# So a is not on the list so no list of numbers from

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Unformatted text preview: path in Figure 1.4 shows how all the elements can be placed on an inﬁnite list. 1 1 2 1 3 1 4 1 5 1 1 2 2 2 3 2 4 2 5 2 1 3 2 3 3 3 4 3 5 3 1 4 2 4 3 4 4 4 5 4 1 5 2 5 3 5 4 5 5 5 ... ... ... ... ... ... ... ... ... ... . . . Figure 1.4: The pmf for the sum of numbers showing for the rolls of two fair dice. So there are many countably inﬁnite sets, but the set of real numbers is deﬁnitely larger than that, as shown by the following population. Proposition 1.5.1 The set of real numbers is not countable. Proof. We will prove that not even the set of numbers in the interval [0, 1] is countable. It is enough to show that for any list of numbers from the interval [0, 1], there is at least one other number in [0, 1] that is not on the list. So, let a1 , a2 , . . . be a list of numbers from the interval [0, 1]. 1.5. COUNTABLY INFINITE SETS 19 Consider the decimal expansions of these numbers: a1 = 0.3439098 · · · a2 = 0.2439465 · · · a3 = 0.9493554 · · · a4 = 0.3343876 · · · a5 = 0.9495249 · · · . Note that for each k , the number in the k th decimal place of ak i...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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