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3.5 Paradox 1.Russell’s paradox A A, A A 。 Russell’s paradox: Let S={A|A A}. The question is, does S S? i.e. S S or S S? If S S, If S S, The statements " S S " and " S S " cannot both be true, thus the contradiction.

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2.Cantor’s paradox 1899,Cantor's paradox, sometimes called the paradox of the greatest cardinal, expresses what its second name would imply--that there is no cardinal larger than every other cardinal. Let S be the set of all sets. |S|? |P (S)| or |P (S)|? |(S)| The Third Crisis in Mathematics

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II Introductory Combinatorics Chapter 4 Introductory Combinatorics Counting

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Combinatorics, is an important part of discrete mathematics. Techniques for counting are important in computer science, especially in the analysis of algorithm. sorting,searching combinatorial algorithms Combinatorics

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existence counting construction optimization existence :Pigeonhole principle Counting techniques for permutation and combinations,and Generating function, and Recurrence relations

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4.1 Pigeonhole principle Dirichlet,1805-1859 shoebox principle

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4.1.1 Pigeonhole principle :Simple Form If n pigeons are assigned to m pigeonholes, and m

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Example 1: Among 13 people there are two who have their birthdays in the same month. Example 2: Among 70 people there are six who have their birthdays in the same month. Example 3:From the integers 1,2,…,2n, we choose n+1 intergers. Show that among the integers chosen there are two such that one of them is divisible by the other. 2ka2ka 2 r a and 2 s a

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Example 4:Given n integers a 1,a 2,…,a n, there exist integers k and l with 0 k

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Concerning Application 5, Show that there exists a succession of (consecutive) days during which the chess master will have played exactly 22 games. (1)The chess master plays few than 12 games at least one week (2)The chess master plays exactly 12 games each week

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Exercise P90 3,7, 8,9 1.From the integers 1,2,…,2n, we choose n+1 intergers. Show that among the integers chosen there are two which are relatively prime. 2.A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers. 3.Show that for any given n+2 integers there exist two of them whose sum, or else whose difference is divisible by 2n. Next: Permutations of sets P79-81, 3.1 circular permutation Combinations of sets,3.2 P83-84

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