q: Explain why or why not Determine whether the following statementsare true and give an explanation or…

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

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: Let a = (a, az); b = (b, ,b2) € R?, define the function %3D %3D d: R? x R? ---> R by d=(a, b) =…

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

Q: Sketch the set of points (r, θ) that satisfy the following conditions: {(r, θ) : −2 ≤ r < 2, 0 ≤ θ ≤…

Q: Let r(t) = (-5t³ – 1, t² + 5t³, 3t²). Find the line (L) tangent to (t) at the point (39, -36, 12)

A: r→(t)=⟨−5t3−1,t2+5t3,3t2⟩Find the line (L) tangent to r→(t) at the point (39,−36,12)

Q: Consider the following mappings on RXR: d1(r, y) = e , d2 = |2.r – 2y|l. and d3 = v2r - yl. None of…

Q: Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i.…

A: To verify that given distance form d is a metric , we have to check for Identity Symmetry…

Question

2. Let
de(x, y) = |x — y|, d₁(x, y) =
for x, y E R. Show that
(a) d₁ is a metric on R;
(b) de and d₁ are not equivalent.
|x - y|
1 + x - y

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Define a theorem corresponding to equivalent metric

VIEW

Step 2: Show the dE is a metric on R

VIEW

Step 3: Show that d1 is a metric on R

VIEW

Step 4: Use inequality

VIEW

Step 5: Prove that d1 is a metric on R

VIEW

Step 6: Use theorem to check dE and d1 are equivalent or not

VIEW

Step 7: Use theorem

VIEW

Solution

VIEW

Step by step

Solved in 8 steps with 8 images

SEE SOLUTION Check out a sample Q&A here

Follow-up Questions

Read through expert solutions to related follow-up questions below.

Follow-up Question

Please write 2(b) again

Solution

by Bartleby Expert

SEE SOLUTION

Similar questions

1d) 1e) 1f)
In a certain geometry, define the distance d(P. P₂) between two points P(x, y,) and P(x, y₂) as d(P₁, P₂)=x₂-x+1₂-yil Define the u-distance between a point P and a line L as d(P, L) = min{d(P, P), P = L} Find the p-distance of the point (1, 0) from the line y=x√3+3.
Prove that the d₁ ("Manhattan") metric on R2 is really a metric.
/Q9. Consider R² with the Euclidean metric. Is (0, 1) = {(x, 0)|0 < x < 1} open, closed, neither, or both? Explain.
I need the answer as soon as possible
Let (X,d) be a metric space. Define f: X x X -> Real Numbers by f(x,y) = d(x,y) / 1+d(x,y) . show that f is a metric on X.
Let X=ℝ2 and define d2,:ℝ2×ℝ2→ℝ by d2((x1 ,y1),(x2,y2)) = max{|x1 - x2|,|y1 - y2|}. a) Verify that d2 is a metric on ℝ2. b.) Draw the neighborhood N((0; 1) for d2, where 0 is the origin in ℝ2.
Let d: R2 × R2 →R given by xỉ + yỉ + x3 + y½ , if (x1,y1) # (x2, Y2) 0, if (x1, Y1) = (x2, Y2) d((x1.y1).(x2.y2))= { (a) Show that d is a metric. (b) Show that (R2, d) is a complete metric space. (c) Show that (R2, d no a connected metric space
Let fiz) = x2 + iy? where x and y are the real and imaginary parts, respectively of z and C is a contour consisting of the line segment from z, = 5i %3D to z, = 6 + 10i. Evaluate Re f(z,) ?
2. For points p = (x1,yı) and q = (x2, y2) € R², the taxicab metric is given by d(p, q) = |#2 – 11| + \y2 – y1l. Verify that this is a valid metric by showing that it satisfies all four necessary criteria of a metric.
Consider the 2-dimensional hyperbolic space with metric ds² dr²+dy² = with y > 0. The geodesic equations are: b. d²x dx² dy = [(#)² - (*)²] = ☐ a. d²x dx² = O and = 0 y dλ 2 dx dy = O and y dλ dλ + d²y O and [(*)² ☐ c. d²x dx² ☐ d²x dx² = dx dy +1 (4+)² = 0 d²y 0 d. ² + 1 = 0 and 2 + } [ ( * )² + (#)'] = • y dλ dλ dx² y dy ग्र 0
Sir, Need this Solution.