MIT18_100BF10_pset2

# MIT18_100BF10_pset2 - (b Let P be the set of all subsets of...

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18.100B : Fall 2010 : Section R2 Homework 2 Due Tuesday, September 21, 1pm Reading: Tue Sept.14 : countability, Euclidean spaces, Rudin 1.32-38, 2.1-17 Thu Sept.16 : metric spaces, Rudin 2.15-28 Notes: We will call N r ( p ) in 2.18(a) an open ball , rather than a neighborhood (which has a different meaning in general topology). The notion of perfect set is not important for us. 1 . (a) Explain in your own words the logic of a proof by contradiction. (b) Show that the set R of real numbers is uncountable. (You can use Theorem 2.14 for inspiration, but be aware that 0 . 9 = 0 . 9999999999 . . . = 1 .) 2 . (a) Let I = { [ p, q ] ; p q, p, q Q } be the set of intervals in R with rational endpoints. Show that I is countable.

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Unformatted text preview: (b) Let P be the set of all subsets of N . Show that P is uncountable. [ Hint : for an alleged bijection f : N → P , consider the set D = { n ∈ N ; n n∈ f ( n ) } .] 3 . Exercise 7, p. 43 of Rudin. 4 . Exercise 8, p. 43 of Rudin. 5 . Exercise 9, p. 43 of Rudin. MIT OpenCourseWare http://ocw.mit.edu 18.100B Analysis I Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT18_100BF10_pset2 - (b Let P be the set of all subsets of...

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