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Unformatted text preview: R = ( R-Q ) ∪ Q so the irrationals must be uncountable since R is.) (c) The set of all English language sentences We can list all the English languate sentences by ﬁrst listing all the ones of length 1 in alphabetical order, then all the ones of length 2, then all those of length 3, etc. Since every sentence has ﬁnite length, we will eventually list all sentences and thus our listing is a surjection from N to this set. Thus, this set is countable. (d) The set of all real numbers that have no 1’s in their decimal representation. Uncountable. The same diagonalization argument we used for R works here...
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This note was uploaded on 08/31/2009 for the course MATH 55 taught by Professor Strain during the Summer '08 term at Berkeley.
- Summer '08