3.3 The Characteristic function of the set  function from universal set to {0,1}  Definition 3.6: Let U be the universal set, and let A  U. The characteristic function of A is defined as a function from U to {0,1} by the following:

 Theorem 3.12: Let A and B be subsets of the universal set. Then, for any x  U, we have  (1)  A (x)  0 if only if A=   (2)  A (x)  1 if only if A=U;  (3)  A (x) ≦  B (x) if only if A  B;  (4)  A (x)   B (x) if only if A=B;  (5)  A∩B (x)=  A (x)  B (x);  (6)  A ∪ B (x)=  A (x)+  B (x)-  A∩B (x);

3.4 Cardinality  Definition 3.7: The empty set is a finite set of cardinality 0. If there is a one-to-one correspondence between A and the set {0,1,2,3,…, n-1}, then A is a finite set of cardinality n.  Definition 3.8: A set A is countably infinite if there is a one-to-one correspondence between A and the set N of natural number. We write |A|=|N|=  0. A set is countable if it is finite or countably infinite.

 Example: The set Z is countably infinite  Proof: f:N→Z,for any n  N,

 The set Q of rational number is countably infinite, i.e. |Q|=|N|=  0.  |[0 ， 1]|   0.  Theorem 3.13: The R of real numbers is not countably infinite. And |R|=|[0 ， 1]|.  Theorem 3.14: The power set P(N) of the set N of natural number is not countably infinite. And |P(N)|=|R|=  1.  Theorem 3.15(Cantor’s Theorem): For any set, the power set of X is cardinally larger than X.  N, P (N),P (P (N)),…

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.

 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

II Introductory Combinatorics Chapter 4 Introductory Combinatorics Counting

 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

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

4.1 Pigeonhole principle  Dirichlet,  shoebox principle

4.1.1 Pigeonhole principle :Simple Form  If n pigeons are assigned to m pigeonholes, and m<n, then at least one pigeonhole contains two or more pigeons.  Theorem 4.1: If n+1 objects are put into n boxes, then at least one box contain tow or more of the objects.

 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. 2ka2ka  2 r  a and 2 s  a

 Example 4:Given n integers a 1,a 2,…,a n, there exist integers k and l with 0  k<l  n such that a k+1 +a k+2 +…+a l is divisible by n.  a 1, a 1 +a 2, a 1 +a 2 +a 3,…,a 1 +a 2 +…+a n.  Example 5:A chess master who has 11 weeks to prepare for a tournament decides to play at least one game every day but, in order not to tire himself, he decides not to play more than 12 games during any calendar week. Show that there exists a succession of (consecutive) days during which the chess master will have played exactly 21 games.

 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

4.1.2 Pigeonhole principle:Strong Form  Theorem 4.2: Let q 1,q 2,…,q n be positive integers. If q 1 +q 2 +…+q n -n+1 objects are put into n boxes, then either the first box contains at least q 1 objects, or the second box contains at least q 2 objects, …, or the nth box contains at least q n objects.  Proof:Suppose that we distribute q 1 +q 2 +…+q n -n+1 objects among n boxes.

 (1)If n(r-1)+1 objects are put into n boxes, then at least one of the boxes contains r or more of the objects. Equivalently,  (2)If the average of n non-negative integers m 1,m 2,…,m n is greater than r-1: (m 1 +m 2 +…+m n )/n>r-1, then at least one of the integers is greater than or equal to r.  Proof:(1)q 1 =q 2 =…=q n =r  q 1 +q 2 +…+q n -n+1=rn-n+1=(r-1)n+1, then at least one of the boxes contains r or more of the objects 。  (2)(m 1 +m 2 +…+m n )>(r-1)n ，  (m 1 +m 2 +…+m n )≥(r-1)n +1

 Example 6:Two disks, one smaller than the other, are each divided into 200 congruent sectors. In the larger disk 100 of the sectors are chosen arbitrarily and painted red; the other 100 of the sectors are painted blue. In the smaller disk each sector is painted either red or blue with no stipulation on the number of red and blue sectors. The small disk is then placed on the larger disk so that their centers coincide. Show that it is possible to align the two disks so that the number of sectors of the small disk whose color matches the corresponding sector of the large disk is at least 100.  if the large disk is fixed in place  there are 200 possible positions for the small disk such that each sector of the small disk is contained in a sector of the large disk.  color matches the corresponding  20000/200=100>100-1  Position with at least 100 color matches

 Example 7:Show that every sequence a 1,a 2,…,a n 2 +1 of n 2 +1 real numbers contains either an increasing subsequence of length n+1 or a decreasing subsequence of length n+1.  Proof:We suppose that there is no increasing subsequence of length n+1 and show that there must be a decreasing subsequence of length n+1.  For each k=1,2, …, n 2 +1 let m k be the length of the longest increasing subsequence which begins with a k.  Let m k 1 =m k 2 =…=m k (n+1) (1  k 1 <k 2 <…<k n+1  n 2 +1).  m k 1 a k 1, m k 2 a k 2, … m k (n+1) a k (n+1),  a k 1,a k 2,…,a k (n+1) (1  k 1 <k 2 <…<k n+1  n 2 +1)  Now show that is a decreasing subsequence

 Permutations of sets, 3.1 P79-81  Combinations of sets,3.2 P83-84  Permutations and Combinations of multisets P82,P85-86

 Exercise P181 4, P90 3,7  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.