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Copy of s3

# Copy of s3 - Solutions to homework 3 1 For each positive...

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Solutions to homework # 3. 1. For each positive integer k , there are only finitely many equations with integer coefficients a 0 , . . . , a n such that | a 0 | + · · · + | a n | + n = k . Each such equation has n k roots. Let A k denote all the roots of all equations with | a 0 | + · · · + | a n | + n = k . Then the set of all algebraic numbers is the union k IN A k . This is a countable union of finite sets, therefore is at most countable. The set of all algebraic numbers contains ZZ, hence is infinite, so it must be countable. 2. The set IR of all real numbers is the (disjoint) union of the sets of all rational and irrational numbers. We know from the lecture that IR is uncountable, whereas Q is countable. If the set of all irrational numbers were countable, then IR would be the union of two countable sets, hence countable. Thus the set of all irrational numbers is uncountable. 3. Here is one of many examples of a bounded set with exactly three limit points: S := { 1 n : n IN } ∪ { 1 + 1 n : n IN } ∪ { 2 + 1 n
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