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2. Let S be a subset of R. We say that S is dense in R, if (D) for any € > 0 and any x R, there is sS- {x} such that |x-s| ≤ €. (i) Write down the negation of (D) as a complete sentence. (ii) Using the result of Problem 1, show that Q is dense in R. (iii) Show that Z is not dense in R. (In other words, show that Z satisfies the negation of (D)).

2. Let S be a subset of R. We say that S is dense in R, if (D) for any € > 0 and any x R, there is sS- {x} such that |x-s| ≤ €. (i) Write down the negation of (D) as a complete sentence. (ii) Using the result of Problem 1, show that Q is dense in R. (iii) Show that Z is not dense in R. (In other words, show that Z satisfies the negation of (D)).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...
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 Let S be a subset of R. We say that S is dense in R, if
(D) for any  > 0 and any x ∈ R, there is s ∈ S − {x} such that |x − s| ≤ .
(i) Write down the negation of (D) as a complete sentence.
(ii) Using the result of Problem 1, show that Q is dense in R.
(iii) Show that Z is not dense in R. (In other words, show that Z satisfies the negation of (D)).

2 2. Let S be a subset of R. We say that S is dense in R, if (D) for any € >0 and any x R, there is s E S- {x} such that |xs| ≤ €. (i) Write down the negation of (D) as a complete sentence. (ii) Using the result of Problem 1, show that Q is dense in R. (iii) Show that Z is not dense in R. (In other words, show that Z satisfies the negation of (D)).
Transcribed Image Text:2 2. Let S be a subset of R. We say that S is dense in R, if (D) for any € >0 and any x R, there is s E S- {x} such that |xs| ≤ €. (i) Write down the negation of (D) as a complete sentence. (ii) Using the result of Problem 1, show that Q is dense in R. (iii) Show that Z is not dense in R. (In other words, show that Z satisfies the negation of (D)).
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