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Kenneth Rosen Edition 7 Exercise 2.5 Question 4 (Page No. 176)

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Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.

  1. integers not divisible by $3$
  2. integers divisible by $5$ but not by $7$
  3. the real numbers with decimal representations consisting of all $1s$
  4. the real numbers with decimal representations of all $1s\: \text{or}\: 9s$
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