# sol09 - CS 330 Discrete Computational Structures Fall...

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CS 330 : Discrete Computational Structures Fall Semester, 2014 Assignment #8 Solutions Due Date: Sunday, Nov 2 Suggested Reading: Rosen 9.1 and 9.5, LLM 9.4 These are the problems that you need to turn in. For more practice, you are encouraged to work on the other problems. Always explain your answers and show your reasoning. 1. [24 Pts] Shana Show that the following sets are countably infinite, by defining a bijection between N (or Z + ) and that set. You do not need to prove that your function is bijective. (a) [12 Pts] the set of integers divisible by 7 Solution: f ( x ) = 7 x/ 2 , if x is even. f ( x ) = - 7( n + 1) / 2 , if x is odd. Given x = 0 , 1 , 2 , 3 , ..., ∈ N , f ( x ) = 0, -7, 7, -14, 14... . (b) [12 Pts] A × Z + where A = { 2 , 3 } Solution: f ( x ) = (2 , ( x + 1) / 2) , if x is odd. f ( x ) = (3 , x/ 2) , if x is even. Given x = 1 , 2 , 3 , ..., ∈ Z + , f ( x ) = (2 , 1) , (3 , 1) , (2 , 2) , (3 , 2) , ... . 2. [12 Pts] Peter Show that if A B and A is uncountable, then B is uncountable. Solution: Suppose B is countable. Then the elements of B can be enumerated in some order b 1 , b 2 , b 3 . . . . This lets us enumerate the elements of A , as follows: the first element of A is the first b i such that b i A , the second element of A is the second b i such that b i A , and so on. Since we can enumerate the elements of A in some order, A is countable. We have proved that if B is countable, then A is countable; by contrapositive, this proves that if A is uncountable, then B is uncountable.

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• Fall '15
• Soma Chaudhuri
• Natural number, Countable set, Georg Cantor, decimal representation

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