For each n E N, we write S" := {(x1, x2, ..., In+1) E R"+1 : E? = 1}. (1) Let f : S2 → Rª be the function f(x, y, 2) = (x – y°, xy, yz, xz). Define an equivalence relation ~ on S? by setting (x1, Y1, 21) ~ (x2, Y2, 2) if and only if f(x1, Y1, z1) = f(x2, Y2, z2). Of the spaces you have encountered in the course thus far, determine which one is the quotient space S²/ ~? [Hint: To determine when (x1, Y1, 21) ~ (x2, Y2, z2) consider the case y1 = 21 = 0 first.]
For each n E N, we write S" := {(x1, x2, ..., In+1) E R"+1 : E? = 1}. (1) Let f : S2 → Rª be the function f(x, y, 2) = (x – y°, xy, yz, xz). Define an equivalence relation ~ on S? by setting (x1, Y1, 21) ~ (x2, Y2, 2) if and only if f(x1, Y1, z1) = f(x2, Y2, z2). Of the spaces you have encountered in the course thus far, determine which one is the quotient space S²/ ~? [Hint: To determine when (x1, Y1, 21) ~ (x2, Y2, z2) consider the case y1 = 21 = 0 first.]
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.3: Subgroups
Problem 23E: 23. Let be the equivalence relation on defined by if and only if there exists an element in ...
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![For each n E N, we write S" :=
{(r1, x2, . .. , Tn+1) E R"+1 : E 2 = 1}.
Li=1
(1) Let f : S2 → R4 be the function
f (x, y, z) = (x2 – y, xy, yz, xz).
Define an equivalence relation - on S? by setting (x1, Y1, 21) ~ (x2, Y2, 22) if and
only if f(x1,Y1, 21) = f(x2, Y2, 22). Of the spaces you have encountered in the
course thus far, determine which one is the quotient space S²/ ~?
[Hint: To determine when (x1, Y1, 21) ~ (x2, Y2, 2) consider the case y1 = 21 = 0
first.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c3ce0fd-4090-405c-80f4-99d74f0446d4%2F6b34600a-2065-485e-9c1d-40e0612e278c%2Fpdvzwzj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For each n E N, we write S" :=
{(r1, x2, . .. , Tn+1) E R"+1 : E 2 = 1}.
Li=1
(1) Let f : S2 → R4 be the function
f (x, y, z) = (x2 – y, xy, yz, xz).
Define an equivalence relation - on S? by setting (x1, Y1, 21) ~ (x2, Y2, 22) if and
only if f(x1,Y1, 21) = f(x2, Y2, 22). Of the spaces you have encountered in the
course thus far, determine which one is the quotient space S²/ ~?
[Hint: To determine when (x1, Y1, 21) ~ (x2, Y2, 2) consider the case y1 = 21 = 0
first.]
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