Define the binary relations =z and =q on IR by setting, for x, y € R, x =zy:⇒x - y € Z x =Qy:⇒x−y € Q₂ where Q is the set of rationals. (a) Prove that each of these is an equivalence relation. (b) Prove that [0,1) is a transversal for =z, i.e. [0, 1) intersects each =z-class in exactly one point. In other words, each =z-class has exactly one representative in [0,1). REMARK: This, in particular, shows that the quotient R/ =z is "canonically identi- fied" with [0, 1). (c) Define a function f : IR→ [0, 1) such that for any x, y € R, ƒ (x) = [x] =z and x =zy ⇒ f(x) = f (y).
Define the binary relations =z and =q on IR by setting, for x, y € R, x =zy:⇒x - y € Z x =Qy:⇒x−y € Q₂ where Q is the set of rationals. (a) Prove that each of these is an equivalence relation. (b) Prove that [0,1) is a transversal for =z, i.e. [0, 1) intersects each =z-class in exactly one point. In other words, each =z-class has exactly one representative in [0,1). REMARK: This, in particular, shows that the quotient R/ =z is "canonically identi- fied" with [0, 1). (c) Define a function f : IR→ [0, 1) such that for any x, y € R, ƒ (x) = [x] =z and x =zy ⇒ f(x) = f (y).
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
![Define the binary relations =z and =q on IR by setting, for x,y € R,
x =zy:⇒x - y € Z
x =QY:⇒x=y € Q₂
where Q is the set of rationals.
(a) Prove that each of these is an equivalence relation.
(b) Prove that [0,1) is a transversal for =z, i.e. [0, 1) intersects each =z-class in exactly
one point. In other words, each =z-class has exactly one representative in [0, 1).
REMARK: This, in particular, shows that the quotient R/ =z is "canonically identi-
fied" with [0, 1).
(c) Define a function f : IR→ [0, 1) such that for any x, y € R, ƒ (x) = [x] =z and
x=zy ⇒ f(x) = f (y).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45d33651-ef87-430c-869b-f81528241937%2F5c0d740b-a75d-49c1-b0c2-4a155d6c5057%2Fafyxspe_processed.png&w=3840&q=75)
Transcribed Image Text:Define the binary relations =z and =q on IR by setting, for x,y € R,
x =zy:⇒x - y € Z
x =QY:⇒x=y € Q₂
where Q is the set of rationals.
(a) Prove that each of these is an equivalence relation.
(b) Prove that [0,1) is a transversal for =z, i.e. [0, 1) intersects each =z-class in exactly
one point. In other words, each =z-class has exactly one representative in [0, 1).
REMARK: This, in particular, shows that the quotient R/ =z is "canonically identi-
fied" with [0, 1).
(c) Define a function f : IR→ [0, 1) such that for any x, y € R, ƒ (x) = [x] =z and
x=zy ⇒ f(x) = f (y).
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