Consider the plane (P) related to x-axis and y-axis system. et R be the binary relation on (P) defined by: I (x, y) R M’(x', y') → x² + y? =x² + y? ) Prove that R is an equivalence relation on the set of points of (P). ) Interprete geometrically the equivalence class of the points A(0, 0) and B(1, 1). ) Describe geometrically the set (P)/R.

Consider the plane (P) related to x-axis and y-axis system. et R be the binary relation on (P) defined by: I (x, y) R M’(x', y') → x² + y? =x² + y? ) Prove that R is an equivalence relation on the set of points of (P). ) Interprete geometrically the equivalence class of the points A(0, 0) and B(1, 1). ) Describe geometrically the set (P)/R.

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

Similar questions

Give the set of limit points A0 of a singleton A = {(5, 2)} on the plane R2 with the discretemetric.
Prove that the system of parabola y2= 4a(x+a) is self orthogonal
Determine the absolute maximum and minimum values of F over the set D, which consists of the closed tirangular region in the xy plane with vertices (0,0), (0,6) and (6,0).
Suppose S is a rectangle in the uv plane with vertices O(0,0), P(delta u, 0), (delta u, delta v), Q(0, delta v). The image of S under the transformation x=g(u,v), y=h(u,v) is a region R in the xy plane. Let O', P' and Q' be the images of O,P,Q, respectively, in the xy plane where O',P',Q' do not all lie on the same line. The coordinates of O' , P', and Q' are (g(0,0),h(0,0)), (g(delta u,0),h(delta u,0)), and (g(0,delta v),h(0,delta v)), respectively. Consider the parallelogram determined by the vectors O'P' and O'Q'. Use the cross product to show that the area of the parallelogram is approximately |J(u,v)|delta(u)delta(v). |J(u,v)| is the jacobian determinant.
Show that except in degenerate cases, (u * v) * w lies in the plane of u and v, whereas u * (v * w) lies in the plane of v and w. What are the degenerate cases?
Is a subset of the plane {(x1, x2); x1 > 0, x1x2 ≥ 1} convex? Justify.
(a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by expressing u directly in terms of x, y, and z before differentiating. Then (b) evaluate ux, u y, and uz at the given point (x, y, z). u = (p - q )/(q - r ), p = x + y + z, q = x - y + z, r = x + y - z; (x, y, z) = (sqrt(3), 2, 1)
Find the image of the rectangular region in the z − plane formed by joining the points0, 0 , 2, 0 , 2, 1 , (0, 1) in the w − plane under the transformation,w = (1 +i )z − 2iInterpret both the regions graphically.
Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)
Let (X, T1) and (Y, T2) be topological spaces. Show that the set {U × Y ∣ U ∈ T1} ∪ {X × V ∣ V ∈ T2} is a subbasis for the product topology of X × Y..
Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = <x, y>z for all x ∈V. First prove that T is linear. Then show that T∗exists, and find an explicit expression for it.
Give an example of a linear operator T on an inner product space V such that N(T) ≠ N(T∗).