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Unformatted text preview: Counting Counting Countable Set Permutations and combinations Pigeonhole Principle Recurrence Relations Countable Set A set A is countable if and only if we can arrange all of its elements in a linear list in a definite order. Definite means that we can specify the first, second, third element, and so on. If the list ended and with the n th element as its last element, it is finite. If the list goes on forever, it is infinite. Proof of Countability The set of all integers is countable. We can arrange all integer in a linear list as follows: 0,1,1,2,2,3,3,... that is: positive k is the (2 k +1)th element, and negative k is the 2 k th element in the list. Is the set of rational numbers countable? Set of Ordered Pairs The set of all objects with the form < i , j > is countable, where i , j are nonnegative integers. <0,0> <1,0> <2,0> <3,0> <4,0> <0,1> <1,1> <2,1> <3,1> <0,2> <1,2> <2,2> <0,3> <1,3> <0,4> ... ... ... ... < 0,0>, <0,1>, <1,0>, <0,2>, <1,1>, <2,0>, <0,3>, ...... So, the set of rational numbers is countable. Real Number Set Is Not Countable The proof by contradiction proceeds as follows: (1) Assume that the interval (0,1) is countably infinite. (2) We may then enumerate the numbers in this interval as a sequence , { r 1, r 2, r 3, ... } (3) Assume, for example, that the decimal expansions of the beginning of the sequence are as follows. r 1 = 0 . 1 0 5 1 1 0 ... r 2 = 0 . 4 1 3 2 0 4 3 ... r 3 = 0 . 8 2 4 5 0 2 6 ... r 4 = 0 . 2 3 3 1 2 6 ... r 5 = 0 . 4 1 0 7 2 4 6 ... r 6 = 0 . 9 9 3 7 8 3 8 ... r 7 = 0 . 0 1 0 5 1 3 ... Real Number Set Is Not Countable The digits we will consider are indicated in red From these digits we define the digits of x as follows. if the n th digit of r n is 0 then the n th digit of x is 1 if the n th digit of r n is not 0 then the n th digit of x is 0 For the example above this will result in the following decimal expansion. x = 0 . 1 0 0 1 0 0 1 ... Is x one of {r1,r2,}? So: (0,1) is not a countable set h&: 2 h&: 2 h&: v A 0 B 0 A 0 B A 0 B  A B, 0 A B A B 0 A 0 B v A B 0 0 A 0 B v R d C& ( : ) & ; X u3& x & u3& y7 &&& F u3& y7 && F & & S C ( C ) & 2& 0, 1, 1, 2, 2, 3, 3, 4, ...... & W (0,1) Lv f : (0,1) R : f ( x ) =tg( x ) a,b( a < b ), [0,1] 0 [a,b] 0 f : [0,1] [ a , b ]: f ( x ) =( b a ) x + a (7 v d u3& ) 2 Some interesting words g a : 2& & ; X u 3&&&&&&&&& & ; X u3&&&&&&&&&& o...
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This note was uploaded on 03/31/2010 for the course SE C0229 taught by Professor Tao during the Spring '08 term at Nanjing University.
 Spring '08
 Tao

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