Download Chapter 3. Introductory Combinatorics

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,
 1
 2 n
f ( n)  
1
 (n  1)
2
n is even number
n is odd number
 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
P100(Sixth) P88(Fifth)
 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,and Generating function, and
Recurrence relations
4.1 Pigeonhole principle
 Dirichlet,1805-1859
 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
 2ra and 2sa
Example 4:Given n integers a1,a2,…,an, there exist
integers k and l with 0k<ln such that
ak+1+ak+2+…+al is divisible by n.
 a1, a1+a2, a1+a2+a3,…,a1+a2+…+an.
 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
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Exercise P103 3,7, 8,9 (Sixth) OR P90 3,7, 8,9(Fifth)
1.Prove:(1)|(0,1)|=|R| (2)|[0,1]|=|R|
2.From the integers 1,2,…,2n, we choose n+1 intergers.
Show that among the integers chosen there are two which
are relatively prime.
3.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.
4.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 P91 (Sixth) OR P79(Fifth)
circular permutation
Combinations of sets, P96 (Sixth) OR P83(Fifth)