Find a set S of subsets of R x R (That is, S C P(R × R)) that satisfies that following properties: (a) S is countable. (b) For every (x, y) = R × R, there exists a set A € S and there exists (a, b) = A such that the distance between (x, y) and (a, b) is less than 1. (You can use the fact that for every real number r € R, there exists a rational number q € Q, with |rq|< . This is still true if is replaced by any other positive number). You must prove that the set S you define does in fact satisfy conditions (a) and (b).

Find a set S of subsets of R x R (That is, S C P(R × R)) that satisfies that following properties: (a) S is countable. (b) For every (x, y) = R × R, there exists a set A € S and there exists (a, b) = A such that the distance between (x, y) and (a, b) is less than 1. (You can use the fact that for every real number r € R, there exists a rational number q € Q, with |rq|< . This is still true if is replaced by any other positive number). You must prove that the set S you define does in fact satisfy conditions (a) and (b).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 29E

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