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(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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.6: Variation
Problem 7E
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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.
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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