A5 - g is n 1 2 3 4 5 6 7 8 :: : g ( n ) b 1 b 2 b 3 b 4 b...

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1. What does it mean to say that two sets A and B have the same cardinality ? What does it mean to say that a set S is countable ? A and B have the same cardinality if there exists f : A ! B that is one-to-one and onto. same cardinality as the set N of natural numbers. 2. Show that the set of even integers ( positive, negative, or zero ) is countable . f : N ! Z by f ( n ) = n 1 if n is odd n if n is even The function f is one-to-one and onto. The table of values of f is n 1 2 3 4 5 6 7 8 :: : f ( n ) 0 2 2 4 4 6 6 8 :: : 3. Let A be a countable set and B a set with exactly 5 elements. Show that A [ B is countable . First we will assume that A and B are disjoint, that is, that A \ B = ; . Let f : N ! A be function that is one-to-one and onto, and let B = f b 1 ;b 2 ;b 3 ;b 4 ;b 5 g g : N ! A [ B by g ( n ) = b n if n ± 5 f ( n 5) if n > 5 The function g is one-to-one and onto. The table of values of
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Unformatted text preview: g is n 1 2 3 4 5 6 7 8 :: : g ( n ) b 1 b 2 b 3 b 4 b 5 f (1) f (2) f (3) :: : We need A and B to be disjoint to avoid duplicates in the second row of the table. What if A and B are not disjoint? Let B = B n A = f x 2 B : x = 2 A g . Then A and B are disjoint and A [ B = A [ B . We can write B = f b 1 ;: :: ;b k g for some k < 5 . De&ne g : N ! A [ B by g ( n ) = & b n if n ± k f ( n & k ) if n > k 4. Let A be the set of rational numbers with odd denominators. Explain why A is countable . Theorem 1.4.11 tells us that Q is countable. Theorem 1.4.12 tells us that any in&nite subset of a countable set is countable. The set A is in&nite because it contains 1 = 1 , 2 = 1 , 3 = 1 , 4 = 1 , . . . . It is a subset of Q , so it is countable....
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This note was uploaded on 05/26/2010 for the course MATH maa 4200 taught by Professor Dr. during the Spring '08 term at Miami University.

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