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Counting - Counting Counting Countable Set Permutations and...

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Counting
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Counting Countable Set Permutations and combinations Pigeonhole Principle Recurrence Relations
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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.
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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?
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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.
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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 . 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 0 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 0 ...
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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
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h °: » “2± h °: » “2± h °: » v A 0 B 0 0 - A 0 B 0 0 0 0 A 0 B - A B, 0 A B A B 0 A 0 B - v 0 A B 0 0 0 A 0 B - v ”R d
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( ‚ °: » ) ± ; ²X u3± x ³± u3± y7 |±±± F ´ u3± y7 |±± F ± ± µ S C ( C ) ‚ ò& » “2± 0, -1, 1, -2, 2, -3, 3, -4, ......
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± W (0,1) L -v 0 f : (0,1) R : f ( x ) =tg( π x- ) a,b( a < b ), [0,1] 0 [a,b] 0 0 f : [0,1] [ a , b ]: f ( x ) =( b - a ) x + a (7 v d u3± ) 2 π
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Some interesting words g æ : » “2± ± ; ² X u 3±±±±±±±±± ± ; ² X u3±±±±±±±±±± o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ° · : » “2 ±± ±±±±±x ² k x ± ± : » k +1 : » ª 1 0 0 ± ± > ±
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Permutations Theorem1,2: Multiplication principle Example: Let A be a set. |A|=n. how many subsets does A have?
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