A) Which property/properties of the equivalence relation R is used to prove the left-to-right direction of Proposition 3.12? (You may choose more than one) B) what about right-to-left?
A) Which property/properties of the equivalence relation R is used to prove the left-to-right direction of Proposition 3.12? (You may choose more than one) B) what about right-to-left?
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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A) Which property/properties of the equivalence relation R is used to prove the left-to-right direction of Proposition 3.12? (You may choose more than one)
B) what about right-to-left?
![each x e A, the equivalence class of x in A is the set [x]R = {y e
A : Rxy}. The quotient of A under R is A/R= {[x]R:x E A}, i.e.,
the set of these equivalence classes.
%3D
The next result vindicates the definition of an equivalence
class, in proving that the equivalence classes are indeed the par-
titions of A:
Proposition 3.12. If R C A is an equivalence relation, then Rxy
iff [x]R = [y]R-
%3D
Proof. For the left-to-right direction, suppose Rxy, and let z E
[x]R. By definition, then, Rxz. Since R is an equivalence relation,
Ryz. (Spelling this out: as Rxy and R is symmetric we have
Ryx, and as Rxz and R is transitive we have Ryz.) So z E [y]R.
Generalising, [x]R C [y]R• But exactly similarly, y]R C [x]R. So
[x]R = [y]R, by extensionality.
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4125d355-cd98-459d-b7ed-b965ceb27b35%2F9bcbbe54-d5b5-4b9f-9a58-e3cccab44d14%2F04jjh7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:each x e A, the equivalence class of x in A is the set [x]R = {y e
A : Rxy}. The quotient of A under R is A/R= {[x]R:x E A}, i.e.,
the set of these equivalence classes.
%3D
The next result vindicates the definition of an equivalence
class, in proving that the equivalence classes are indeed the par-
titions of A:
Proposition 3.12. If R C A is an equivalence relation, then Rxy
iff [x]R = [y]R-
%3D
Proof. For the left-to-right direction, suppose Rxy, and let z E
[x]R. By definition, then, Rxz. Since R is an equivalence relation,
Ryz. (Spelling this out: as Rxy and R is symmetric we have
Ryx, and as Rxz and R is transitive we have Ryz.) So z E [y]R.
Generalising, [x]R C [y]R• But exactly similarly, y]R C [x]R. So
[x]R = [y]R, by extensionality.
%3D
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