3.3 The Characteristic function of the set

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• 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

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• 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 AU
• (3)?A(x)??B(x) if only if A?B
• (4)?A(x) ? ?B(x) if only if AB
• (5)?AnB(x)?A(x)??B(x)
• (6)?A?B(x)?A(x)?B(x)-?AnB(x)

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• 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
  AN?0. A set is countable if it is finite or
  countably infinite.

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• Example The set Z is countably infinite
• Proof fN?Z,for any n?N,

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• The set Q of rational number is countably
  infinite, i.e. QN?0.
• 0,1? ?0.
• Theorem 3.13 The R of real numbers is not
  countably infinite. And R0,1.
• 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(Cantors Theorem) For any set, the
  power set of X is cardinally larger than X.
• N, P (N),P (P (N)),

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• 3.5 Paradox
• 1.Russells paradox
• A?A, A?A?
• Russells paradox Let SAA?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.Cantors 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
4Introductory 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
• Generating function
• 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
  mltn, then at least one pigeonhole contains two or
  more pigeons.
• Theorem 4.1 If n1 objects are put into n
  boxes, then at least one box contain tow or more
  of the objects.

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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 3From the integers 1,2,,2n, we choose
  n1 intergers. Show that among the integers
  chosen there are two such that one of them is
  divisible by the other.
• 2k?a
• 2r?a and 2s?a

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• Example 4Given n integers a1,a2,,an, there
  exist integers k and l with 0?kltl?n such that
  ak1ak2al is divisible by n.
• a1, a1a2, a1a2a3,,a1a2an.
• Example 5A 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.

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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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4.1.2 Pigeonhole principleStrong Form

• Theorem 4.2 Let q1,q2,,qn be positive integers.
  If q1q2qn-n1 objects are put into n boxes,
  then either the first box contains at least q1
  objects, or the second box contains at least q2
  objects, , or the nth box contains at least qn
  objects.
• ProofSuppose that we distribute q1q2qn-n1
  objects among n boxes.

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• (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
  m1,m2,,mn is greater than r-1
  (m1m2mn)/ngtr-1, then at least one of the
  integers is greater than or equal to r.
• Proof(1)q1q2qnr
• q1q2qn-n1rn-n1(r-1)n1, then at least one
  of the boxes contains r or more of the objects?
• (2)(m1m2mn)gt(r-1)n,
• (m1m2mn)(r-1)n 1

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• Example 6Two 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
• 20000/200100gt100-1
• Position with at least 100 color matches

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• Example 7Show that every sequence a1,a2,,an21
  of n21 real numbers contains either an
  increasing subsequence of length n1 or a
  decreasing subsequence of length n1.
• ProofWe suppose that there is no increasing
  subsequence of length n1 and show that there
  must be a decreasing subsequence of length n1.
• For each k1,2, , n21 let mk be the length of
  the longest increasing subsequence which begins
  with ak.
• Let mk1mk2mk(n1)(1?k1ltk2ltltkn1?n21).
• mk1 ak1, mk2 ak2, mk(n1) ak(n1),
• ak1,ak2,,ak(n1)(1?k1ltk2ltltkn1?n21)
• Now show that is a decreasing subsequence

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

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• Exercise P181 4, P90 3,7
• 1.From the integers 1,2,,2n, we choose n1
  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 n2 integers there
  exist two of them whose sum, or else whose
  difference is divisible by 2n.