Thus we could construct a list of the triplets by

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Thus we could construct a list of the triplets by first writing down all those that sum to 3, then to 4, then to 5, and so on; such a list would begin (1 , 1 , 1) , (2 , 1 , 1) , (1 , 2 , 1) , (1 , 1 , 2) , (3 , 1 , 1) , (2 , 2 , 1) , (2 , 1 , 2) , (1 , 3 , 1) , (1 , 2 , 2) , (1 , 1 , 3) , (4 , 1 , 1) , (3 , 2 , 1) , . . . These are not the only possible lists. 8. If we had n circles, then we could color them in 2 n ways. For example, if there were four circles to be colored, then we could color them in 16 ways. 2 n is so much larger than n that it starts to seem reasonable that maybe 2 | N | is larger than | N | , where | N | is the cardinality of the natural numbers. In fact it is; the set of all possible circle colorings has the same cardinality as the set of real numbers, which is much larger than the cardinality of the naturals. (Circle colorings, in fact, are secretly just binary expansions of real numbers.) 2
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  • Summer '09
  • Lugo
  • Math, Real Numbers, Decimal, Natural number, Rational number, Countable set

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