Consider the continuous-time dynamical system given by the ODE & = –x. (c) Determine whether the following sets M CR are invariant, positively invariant,negatively invariant, or neither. Prove your statement.i. M = {0}ii. M = {2}ii. М %—D [0, 2]= [0, 2]iv. M = (-2,2)v. M%3D (0, о0)

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Answered: (c) Determine whether the following…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Let A := {1,2,3,4}: B and T:= {(a, b) E A x B | +¹ € Z}. (a) Find A x B. (b) Find T. (c) Check what property of Relations thus T have. Provide a brief explana- tion for each property.
Don’t forget to write your ASSUMPTION, what you NEED TO SHOW, and your CONCLUSION.
. Tell whether the following sets are open: (Justify your answer.) (a) (3,5) U {6} (b) (-∞, 0) U (0, 1) (c) {1, 2, 3, 4, 5, 6, 7, 8, 9) (d) (-∞, 0) U[0, 1) (e) Z (f) (-∞,0) U (0, 1] (h) R - {1, 2, 3} (j) { //: neN} U {0} (1) Qn (0,1) (g) (-∞,0) U [0, 1] (i) { //:ne N} (k) Q
Let X be a superset of Y. Let Z be a set. Using an example of your own, state whether thefollowing are correct or incorrect. If incorrect, correct by changing the right hand side ofthe equation.(i) X n Y = Y(ii) X u Y = Ø (iii) X n Z ⊂ Y u Z
How do you solve this question on discrete mathematics: 2. Let A be the set of all points in three-dimensional space. Define R as the relation that two points will have the same z coordinate. (a) Give three examples that are related to P(4,6,2). (b) Prove or disprove that R is reflexive. (c) Prove or disprove that R is symmetric. (d) Prove or disprove that R is transitive. (e) Is R an equivalence relation? (f) What space do all points related to each other (e.g. all points withz= 2) describe?
38
Let U = (−∞, ∞) be the universal set and M = (−6, 5] and W = (−3, 9]. In interval notation, determine the following. 1. M ∪ W2. M ∩ W 3. M − W4. W − M
Only Need Part(d), thank you!

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(c) Determine whether the following sets M C R are invariant, positively invariant, negatively invariant, or neither. Prove your statement. i. M = {0} ii. M = {2} iii. M = [0, 2] iv. M = (-2,2) v. M = (0, ∞)