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 eA : Rxy}. The quotient of A under R is A/R= {[x]R:x E A}, i.e.,the set of these equivalence classes.%3DThe next result vindicates the definition of an equivalenceclass, 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 Rxyiff [x]R = [y]R-%3DProof. 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 haveRyx, 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

If you observe carefully, you will see that they have used symmetric and transitive properties of…

Answered: A) Which property/properties of the…

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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Question

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

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