Suppose that we define a function d that maps R2 to R by d(x,y) = { 0,. if x = y 1, if x is not equal to y }. a. Prove that d is a metric on R. b. Prove that for any a in R, {a} is an open set under the metric d. c. Let X = R with the usual Euclidean distance and Y = R with the metric d defined above. Prove that f maps X to Y defined by f(x) = x is not continuous.

Q: Question 5. In this problem we prove that a straight line is the shortest curve between two points…

A: We have to prove ,  γ..u ≤γ.  We will use Cauchy-Schwarz inequality to prove the above.

Q: Let F = (y – x, ry) and let C be the line segment from (2, –3) to (1, –1). %3D Evaluate F. dr.

A: We wil first equation of parametric equation of line segment

Q: 3. This question concerns the L metric d and the L¹ metric e on a) Calculate d(f.g) and e(f.g),…

A: Consider L∞ metric d and L1metric e on C[0,1]. (i) To Find: d(f,g) , e(f,g) for  f(x)=1+x , g(x)=x2.…

Q: ii) The zero-lift AOA of the aerofoil.

A: As per the question, our objective is to calculate the zero lift angle of attack (AOAzero lift) and…

Q: 1 2 3 4 2 1 3 4. n n

A:

Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.

A: Euclidean metric dEuc((a,b),(c,d))=√((a-c)2+(b-d)2) Maximum metric dMax((a,b),(c,d))=max{|a-c| ,…

Q: Let A, B and C be three points in a n. g. lying on a circle with center D. Show that: if D is in…

A: Saccheri theorem states that the sum of angles of a triangle is atmost 180°.

Q: Let N R R be any norm on Rn. Show that N is a convex function (give all the details for the proof).

A: Given N :R^n→R be a norm on R^n.      To show : N is a convex.     Definition of norm :…

Q: Exercise 5. 1) Show that the metrics dp, p ≥ 1 and dmax on Rª are all Lipschitz equivalent. 2) If X…

A:

Q: 1. Let A in R (standard metric) be defined as follows: A={1/n|n € N} U 0 . Is the set B = {b = a…

A:

Q: metric on R² is d: R² x R² → R$, given by fo 0 ifr=y 1 if x #y. d(x, y) =

A:

Q: Let (t) = (t, t, tº, tª) for all t € R. Define C = Þ(R). Let p = (1, 1, 1, 1). Derive the tangent…

A: Let  for all Let the point be We need to calculate the tangent and normal spaces to C at p.

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…

A:

Q: 1. Vector Functions and Space Curves a) Find the domain of the vector function: r(t) = (√4 — t²,…

A:

Q: c) Find the characteristic polynomial of each of the restricted maps Tw and Tu. d) Determine whether…

A: Given that,  V=ℝ4 and T be the linear operator on V defined by, Tx,y,z,t=3t, -y+4z, 5y, x-2t c) To…

Q: Suppose E ⊂ Real Numbers. With respect to the Euclidean metric, is it possible that E' = { 1/n: n ∈…

A:

Q: Draw the unit “circle” around the origin in the Euclidean metric, the taxicab metric, the hub…

A: The unit circle around the origin in Euclidean metric is The unit circle around the origin in…

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.