(1) Show that A × B c X × Y. (2) Prove or disprove: (X \ A) × (Y\B) = (X × Y)\(A × B). (If disproved, what does hold?) (3) Show that (A × B) N (C × D) = (AnC) × (BN D). (4) Can we replace n by U in the above statement? (If not, what does hold?) %3D
(1) Show that A × B c X × Y. (2) Prove or disprove: (X \ A) × (Y\B) = (X × Y)\(A × B). (If disproved, what does hold?) (3) Show that (A × B) N (C × D) = (AnC) × (BN D). (4) Can we replace n by U in the above statement? (If not, what does hold?) %3D
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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Topic Video
Question
Solve problem 2.54 and theorem 2.55 in detail please. Please only attempt the questions when you know the right answer please.
![Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is the
set of ordered pairs given by
X × Y := {(x, y) | x € X ^ y € Y }.
Exercise 2.51. Let A, C C X and B, D CY.
(1) Show that A × B c X ×Y.
(2) Prove or disprove: (X \ A) × (Y \B) = (X ×Y)\(A × B). (If disproved, what does
hold?)
(3) Show that (A × B) (C x D) = (AnC) × (Bn D).
(4) Can we replace n by U in the above statement? (If not, what does hold?)
Definition 2.52. Let (X, Tx) and (Y, Ty) be spaces. The product topology on X × Y is the
topology generated by the basis
B := {U × V c X × Y | U e Tx ^V € Ty}.
Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52
is a basis for a topology.
Problem 2.54. Give an example to show that the basis for the product topology on X × Y
is just a basis, and not generally a topology.
Theorem 2.55. Let X and Y be spaces, АсХ, ВСҮ, аnd give X xY the product
topology. Then Ax В — А x В.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e9ab80-64ca-4d50-9be9-589e5309635d%2Fe4a86f4d-81cd-484b-8316-726f00ed8fb7%2Fzv26h85_processed.png&w=3840&q=75)
Transcribed Image Text:Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is the
set of ordered pairs given by
X × Y := {(x, y) | x € X ^ y € Y }.
Exercise 2.51. Let A, C C X and B, D CY.
(1) Show that A × B c X ×Y.
(2) Prove or disprove: (X \ A) × (Y \B) = (X ×Y)\(A × B). (If disproved, what does
hold?)
(3) Show that (A × B) (C x D) = (AnC) × (Bn D).
(4) Can we replace n by U in the above statement? (If not, what does hold?)
Definition 2.52. Let (X, Tx) and (Y, Ty) be spaces. The product topology on X × Y is the
topology generated by the basis
B := {U × V c X × Y | U e Tx ^V € Ty}.
Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52
is a basis for a topology.
Problem 2.54. Give an example to show that the basis for the product topology on X × Y
is just a basis, and not generally a topology.
Theorem 2.55. Let X and Y be spaces, АсХ, ВСҮ, аnd give X xY the product
topology. Then Ax В — А x В.
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