hw4-solutions

# Tation of this number that ends in infinitely many 9s

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tation of this number that ends in infinitely many 9s; for example, we write 2 . 4599999 . . . instead of 2 . 46.) 5. Map the natural number n to the even number 2 n , for each n . 6. The cardinality of the balls in at least one of the barrels must be infinite. If the cardinality of the balls in each barrel was finite, then the cardinality of all the balls put together would also be finite. 7. The set of all triplets of natural numbers is countable. One way to see this is to remember that a product of two countable sets is countable. Thus, the set of all pairs of natural numbers is countable (as we saw in class), and the product of this set with the natural numbers is countable; the latter set is the triplets. We could construct a list of the triplets by first generating a list of the pairs of natural numbers, (1 , 1) , (2 , 1) , (1 , 2) , (3 , 1) , (2 , 2) , (1 , 3) , (4 , 1) , (3 , 2) , (2 , 3) , (1 , 4) . . . and then creating a list which consisted of the first element of this list, followed by 1; then the second element, followed by 1; then the first element, followed by 2; then the third element, followed by 1; and so on. This gives (1 , 1 , 1) , (1 , 2 , 1) , (1 , 1 , 2) , (2 , 1 , 1) , (1 , 2 , 2) , (1 , 1 , 3) , (1 , 3 , 1) , (2 , 1 , 2) , (1 , 2 , 3) , (1 , 1 , 4) , Another way to see this is to note that there are finitely many triplets of natural numbers summing to each of 3 , 4 , 5 , 6 , . . . , and the union of countably many finite sets is countable.
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