a. Prove that ~ is an equivalence relation on X and describe the set X/ ~ of equivalenceclasses of X induced by ~.b. Let S1 = {x ∈ R2 : ||x|| = 1} ⊂ X, and endow S1 with the same equivalencerelation ~ as above. Describe the set S1/~ of equivalence classes of S1 induced by ~.c. Given a equivalence class [x]∈ S1/ ~, choose its representative x to be the largestelement of the equivalence class with respect to the dictionary order. Sketch a graph of all therepresentative points of all equivalence classes of S1/ ~. Let X = R? – (0, 0) and - be defined over X by settingX ~y A y = txfor some non-zero t e R.

The given set is X=ℝ2-0,0 and ~ be defined over X by setting x~y⇔y=tx or some non zero t∈ℝ. (a). To…

Answered: Let X = R2 - (0, 0) and ~ be defined…

Math

Advanced Math

Let X = R2 - (0, 0) and ~ be defined over X by setting X~y + y = tx for some non-zero t €R.

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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Question

a. Prove that ~ is an equivalence relation on X and describe the set X/ ~ of equivalence
classes of X induced by ~.

b. Let S¹ = {x ∈ R² : ||x|| = 1} ⊂ X, and endow S¹ with the same equivalence
relation ~ as above. Describe the set S¹/~ of equivalence classes of S¹induced by ~.

c. Given a equivalence class [x]∈ S¹/ ~, choose its representative x to be the largest
element of the equivalence class with respect to the dictionary order. Sketch a graph of all the
representative points of all equivalence classes of S¹/ ~.

Let X = R? – (0, 0) and - be defined over X by setting
X ~y A y = tx
for some non-zero t e R.

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