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Unformatted text preview: 1.2 Infinite Sets 19 Theorem 1.2.5. N is “the smallest” infinite set, i.e., every subset of N is either finite or countable. Proof. Homework. Write it out, cross ’em off. Theorem 1.2.6. If A is countable and B is countable, then A ∪ B is countable. Proof. Since A = { a 1 ,a 2 ,a 3 ,... } and B = { b 1 ,b 2 ,b 3 ,... } , we can write A ∪ B = { a 1 ,b 1 ,a 2 ,b 2 ,a 3 ,b 3 ,... } . That was a direct proof. Theorem 1.2.7. Suppose A 1 ,A 2 ,A 3 ,...A n are countable. Then the union is also count able: n [ k =1 A k = A 1 ∪ ··· ∪ A n = { x . . . x ∈ A k , for some n } . Proof. Homework. Use induction and the previous result. What if we take an infinite union? Theorem 1.2.8. Suppose A 1 ,A 2 ,A 3 ,... is a countable sequence of countable sets. Then the union is also countable: ∞ [ k =1 A k = { x . . . ∃ k such that x ∈ A k } . Proof. If we write A i = { a i 1 ,a i, 2 ,... } , then: (grid)....
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 Fall '11
 Wong
 Sets, Natural number, Rational number, Countable set

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