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  • 7/27/2019 Discrete Mathematics - Dr. J. Saxl - University of Cambridge

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    Discrete Mathematics

    Dr. J. Saxl

    Michlmas 1995

    These notes are maintained by Paul Metcalfe.

    Comments and corrections to [email protected] .

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    Revision: 2.3

    Date: 1999/10/21 11:21:05

    The following people have maintained these notes.

    date Paul Metcalfe

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    Contents

    Introduction v

    1 Integers 1

    1.1 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 The division algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.3 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.4 Applications of the Euclidean algorithm . . . . . . . . . . . . . . . . 4

    1.4.1 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . 51.5 Complexity of Euclidean Algorithm . . . . . . . . . . . . . . . . . . 6

    1.6 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.6.1 Uniqueness of prime factorisation . . . . . . . . . . . . . . . 7

    1.7 Applications of prime factorisation . . . . . . . . . . . . . . . . . . . 7

    1.8 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.9 Solving Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.9.1 Systems of congruences . . . . . . . . . . . . . . . . . . . . 9

    1.10 E ulers Phi Function . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    1.10.1 Public Key Cryptography . . . . . . . . . . . . . . . . . . . . 10

    2 Induction and Counting 11

    2.1 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . 11

    2.2 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    2.3 Strong Principle of Mathematical Induction . . . . . . . . . . . . . . 12

    2.4 Recursive Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    2.5 Selection and Binomial Coefficients . . . . . . . . . . . . . . . . . . 13

    2.5.1 Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    2.5.2 Some more identities . . . . . . . . . . . . . . . . . . . . . . 14

    2.6 Special Sequences of Integers . . . . . . . . . . . . . . . . . . . . . 16

    2.6.1 Stirling numbers of the second kind . . . . . . . . . . . . . . 16

    2.6.2 Generating Functions . . . . . . . . . . . . . . . . . . . . . . 16

    2.6.3 Catalan numbers . . . . . . . . . . . . . . . . . . . . . . . . 17

    2.6.4 Bell numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    2.6.5 Partitions of numbers and Young diagrams . . . . . . . . . . 18

    2.6.6 Generating function for self-conjugate partitions . . . . . . . 20

    3 Sets, Functions and Relations 23

    3.1 Sets and indicator functions . . . . . . . . . . . . . . . . . . . . . . . 23

    3.1.1 De Morgans Laws . . . . . . . . . . . . . . . . . . . . . . . 24

    3.1.2 Inclusion-Exclusion Principle . . . . . . . . . . . . . . . . . 24

    iii

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    iv CONTENTS

    3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    3.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    3.3.1 Stirling numbers of the first kind . . . . . . . . . . . . . . . . 27

    3.3.2 Transpositions and shuffles . . . . . . . . . . . . . . . . . . . 27

    3.3.3 Order of a permutation . . . . . . . . . . . . . . . . . . . . . 28

    3.3.4 Conjugacy classes in Sn

    . . . . . . . . . . . . . . . . . . . . 28

    3.3.5 Determinants of an n n matrix . . . . . . . . . . . . . . . . 283.4 Binary Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    3.5 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    3.5.1 Products of posets . . . . . . . . . . . . . . . . . . . . . . . 30

    3.5.2 Eulerian Digraphs . . . . . . . . . . . . . . . . . . . . . . . 30

    3.6 Countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    3.7 Bigger sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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    Introduction

    These notes are based on the course Discrete Mathematics given by Dr. J. Saxl in

    Cambridge in the Michlmas Term 1995. These typeset notes are totally unconnected

    with Dr. Saxl.

    Other sets of notes are available for different courses. At the time of typing these

    courses were:

    Probability Discrete Mathematics

    Analysis Further Analysis

    Methods Quantum MechanicsFluid Dynamics 1 Quadratic Mathematics

    Geometry Dynamics of D.E.s

    Foundations of QM Electrodynamics

    Methods of Math. Phys Fluid Dynamics 2

    Waves (etc.) Statistical Physics

    General Relativity Dynamical Systems

    Physiological Fluid Dynamics Bifurcations in Nonlinear Convection

    Slow Viscous Flows Turbulence and Self-Similarity

    Acoustics Non-Newtonian Fluids

    Seismic Waves

    They may be downloaded from

    http://www.istari.ucam.org/maths/ or

    http://www.cam.ac.uk/CambUniv/Societies/archim/notes.htm

    or you can email [email protected] to get a copy of the

    sets you require.

    v

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    Copyright (c) The Archimedeans, Cambridge University.

    All rights reserved.

    Redistribution and use of these notes in electronic or printed form, with or without

    modification, are permitted provided that the following conditions are met:

    1. Redistributions of the electronic files must retain the above copyright notice, thislist of conditions and the following disclaimer.

    2. Redistributions in printed form must reproduce the above copyright notice, this

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    3. All materials derived from these notes must display the following acknowledge-

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    This product includes notes developed by The Archimedeans, Cambridge

    University and their contributors.

    4. Neither the name of The Archimedeans nor the names of their contributors may

    be used to endorse or promote products derived from these notes.

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    THESE NOTES ARE PROVIDED BY THE ARCHIMEDEANS AND CONTRIB-

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    Chapter 1

    Integers

    Notation. The natural numbers, which we will denote by N , are

    f 1 2 3 : : : g :

    The integers Z are

    f : : : ; 2 ; 1 0 1 2 : : : g :

    We will also use the non-negative integers, denoted either byN

    0

    orZ

    +

    , which isN

    f 0 g . There are also the rational numbers Q and the real numbers R .

    Given a set S , we write x 2 S if x belongs to S , and x =2 S otherwise.

    There are operations + and on Z . They have certain nice properties which we

    will take for granted. There is also ordering.N

    is said to be well-ordered, which

    means that every non-empty subset of N has a least element. The principle of induction

    follows from well-ordering.

    Proposition (Principle of Induction). Let P ( n ) be a statement about n for each n 2

    N . Suppose P ( 1 ) is true and P ( k ) true implies thatP ( k + 1 ) is true for each k 2 N .

    ThenP

    is true for alln

    .

    Proof. SupposeP

    is not true for alln

    . Then consider the subsetS

    ofN

    of all numbers

    k for which P is false. Then S has a least element l . We know that P ( l ; 1 ) is true

    (sincel > 1

    ), so thatP ( l )

    must also be true. This is a contradiction andP

    holds for all

    n .

    1.1 Division

    Given two integersa

    ,b 2 Z

    , we say thata

    dividesb

    (and writea j b

    ) ifa 6= 0

    andb = a

    q for some q

    2Z ( a is a divisor of b ). a is a proper divisorof b if a is not

    1

    or b

    .

    Note. If a j b and b j c then a j c , for if b = q1

    a and c = q2

    b for q1

    , q2

    2 Z then

    c = ( q

    1

    q

    2

    ) a

    . Ifd j a

    andd j b

    thend j a x + b y

    . The proof of this is left as an exercise.

    1

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    1.3. THE EUCLIDEAN ALGORITHM 3

    a = q

    1

    b + r

    1

    with0 < r

    1

    < b

    b = q

    2

    r

    1

    + r

    2

    with 0 < r2

    < r

    1

    r

    1

    = q

    3

    r

    2

    + r

    3

    with0 < r

    3

    < r

    2

    ...

    r

    i ; 2

    = q

    i

    r

    i ; 1

    + r

    i

    with0 < r

    i

    < r

    i ; 1

    ...

    r

    n ; 3

    = q

    n ; 1

    r

    n ; 2

    + r

    n ; 1

    with0 < r

    n

    < r

    n ; 1

    r

    n ; 2

    = q

    n

    r

    n ; 1

    + 0 :

    This process must terminate as b > r1

    > r

    2

    > > r

    n ; 1

    > 0 . Using Lemma

    (1.2),( a b ) = ( b r

    1

    ) = = ( r

    n ; 2

    r

    n ; 1

    ) = r

    n ; 1

    . So( a b )

    is the last non-zero

    remainder in this process.

    We now wish to findx

    0

    andy

    0

    2 Zwith

    ( a b ) = a x

    0

    + b y

    0

    . We can do this by

    backsubstitution.

    2 1 = 6 3

    ;1

    4 2

    = 6 3 ; ( 2 3 1 ; 3 6 3 )

    = 4 6 3 ; 2 3 1

    = 4 ( 5 2 5 ; 2 2 3 1 ) ; 2 3 1

    = 4 5 2 5 ; 9 2 3 1 :

    This works in general but can be confusing and wasteful. These numbers can be

    calculated at the same time as ( a b ) if we know we shall need them.

    We introduceA

    i

    andB

    i

    . We putA

    ; 1

    = B

    0

    = 0

    andA

    0

    = B

    ; 1

    = 1

    . We

    iteratively define

    A

    i

    = q

    i

    A

    i ; 1

    + A

    i ; 2

    B

    i

    = q

    i

    B

    i ; 1

    + B

    i ; 2

    :

    Now consider a Bj

    ;b A

    j

    .

    Lemma 1.3.

    a B

    j

    ; b A

    j

    = ( ; 1 )

    j + 1

    r

    j

    :

    Proof. We shall do this using strong induction. We can easily see that (1.3) holds for

    j = 1 and j = 2 . Now assume we are at i

    2 and we have already checked that

    r

    i ; 2

    = ( ; 1 )

    i ; 1

    ( a B

    i ; 2

    ; b A

    i ; 2

    )

    andr

    i ; i

    = ( ; 1 )

    i

    ( a B

    i ; 1

    ; b A

    i ; 1

    )

    . Now

    r

    i

    = r

    i ; 2

    ; q

    i

    r

    i ; 1

    = ( ; 1 )

    i ; 1

    ( a B

    i ; 2

    ; b A

    i ; 2

    ) ; q

    i

    ( ; 1 )

    i

    ( a B

    i ; 1

    ; b A

    i ; 1

    )

    = ( ; 1 )

    i + 1

    ( a B

    i

    ; b A

    i

    ) , using the definition of Ai

    and Bi

    .

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    4 CHAPTER 1. INTEGERS

    Lemma 1.4.

    A

    i

    B

    i + 1

    ; A

    i + 1

    B

    i

    = ( ; 1 )

    i

    Proof. This is done by backsubstitution and using the definition of Ai

    andB

    i

    .

    An immediate corollary of this is that ( Ai

    B

    i

    ) = 1 .

    Lemma 1.5.

    A

    n

    =

    a

    ( a b )

    B

    n

    =

    b

    ( a b )

    :

    Proof. (1.3) fori = n

    givesa B

    n

    = b A

    n

    . Therefore a( a b )

    B

    n

    =

    b

    ( a b )

    A

    n

    . Now a( a b )

    and b( a b )

    are coprime. An

    and Bn

    are coprime and thus this lemma is therefore an

    immediate consequence of the following theorem.

    Theorem 1.2. If d j c e and ( c d ) = 1 then d j e .

    Proof. Since ( c d ) = 1 we can write 1 = c x + d y for some x , y 2 Z . Then e =

    e c x + e d y

    andd j e

    .

    Definition. The least common multiple (lcm) ofa and b (written a b ] ) is the integerl

    such that

    1.a j l

    andb j l

    ,

    2. ifa j l

    0 andb j l

    0 thenl j l

    0 .

    It is easy to show that a b ] =

    a b

    ( a b )

    .

    1.4 Applications of the Euclidean algorithm

    Takea

    ,b

    andc 2 Z

    . Suppose we want to find all the solutionsx

    ,y 2 Z

    ofa x + b y = c

    .

    A necessary condition for a solution to exist is that ( a b ) j c , so assume this.

    Lemma 1.6. If ( a b ) j c then a x + b y = c has solutions in Z .

    Proof. Take x 0 and y 0 2 Z such that a x 0 + b y 0 = ( a b ) . Then if c = q ( a b ) then if

    x

    0

    = q x

    0 andy

    0

    = q y

    0 ,a x

    0

    + b y

    0

    = c

    .

    Lemma 1.7. Any other solution is of the form x = x0

    +

    b k

    ( a b )

    ,y = y

    0

    ;

    a k

    ( a b )

    for

    k 2 Z .

    Proof. These certainly work as solutions. Now supposex

    1

    andy

    1

    is also a solution.

    Then a( a b )

    ( x

    0

    ; x

    1

    ) = ;

    b

    ( a b )

    ( y

    0

    ; y

    1

    )

    . Since a( a b )

    and b( a b )

    are coprime we havea

    ( a b )

    j ( y

    0

    ; y

    1

    )

    and b( a b )

    j ( x

    0

    ; x

    1

    )

    . Say thaty

    1

    = y

    0

    ;

    a k

    ( a b )

    ,k 2 Z

    . Then

    x

    1

    = x

    0

    +

    b k

    ( a b )

    .

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    1.4. APPLICATIONS OF THE EUCLIDEAN ALGORITHM 5

    1.4.1 Continued Fractions

    We return to5 2 5

    and2 3 1

    . Note that

    5 3 5

    2 3 1

    = 2 +

    6 3

    2 3 1

    = 2 +

    1

    2 3 1

    6 3

    = 2 +

    1

    3 +

    4 2

    6 3

    = 2 +

    1

    3 +

    1

    1 +

    1

    2

    :

    Notation.5 3 5

    2 3 1

    = 2 +

    1

    3 +

    1

    1 +

    1

    2

    = 2 3 1 2 ] = 2 3 1 2 :

    Note that2

    ,3

    ,1

    and2

    are just theq

    i

    s in the Euclidean algorithm. The rationala

    b

    > 0 is written as a continued fraction

    a

    b

    = q

    1

    +

    1

    q

    2

    +

    1

    q

    3

    +

    : : :

    1

    q

    n

    with all theq

    i

    2 N

    0

    ,q

    i

    1for

    1 < i < n

    andq

    n

    2.

    Lemma 1.8. Every rational ab

    with a and b 2 N has exactly one expression in this

    form.

    Proof. Existance follows immediately from the Euclidean algorithm. As for unique-ness, suppose that

    a

    b

    = p

    1

    +

    1

    p

    2

    +

    1

    p

    3

    +

    : : :

    1

    p

    m

    with the pi

    s as before. Firstly p1

    = q

    1

    as both are equal to b ab

    c . Since 1p

    2

    +

    1

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    6 CHAPTER 1. INTEGERS

    1.5 Complexity of Euclidean Algorithm

    Givena

    andb

    , how many steps does it take to find( a b )

    . The Euclidean algorithm is

    good.

    Proposition. The Euclidean algorithm will find( a b ) , a > b in fewer than 5 d ( b ) steps,

    whered ( b )

    is the number of digits ofb

    in base1 0

    .

    Proof. We look at the worst case scenario. What are the smallest numbers needingn

    steps. In this case qi

    = 1 for 1 i < n and qn

    = 2 . Using these qi

    s to calculate An

    andB

    n

    we find the Fibonacci numbers, that is the numbers such thatF

    1

    = F

    2

    = 1

    ,

    F

    i + 2

    = F

    i + 1

    + F

    i

    . We getA

    n

    = F

    n + 2

    andB

    n

    = F

    n + 1

    . So ifb < F

    n + 1

    then fewer

    thann

    steps will do. Ifb

    hasd

    digits then

    b 1 0

    d

    ; 1

    1

    p

    5

    1 +

    p

    5

    2

    !

    5 d + 2

    ; 1 < F

    5 d + 2

    as

    F

    n

    =

    1

    p

    5

    "

    1 +

    p

    5

    2

    !

    n

    ;

    1 ;

    p

    5

    2

    !

    n

    #

    : This will be shown later.

    1.6 Prime Numbers

    A natural number p is a prime iff p > 1 and p has no proper divisors.

    Theorem 1.3. Any natural number n > 1 is a prime or a product of primes.

    Proof. If n is a prime then we are finished. If n is not prime then n = n1

    n

    2

    with n1

    andn

    2 proper divisors. Repeat withn

    1 andn

    2 .

    Theorem 1.4 (Euclid). There are infinitely many primes.

    Proof. Assume not. Then letp

    1

    p

    2

    : : : p

    n

    be all the primes. Form the numberN =

    p

    1

    p

    2

    : : : p

    n

    + 1 . Now N is not divisible by any of the pi

    but N must either be prime

    or a product of primes, giving a contradiction.

    This can be made more precise. The following argumentof Erdos shows that thek

    th

    smallest prime pk

    satisfies pk

    4

    k ; 1

    + 1 . Let M be an integer such that all numbers

    Mcan be written as the product of the powers of the first

    k

    primes. So any such

    number can be written

    m

    2

    p

    i

    1

    1

    p

    i

    2

    2

    : : : p

    i

    k

    k

    with i1

    : : : i

    k

    2 f 0 1 g . Now m p

    M , so there are at mostp

    M 2

    k possible num-

    bers less thanM

    . HenceM 2

    k

    p

    M

    , orM 4

    k . Hencep

    k + 1

    4

    k

    + 1

    .

    A much deeper result (which will not be proved in this course!) is the Prime Num-

    ber Theorem, that pk

    k l o g k .

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    1.7. APPLICATIONS OF PRIME FACTORISATION 7

    1.6.1 Uniqueness of prime factorisation

    Lemma 1.9. If p j a b , a b 2 N then p j a and/or p j b .

    Proof. If p - a then ( p a ) = 1 and so p j b by theorem (1.2).

    Theorem 1.5. Every natural number > 1 has a unique expression as the product ofprimes.

    Proof. The existence part is theorem (1.3). Now suppose n = p1

    p

    2

    : : : p

    k

    = q

    1

    q

    2

    : : : q

    l

    with thep

    i

    s andq

    j

    s primes. Thenp

    1

    j q

    1

    : : : q

    l

    , sop

    1

    = q

    j

    for somej

    . By renumber-

    ing (if necessary) we can assume that j = 1 . Now repeat with p2

    : : : p

    k

    and q2

    : : : q

    l

    ,

    which we know must be equal.

    There are perfectly nice algebraic systems where the decomposition into primes

    is not unique, for instanceZ

    p

    ; 5

    = f a + b

    p

    ; 5 : a b 2 Z g, where

    6 = ( 1 +

    p

    ; 5 ) ( 1 ;

    p

    ; 5 ) = 2 3 and 2 3 and 1 p

    ; 5 are each prime. Or alternatively,

    2 Z = fall even numbers

    g, where prime means not divisible by

    4

    .

    1.7 Applications of prime factorisation

    Lemma 1.10. If n 2 N is not a square number thenp

    n is irrational.

    Proof. Supposep

    n =

    a

    b

    , with( a b ) = 1

    . Thenn b

    2

    = a

    2 . Ifb > 1

    then letp

    be a

    prime dividingb

    . Thusp j a

    2 and sop j a

    , which is impossible as( a b ) = 1

    . Thus

    b = 1

    andn = a

    2 .

    This lemma can also be stated: ifn 2 N

    withp

    n 2 Qthen

    p

    n 2 N.

    Definition. A real number is algebraic if it satisfies a polynomial equation with co-efficients in

    Z

    .

    Real numbers which are not algebraic are transcendental (for instance and e ).

    Most reals are transcendental.

    If the rational ab

    ( with( a b ) = 1

    ) satisfies a polynomial with coefficients inZ

    then

    c

    n

    a

    n

    + c

    n ; 1

    a

    n ; 1

    b + : : : b

    n

    c

    0

    = 0

    sob j c

    n

    anda j c

    0

    . In particular ifc

    n

    = 1

    thenb = 1

    , which is stated as algebraic

    integers which are rational are integers.

    Note that if a = p 11

    p

    2

    2

    : : : p

    k

    k

    and b = p

    1

    1

    p

    2

    2

    : : : p

    k

    k

    with i

    i

    2 N

    0

    then

    ( a b ) = p

    1

    1

    p

    2

    2

    : : : p

    k

    k

    and a b ] = p 11

    p

    2

    2

    : : : p

    k

    k

    , i

    = m i n

    f

    i

    i

    gand

    i

    =

    m a x f

    i

    i

    g.

    Major open problems in the area of prime numbers are the Goldbach conjecture

    (every even number greater than two is the sum of two primes) and the twin primes

    conjecture (there are infinitely many prime pairs p and p + 2 ).

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    8 CHAPTER 1. INTEGERS

    1.8 Modular Arithmetic

    Definition. Ifa

    andb 2 Z

    ,m 2 N

    we say thata

    andb

    are congruent mod(ulo)m

    if m j a ; b . We write a b ( m o d m ) .

    It is a bit like=

    but less restrictive. It has some nice properties:

    a a ( m o d m ) ,

    if

    a b ( m o d m )then

    b a ( m o d m ),

    if

    a b ( m o d m )and

    b c ( m o d m )then

    a c ( m o d m ).

    Also, ifa

    1

    b

    1

    ( m o d m )

    anda

    2

    b

    2

    ( m o d m )

    a

    1

    + a

    2

    b

    1

    + b

    2

    ( m o d m ) ,

    a

    1

    a

    2

    b

    1

    a

    2

    b

    1

    b

    2

    ( m o d m )

    .

    Lemma 1.11. For a fixed m2

    N , each integer is congruent to precisely one of the

    integers

    f 0 1 : : : m ; 1 g :

    Proof. Take a 2 Z . Then a = q m + r for q r 2 Z and 0 r < m . Then a r

    ( m o d m )

    .

    If0 r

    1

    < r

    2

    < m

    then0 < r

    2

    ; r

    1

    < m

    , som - r

    2

    ; r

    1

    and thusr

    1

    6 r

    2

    ( m o d m ) .

    Example. No integer congruent to 3 ( m o d 4 ) is the sum of two squares.

    Solution. Every integer is congruent to one of0 1 2 3 ( m o d 4 )

    . The square of any

    integer is congruent to0

    or1 ( m o d 4 )

    and the result is immediate.

    Similarly, using congruence modulo 8, no integer congruent to7 ( m o d 8 )

    is the

    sum of 3 squares.

    1.9 Solving Congruences

    We wish to solve equations of the forma x b ( m o d m )

    givena b 2 Z

    andm 2 N

    for x 2 Z . We can often simplify these equations, for instance 7 x 3 ( m o d 5 )

    reduces tox 4 ( m o d 5 )

    (since2 1 1

    and9 4 ( m o d 5 )

    ).

    This equations are not always soluble, for instance 6 x 4 ( m o d 9 ) , as 9 - 6 x ; 4

    for anyx 2 Z

    .

    How to do it

    The equation a x b ( m o d m ) can have no solutions if( a m ) - b since then m - a x ; b

    for anyx 2 Z

    . So assume that( a m ) j b

    .We first consider the case ( a m ) = 1 . Then we can find x

    0

    and y0

    2 Z such

    thata x

    0

    + m y

    0

    = b

    (use the Euclidean algorithm to getx

    0 andy

    0

    2 Zsuch that

    a x

    0

    + m y

    0

    = 1 ). Then put x0

    = b x

    0 so a x0

    b ( m o d m ) . Any other solution is

    congruent tox

    0

    ( m o d m )

    , asm j a ( x

    0

    ; x

    1

    )

    and( a m ) = 1

    .

    So if ( a m ) = 1 then a solution exists and is unique modulo m .

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    1.10. EULERS PHI FUNCTION 9

    1.9.1 Systems of congruences

    We consider the system of equations

    x a m o d m

    x b m o d n :

    Our main tool will be the Chinese Remainder Theorem.

    Theorem 1.6 (Chinese Remainder Theorem). Assume m n 2 N are coprime and

    leta b 2 Z

    . Then9 x

    0

    satisfying simultaneouslyx

    0

    a ( m o d m )and

    x

    0

    b

    ( m o d n ) . Moreover the solution is unique up to congruence modulo m n .

    Proof. Writec m + d n = 1

    withm n 2 Z

    . Thenc m

    is congruent to0

    modulom

    and 1 modulo n . Similarly d n is congruent to 1 modulo m and 0 modulo n . Hence

    x

    0

    = a d n + b c m

    satifiesx

    0

    a ( m o d m )and

    x

    0

    b ( m o d n ). Any other solution

    x

    1

    satisfies x0

    x

    1

    both modulo m and modulo n , so that since ( m n ) = 1 , m n j

    x

    0

    ; x

    1

    andx

    1

    x

    0

    ( m o d m n )

    .

    Finally, if1

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    10 CHAPTER 1. INTEGERS

    An immediate corollary of this is that for anyn 2 N

    ,

    ( n ) = n

    Y

    p j n

    p prime

    1

    ;

    1

    p

    :

    Theorem 1.8 (Fermat-Euler Theorem). Takea

    ,m 2 N

    such that( a m ) = 1

    . Thena

    ( m )

    1 ( m o d m ).

    Proof. Multiply each residue ri

    by a and reduce modulo m . The ( m ) numbers thus

    obtained are prime tom

    and are all distinct. So the ( m )

    new numbers are just

    r

    1

    : : : r

    ( m )

    in a different order. Therefore

    r

    1

    r

    2

    : : : r

    ( m )

    a r

    1

    a r

    2

    : : : a r

    ( m )

    ( m o d m )

    a

    ( m )

    r

    1

    r

    2

    : : : r

    ( m )

    ( m o d m ) :

    Since( m r

    1

    r

    2

    : : : r

    ( m )

    ) = 1

    we can divide to obtain the result.

    Corollary (Fermats Little Theorem). If p is a prime and a2

    Z such that p - a then

    a

    p ; 1

    1 ( m o d p ) .

    This can also be seen as a consequence of Lagranges Theorem, sinceU

    m

    is a group

    under multiplication modulo m .

    Fermats Little Theorem can be used to check thatn 2 N

    is prime. If9 a

    coprime

    to n such that a n ; 1 6 1 ( m o d n ) then n is not prime.

    1.10.1 Public Key Cryptography

    Private key cryptosystems rely on keeping the encoding key secret. Once it is known

    the code is not difficult to break. Public key cryptography is different. The encoding

    keys are public knowledge but decoding remains impossible except to legitimate

    users. It is usually based of the immense difficulty of factorising sufficiently large

    numbers. At present 150 200 digit numbers cannot be factorised in a lifetime.

    We will study the RSA system of Rivest, Shamir and Adleson. The user A (forAlice) takes two large primes

    p

    A

    andq

    A

    with> 1 0 0

    digits. She obtainsN

    A

    = p

    A

    q

    A

    and chooses at random

    A

    such that(

    A

    ( N

    A

    ) ) = 1

    . We can ensure thatp

    A

    ; 1and

    q

    A

    ;1 have few factors. Now A publishes the pair N

    A

    and A

    .

    By some agreed methodB

    (for Bob) codes his message for Alice as a sequence of

    numbers M < NA

    . Then B sends A the number M A ( m o d NA

    ) . When Alice wants

    to decode the message she choosesd

    A

    such thatd

    A

    A

    1 ( m o d ) ( N

    A

    )

    . Then

    M

    A

    d

    A

    M ( m o d N

    A

    ) since M ( N A ) 1 . No-one else can decode messages to

    Alice since they would need to factoriseN

    A

    to obtain ( N

    A

    )

    .

    If Alice and Bob want to be sure who is sending them messages, then Bob could

    send AliceE

    A

    ( D

    B

    ( M ) )

    and Alice could applyE

    B

    D

    A

    to get the message if its

    from Bob.

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    Chapter 2

    Induction and Counting

    2.1 The Pigeonhole Principle

    Proposition (The Pigeonhole Principle). If n m + 1 objects are placed into n boxes

    then some box contains more thanm

    objects.

    Proof. Assume not. Theneach box has at most m objects so the total number of objects

    isn m

    a contradiction.

    A few examples of its use may be helpful.

    Example. In a sequence of at leastk l + 1 distinct numbers there is either an increasing

    subsequence of length at leastk + 1

    or a decreasing subsequence of length at leastl + 1

    .

    Solution. Let the sequence be c1

    c

    2

    : : : c

    k l + 1

    . For each position let ai

    be the length

    of the longest increasing subsequence starting withc

    i

    . Letd

    j

    be the length of the

    longest decreasing subsequence starting with cj

    . If ai

    k and di

    l then there are

    only at mostk l

    distinct pairs( a

    i

    d

    j

    )

    . Thus we havea

    r

    = a

    s

    andd

    r

    = d

    s

    for some

    1 r < s k l + 1

    . This is impossible, for ifc

    r

    < c

    s thena

    r

    > a

    s and ifc

    r

    > c

    s

    thend

    r

    > d

    s

    . Hence either somea

    i

    > k

    ord

    j

    > l

    .

    Example. In a group of 6 people any two are either friends or enemies. Then there

    are either 3 mutual friends or 3 mutual enemies.

    Solution. Fix a person X . Then X has either 3 friends or 3 enemies. Assume the

    former. If a couple of friends ofX

    are friends of each other then we have 3 mutual

    friends. Otherwise, X s 3 friends are mutual enemies.

    Dirichlet used the pigeonhole principle to prove that for any irrational

    there are

    infinitely many rationals pq

    satisfying

    ;

    p

    q

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    12 CHAPTER 2. INDUCTION AND COUNTING

    Proposition (Principle of Induction). Let P ( n ) be a statement aboutn for each n 2

    N

    0

    . Suppose P ( k0

    ) is true for some k0

    2N

    0

    and P ( k ) true implies that P ( k + 1 ) is

    true for eachk 2 N

    . ThenP ( n )

    is true for alln 2 N

    0

    such thatn k

    0

    .

    The favourite example is the Tower of Hanoi. We have n rings of increasing radius

    and 3 vertical rods (A

    ,B

    andC

    ) on which the rings fit. The rings are initially stacked

    in order of size on rodA

    . The challenge is to move the rings fromA

    toB

    so that alarger ring is never placed on top of a smaller one.

    We write the number of moves required to moven

    rings asT

    n

    and claim that

    T

    n

    = 2

    n

    ; 1for

    n 2 N

    0

    . We note thatT

    0

    = 0 = 2

    0

    ; 1, so the result is true for

    n = 0

    .

    We takek > 0

    and suppose we havek

    rings. Now the only way to move the largest

    ring is to move the other k;

    1 rings onto C (in Tk ; 1

    moves). We then put the largest

    ring on rodB

    (in1

    move) and move thek ; 1

    smaller rings on top of it (inT

    k ; 1

    moves

    again). Assume that Tk ; 1

    = 2

    k ; 1

    ; 1 . Then Tk

    = 2 T

    k ; 1

    + 1 = 2

    k

    ; 1 . Hence the

    result is proven by the principle of induction.

    2.3 Strong Principle of Mathematical Induction

    Proposition (Strong Principle of Induction). If P ( n ) is a statement aboutn for eachn 2 N

    0

    ,P ( k

    0

    )

    is true for somek

    0

    2 N

    0

    and the truth ofP ( k )

    is implied by the truth

    of P ( k0

    ) , P ( k0

    + 1 ) , : : : , P ( k ; 1 ) then P ( n ) is true for all n 2 N0

    such that n k0

    .

    The proof is more or less as before.

    Example (Evolutionary Trees). Every organism can mutate and produce2

    new ver-

    sions. Then n mutations are required to produce n + 1 end products.

    Proof. Let P ( n ) be the statement n mutations are required to produce n + 1 end

    products.P

    0

    is clear. Consider a tree withk + 1

    end products. The first mutation (the

    root) produces 2 trees, say with k1

    + 1 and k2

    + 1 end products with k1

    k

    2

    < k . Then

    k + 1 = k

    1

    + 1 + k

    2

    + 1

    sok = k

    1

    + k

    2

    + 1

    . If bothP ( k

    1

    )

    andP ( k

    2

    )

    are true then

    there arek

    1 mutations on the left andk

    2 on the right. So in total we havek

    1

    + k

    2

    + 1

    mutations in our tree andP ( k )

    is true isP ( k

    1

    )

    andP ( k

    2

    )

    are true. HenceP ( n )

    is true

    for alln 2 N

    0

    .

    2.4 Recursive Definitions

    (Or in other words) Defining f ( n ) , a formula or functions, for all n 2 N0

    with n k0

    by definingf ( k

    0

    )

    and then defining fork > k

    0

    ,f ( k )

    in terms off ( k

    0

    )

    ,f ( k

    0

    + 1 )

    ,

    : : : , f ( k ; 1 ) .

    The obvious example is factorials, which can be defined by n ! = n ( n ; 1 ) ! for

    n 1and

    0 ! = 1

    .

    Proposition. The number of ways to order a set ofn

    points isn !

    for alln 2 N

    0 .

    Proof. This is true for n = 0 . So, to order an n -set, choose the 1 st element in n ways

    and then order the remainingn ; 1

    -set in( n ; 1 ) !

    ways.

    Another example is the Ackermann function, which appears on example sheet 2.

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    2.5. SELECTION AND BINOMIAL COEFFICIENTS 13

    2.5 Selection and Binomial Coefficients

    We define a set of polynomials form 2 N

    0

    as

    x

    m

    = x ( x ; 1 ) ( x ; 2 ) : : : ( x ; m + 1 )

    which is pronounced x

    to them

    falling. We can do this recursively byx

    0

    = 1

    and

    x

    m

    = ( x ; m + 1 ) x

    m ; 1 for m > 0 . We also define x to the m rising by

    x

    m

    = x ( x + 1 ) ( x + 2 ) : : : ( x + m ; 1 ) :

    We further define;

    x

    m

    (read x

    choosem

    ) by

    x

    m

    =

    x

    m

    m !

    :

    It is also convienient to extend this definition to negativem

    by;

    x

    m

    = 0

    ifm 0

    ,S

    n

    = 2 S

    n ; 1

    . (Pick a point in then

    -set and observe that

    there are Sn ; 1

    subsets not containing it and Sn ; 1

    subsets containing it.

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    14 CHAPTER 2. INDUCTION AND COUNTING

    3.( 1 ; 1 )

    n

    = 0 =

    P

    k

    ;

    n

    k

    ( ; 1 )

    k so in any finite set the number of subsets of

    even sizes equals the number of subsets of odd sizes.

    It also gives us another proof of Fermats Little Theorem: if p is prime then a p a

    ( m o d p )

    for alla 2 N

    0

    .

    Proof. It is done by induction ona

    . It is obviously true whena = 0

    , so takea > 0

    and

    assume the theorem is true for a ; 1 . Then

    a

    p

    = ( ( a ; 1 ) + 1 )

    p

    ( a ; 1 )

    p

    + 1 m o d p

    as

    p

    k

    0 ( m o d p )unless

    k = 0

    ork = p

    a ; 1 + 1 m o d p

    a m o d p

    2.5.1 Selections

    The number of ways of choosing m objects out of n objects is

    ordered unordered

    no repeats n m;

    n

    m

    repeats n m;

    n ; m + 1

    m

    The only entry that needs justification is;

    n ; m + 1

    m

    . But there is a one-to-one cor-

    respondance betwen the set of ways of choosing m out of n unordered with possible

    repeats and the set of all binary strings of lengthn + m ; 1

    withm

    zeros andn ; 1

    ones. For suppose there are mi

    occurences of element i , mi

    0 . Then

    n

    X

    i = 1

    m

    i

    = m $ 0 : : : 0

    | { z }

    m

    1

    1 0 : : : 0

    | { z }

    m

    2

    1 : : : 1 0 : : : 0

    | { z }

    m

    n

    :

    There are;

    n ; m + 1

    m

    such strings (choosing where to put the1

    s).

    2.5.2 Some more identities

    Proposition.

    n

    k

    =

    n

    n ; k

    n 2 N

    0

    k 2 Z

    Proof. For: choosing a k -subset is the same as choosing an n ; k -subset to reject.

    Proposition.

    n

    k

    =

    n ; 1

    k ; 1

    +

    n ; 1

    k

    n 2 N

    0

    k 2 Z

    Proof. This is trivial if n 0 . Choose a special

    element in then

    -set. Anyk

    -subset will either contain this special element (there are;

    n ; 1

    k ; 1

    such) or not contain it (there are;

    n ; 1

    k

    such).

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    2.5. SELECTION AND BINOMIAL COEFFICIENTS 15

    In fact

    Proposition.

    x

    k

    =

    x ; 1

    k ; 1

    +

    x ; 1

    k

    k 2 Z

    Proof. Trivial if k

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    16 CHAPTER 2. INDUCTION AND COUNTING

    2.6 Special Sequences of Integers

    2.6.1 Stirling numbers of the second kind

    Definition. The Stirling number of the second kind, S ( n k ) , n k 2 N0

    is defined

    as the number of partitions of f 1 : : : n g into exactly k non-empty subsets. Also

    S ( n 0 ) = 0

    ifn > 0

    and1

    ifn = 0

    .Note that S ( n k ) = 0 if k > n , S ( n n ) = 1 for all n , S ( n n ; 1 ) =

    ;

    n

    2

    and

    S ( n 2 ) = 2

    n ; 1

    ; 1.

    Lemma 2.1. A recurrence: S ( n k ) = S ( n ; 1 k ; 1 ) + k S ( n ; 1 k ) .

    Proof. In any partition off 1 : : : n g

    , the elementn

    is either in a part on its own (S ( n ;

    1 k

    ;1 ) such) or with other things ( k S ( n

    ;1 k ) such).

    Proposition. For n 2 N0

    ,x

    n

    =

    P

    k

    S ( n k ) x

    k .

    Proof. Proof is by induction on n . It is clearly true when n = 0 , so take n > 0 and

    assume the result is true forn ; 1

    . Then

    x

    n

    = x x

    n ; 1

    = x

    X

    k

    S ( n ; 1 k ) x

    k

    =

    X

    k

    S ( n ; 1 k ) x

    k

    ( x ; k + k )

    =

    X

    k

    S ( n ; 1 k ) x

    k + 1

    +

    X

    k

    k S ( n ; 1 k ) x

    k

    =

    X

    k

    S ( n ; 1 k ; 1 ) x

    k

    +

    X

    k

    k S ( n ; 1 k ) x

    k

    =

    X

    k

    S ( n k ) x

    k as required.

    2.6.2 Generating Functions

    Recall the Fibonacci numbers,F

    n

    such thatF

    1

    = F

    2

    = 1

    andF

    n + 2

    = F

    n + 1

    + F

    n

    .

    Suppose that we wish to obtain a closed formula.

    First method

    Try a solution of the form Fn

    =

    n . Then we get 2 ; ; 1 = 0 and = 1 p

    5

    2

    . We

    then take

    F

    n

    = A

    1 +

    p

    5

    2

    !

    n

    + B

    1

    ;

    p

    5

    2

    !

    n

    and use the initial conditions to determine A and B . It turns out that

    F

    n

    =

    1

    p

    5

    "

    1 +

    p

    5

    2

    !

    n

    ;

    1 ;

    p

    5

    2

    !

    n

    #

    :

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    2.6. SPECIAL SEQUENCES OF INTEGERS 17

    Note that 1 +p

    5

    2

    > 1

    and

    1 ;

    p

    5

    2

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    18 CHAPTER 2. INDUCTION AND COUNTING

    SinceC ( 0 ) = 1

    we must have the;

    sign. From the binomial theorem

    ( 1 ; 4 z )

    1

    2

    =

    X

    k 0

    1

    2

    k

    ( ; 4 )

    k

    z

    k

    :

    Thus Cn

    = ;

    1

    2

    ;

    1

    2

    n + 1

    ( ; 4 )

    n + 1 . Simplifying this we obtain Cn

    =

    1

    n + 1

    ;

    2 n

    n

    and note

    the corollary that ( n + 1 )j

    ;

    2 n

    n

    .

    Other possible definitions forC

    n

    are:

    The number of ways of bracketing

    n + 1

    variables.

    The number of sequences of length

    2 n

    withn

    each of 1

    such that all partial

    sums are non-negative.

    2.6.4 Bell numbers

    Definition. The Bell number Bn

    is the number of partitions off 1 : : : n g .

    It is obvious from the definitions that Bn

    =

    P

    k

    S ( n k ) .

    Lemma 2.3.

    B

    n + 1

    =

    X

    0 k n

    n

    k

    B

    k

    Proof. For, put the element n + 1 in with a k -subset of f 1 : : : n g for k = 0 to k =

    n

    .

    There isnt a nice closed formula for Bn

    , but there is a nice expression for its

    exponential generating function.

    Definition. The exponential generating function that is associated with the sequence

    ( a

    n

    )

    n 2 N

    0

    is

    A ( z ) =

    X

    n

    a

    n

    n !

    z

    n

    :

    If we have A ( z )

    and B ( z )

    (with obvious notation) and A ( z )

    B ( z ) =

    P

    n

    c

    n

    n !

    z

    n then

    c

    n

    =

    P

    k

    ;

    n

    k

    a

    k

    b

    n ; k

    , the exponential convolution of ( an

    )

    n 2 N

    0

    and ( bn

    )

    n 2 N

    0

    .

    HenceB

    n + 1

    is the coefficient ofz

    n in the exponential convolution of the sequences

    1 1 1 1 : : : and B0

    B

    1

    B

    2

    : : : . Thus B ( z ) 0 = e z B ( z ) . (Shifting is achieved by

    differentiation for exponential generating functions.) Therefore B ( z ) = e

    e

    z

    + C and

    using the condition B ( 0 ) = 1

    we find thatC = ; 1

    . So

    B ( z ) = e

    e

    z

    ; 1

    :

    2.6.5 Partitions of numbers and Young diagrams

    Forn 2 N

    letp ( n )

    be the number of ways to writen

    as the sum of natural numbers.

    We can also definep ( 0 ) = 1

    .

    For instance, p ( 5 ) = 7 :

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    2.6. SPECIAL SEQUENCES OF INTEGERS 19

    5 4 + 1 3 + 2 3 + 1 + 1

    Notation5 4 1

    3

    3 2 3 1

    2

    2 + 2 + 1 2 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1

    Notation 2 2 1 2 1 3 1 5

    These partitions of n are usefully pictured by Young diagrams.

    The conjugate partition is obtained by taking the mirror image in the main diag-

    onal of the Young diagram. (Or in other words, consider columns instead of rows.)

    By considering (conjugate) Young diagrams this theorem is immediate.

    Theorem 2.2. The number of partions ofn

    into exactlyk

    parts equals the number ofpartitions of n with largest part k .

    We now define an ordinary generating function forp ( n )

    P ( z ) = 1 +

    X

    n 2 N

    p ( n ) z

    n

    :

    Proposition.

    P ( z ) =

    1

    1 ; z

    1

    1 ; z

    2

    1

    1 ; z

    3

    =

    Y

    k 2 N

    1

    1 ; z

    k

    :

    Proof. The RHS is ( 1 + z + z 2 + : : : ) ( 1 + z 2 + z 4 + : : : ) ( 1 + z 3 + z 6 : : : ) : : : .

    We get a termz

    n whenever we selectz

    a

    1 from the first bracket,z

    2 a

    2 from the

    second,z

    3 a

    3 from the third and so on, andn = a

    1

    + 2 a

    2

    + 3 a

    3

    + : : :

    , or in other words

    1

    a

    1

    2

    a

    2

    3

    a

    3

    : : : is a partition of n . There are p ( n ) of these.

    We can similarly prove these results.

    Proposition. The generating function Pm

    ( z )

    of the sequencep

    m

    ( n )

    of partitions ofn

    into at most m parts (or the generating function for the sequence pm

    ( n ) of partitions

    ofn

    with largest part m

    ) satisfies

    P

    m

    ( z ) =

    1

    1 ; z

    1

    1 ; z

    2

    1

    1 ; z

    3

    : : :

    1

    1 ; z

    m

    :

    Proposition. The generating function for the number of partitions into odd parts is

    1

    1 ; z

    1

    1 ; z

    3

    1

    1 ; z

    5

    : : : :

    Proposition. The generating function for the number of partitions into unequal parts

    is

    ( 1 + z ) ( 1 + z

    2

    ) ( 1 + z

    3

    ) : : : :

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    20 CHAPTER 2. INDUCTION AND COUNTING

    Theorem 2.3. The number of partitions ofn into odd parts equals the number of par-

    titions ofn into unequal parts.

    Proof.

    ( 1 + z ) ( 1 + z

    2

    ) ( 1 + z

    3

    ) : : : =

    1 ; z

    2

    1

    ;z

    1 ; z

    4

    1

    ;z

    2

    1 ; z

    6

    1

    ;z

    3

    : : :

    =

    1

    1 ; z

    1

    1 ; z

    3

    1

    1 ; z

    5

    : : :

    Theorem 2.4. The number of self-conjugate partitions ofn

    equals the number of par-

    titions ofn into odd unequal parts.

    Proof. Consider hooks along the main diagonal like this.

    This process can be reversed, so there is a one-to-one correspondance.

    2.6.6 Generating function for self-conjugate partitions

    Observe that any self-conjugate partition consists of a largest k k subsquare and twice

    a partition of 12

    ( n ; k

    2

    )

    into at mostk

    parts. Now

    1

    ( 1 ; z

    2

    ) ( 1 ; z

    4

    ) : : : ( 1 ; z

    2 m

    )

    is the generating function for partitions ofn

    into even parts of size at most2 m

    , or

    alternatively the generating function for partitions of 12

    n into parts of size

    m . We

    deduce thatz

    l

    ( 1 ; z

    2

    ) ( 1 ; z

    4

    ) : : : ( 1 ; z

    2 m

    )

    is the generating function for partitions of 12

    ( n ; l )into at most

    m

    parts. Hence the

    generating function for self-conjugate partitions is

    1 +

    X

    k 2 N

    z

    k

    2

    ( 1 ; z

    2

    ) ( 1 ; z

    4

    ) : : : ( 1 ; z

    2 k

    )

    :

    Note also that this equals

    Y

    k 2 N

    0

    ( 1 + z

    2 k + 1

    )

    as the number of self-conjugate partitions ofn

    equals the number of partitions ofn

    into

    unequal odd parts.

    In fact in any partition we can consider the largestk k

    subsquare, leaving two

    partitions of at most k parts, one of ( n ; k 2 ; j ) , the other of j for some j . The number

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    2.6. SPECIAL SEQUENCES OF INTEGERS 21

    of these two lots are the coefficients ofz

    n ; k

    2

    ; j andz

    j inQ

    k

    i = 1

    1

    1 ; z

    i

    respectively.

    Thus

    P ( z ) = 1 +

    X

    k 2 N

    z

    k

    2

    ( ( 1

    ;z ) ( 1

    ;z

    2

    ) : : : ( 1

    ;z

    k

    ) )

    2

    :

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    22 CHAPTER 2. INDUCTION AND COUNTING

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    Chapter 3

    Sets, Functions and Relations

    3.1 Sets and indicator functions

    We fix some universal set S . We write P ( S ) for the set of all subsets ofS the power

    set ofS

    . IfS

    is finite withj S j = m

    (the number of elements), thenj P ( S ) j = 2

    m

    .Given a subset

    A

    ofS

    (A S

    ) we define the complement A

    ofA

    inS

    as

    A = f s 2 S : s =2 A g .

    Given two subsetsA

    ,B

    ofS

    we can define various operations to get new subsets

    ofS

    .

    A \ B = f s 2 S : s 2 A and s 2 B g

    A B = f s 2 S : s 2 A(inclusive) or

    s 2 B g

    A n B = f s 2 A : s =2 B g

    A B = f s 2 S : s 2 A(exclusive) or

    s 2 B gthe symmetric difference

    = ( A B ) n ( A \ B )

    = ( A n B ) ( B n A ) :

    The indicator functionI

    A

    of the subsetA

    ofS

    is the functionI

    A

    : S 7! f 0 1 g

    defined by

    I

    A

    ( s ) =

    (

    1 x 2 A

    0 otherwise.

    It is also known as the characteristic function A

    . Two subsets A and B of S are equal

    iffI

    A

    ( s ) = I

    B

    ( s ) 8 s 2 S. These relations are fairly obvious:

    I

    A

    = 1 ; I

    A

    I

    A \ B

    = I

    A

    I

    B

    I

    A B

    = I

    A

    + I

    B

    I

    A B

    = I

    A

    + I

    B

    m o d 2 :

    Proposition. A ( B C ) = ( A B ) C .

    23

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    24 CHAPTER 3. SETS, FUNCTIONS AND RELATIONS

    Proof. For, modulo 2,

    I

    A ( B C )

    = I

    A

    + I

    B C

    = I

    A

    + I

    B

    + I

    C

    = I

    A B

    + I

    C

    = I

    ( A B ) C

    m o d 2 :

    Thus P ( S ) is a group under . Checking the group axioms we get:

    Given A B 2 P ( S ) , A B 2 P ( S ) closure,

    A ( B C ) = ( A B ) C associativity,

    A = Afor all

    A 2 P ( S ) identity,

    A A = for all

    A 2 P ( S ) inverse.

    We note that A B = B A so that this group is abelian.

    3.1.1 De Morgans LawsProposition. 1. A \ B = A B

    2. A B = A \ B

    Proof.

    I

    A \ B

    = 1 ; I

    A \ B

    = 1 ; I

    A

    I

    B

    = ( 1 ; I

    A

    ) + ( 1 ; I

    B

    ) ; ( 1 ; I

    A

    ) ( 1 ; I

    B

    )

    = I

    A

    + I

    B

    ; I

    A \

    B

    = I

    A

    B

    :

    We prove 2 by using 1 on

    A and

    B .

    A more general version of this is: Suppose A1

    : : : A

    n

    S . Then

    1.T

    n

    i = 1

    A

    i

    =

    S

    n

    i = 1

    A

    i

    2.S

    n

    i = 1

    A

    i

    =

    T

    n

    i = 1

    A

    i

    .

    These can be proved by induction onn

    .

    3.1.2 Inclusion-Exclusion Principle

    Note thatj A j =

    P

    s 2 S

    I

    A

    ( s )

    .

    Theorem 3.1 (Principle of Inclusion-Exclusion). Given A1

    : : : A

    n

    Sthen

    j A

    1

    A

    n

    j =

    X

    6= J f 1 : : : n g

    ( ; 1 )

    j J j ; 1

    j A

    J

    j where

    A

    J

    =

    \

    i 2 J

    A

    i

    .

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    3.1. SETS AND INDICATOR FUNCTIONS 25

    Proof. We considerA

    1

    A

    n

    and note that

    I

    A

    1

    A

    n

    = I

    A

    1

    \ \

    A

    n

    = I

    A

    1

    I

    A

    2

    : : : I

    A

    n

    = ( 1 ; I

    A

    1

    ) ( 1 ; I

    A

    2

    ) : : : ( 1 ; I

    A

    n

    )

    =

    X

    J f 1 : : : n g

    ( ; 1 )

    j J j

    I

    A

    J

    Summing over s2

    S we obtain the result

    A

    1

    A

    n

    =

    X

    J f 1 : : : n g

    ( ; 1 )

    j J j

    j A

    J

    j

    which is equivalent to the required result.

    Just for the sake of it, well prove it again!

    Proof. For eachs 2 S

    we calculate the contribution. Ifs 2 S

    buts

    is in noA

    i

    then

    there is a contribution 1 to the left. The only contribution to the right is + 1 when J = .

    Ifs 2 S

    andK = f i 2 f 1 : : : n g : s 2 A

    i

    gis non-empty then the contribution to the

    right isP

    I K

    ( ; 1 )

    j I j

    =

    P

    k

    i = 0

    ;

    k

    i

    ( ; 1 )

    i

    = 0

    , the same as on the left.

    Example (Eulers Phi Function).

    ( m ) = m

    Y

    p

    primep j m

    1 ;

    1

    p

    :

    Solution. Letm =

    Q

    n

    i = 1

    p

    a

    i

    i

    , where thep

    i

    are distinct primes anda

    i

    2 N. Let

    A

    i

    be

    the set of integers less than m which are divisible by pi

    . Hence ( m ) =

    T

    n

    i = 1

    A

    i

    .

    Nowj A

    i

    j =

    m

    p

    i

    , in fact forJ f 1 : : : m g

    we havej A

    J

    j =

    m

    Q

    i 2 J

    p

    i

    . Thus

    ( m ) = m ;

    m

    p

    1

    ;

    m

    p

    2

    ; ;

    m

    p

    n

    +

    m

    p

    1

    p

    2

    +

    m

    p

    1

    p

    3

    + +

    m

    p

    2

    p

    3

    + +

    m

    p

    n ; 1

    p

    n

    ...

    + ( ; 1 )

    n

    m

    p

    1

    p

    2

    : : : p

    n

    = m

    Y

    p primep j m

    1 ;

    1

    p

    as required.

    Example (Derangements). Suppose we have n psychologists at a meeting. Leaving

    the meeting they pick up their overcoats at random. In how many ways can this be

    done so that none of them has his own overcoat. This number isD

    n

    , the number of

    derangements of n objects.

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    26 CHAPTER 3. SETS, FUNCTIONS AND RELATIONS

    Solution. LetA

    i

    be the number of ways in which psychologisti

    collects his own coat.

    Then Dn

    =

    A

    1

    \ \

    A

    n

    . If J f

    1 : : : n

    gwith

    jJ

    j= k then

    jA

    J

    j= ( n

    ;k ) ! .

    Thus

    A

    1

    \ \

    A

    n

    = n ! ;

    n

    1

    ( n ; 1 ) ! +

    n

    2

    ( n ; 2 ) ! ; : : :

    = n !

    n

    X

    k = 0

    ( ; 1 )

    k

    k !

    :

    Thus Dn

    is the nearest integer to n ! e ; 1 , since D nn !

    !e

    ; 1 as n! 1

    .

    3.2 Functions

    Let A B be sets. A function (or mapping, or map) f : A 7! B is a way to associate

    a unique imagef ( a ) 2 B

    with eacha 2 A

    . IfA

    andB

    are finite withj A j = m

    and

    j B j = n then the set of all functions from A to B is finite with n m elements.

    Definition. The function f : A7!

    B is injective (or one-to-one) if f ( a1

    ) = f ( a

    2

    )

    implies thata1

    = a

    2

    for all a1

    a

    2

    2 A .

    The number of injective functions from an m -set to an n -set is n m .

    Definition. The function f : A7!

    B is surjective (or onto) if each b2

    B has at least

    one preimagea 2 A

    .

    The number of surjective functions from an m -set to an n -set is n ! S ( m n ) .

    Definition. The function f : A 7! B is bijective if it is both injective and surjective.

    IfA

    andB

    are finite thenf : A 7! B

    can only be bijective ifj A j = j B j

    . If

    j A j = j B j

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    3.3. PERMUTATIONS 27

    3.3 Permutations

    A permutation ofA

    is a bijectionf : A 7! A

    . One notation is

    f =

    1 2 3 4 5 6 7 8

    1 3 4 2 8 7 6 5

    :

    The set of permutations ofA

    is a group under composition, the symmetric group

    s y m A . If j A j = n then s y m A is also denoted Sn

    and j s y m A j = n ! . Sn

    is not abelian

    you can come up with a counterexample yourself. We can also think of permutations

    as directed graphs, in which case the following becomes clear.

    Proposition. Any permutation is the product of disjoint cycles.

    We have a new notation for permutations, cycle notation. 1 For our functionf

    above, we write

    f = ( 1 ) ( 2 3 4 ) ( 5 8 ) ( 6 7 ) = ( 2 3 4 ) ( 5 8 ) ( 6 7 ) :

    3.3.1 Stirling numbers of the first kind

    Definition. s ( n k ) is the number of permutations off

    1 : : : n

    gwith precisely k cycles

    (including fixed points).

    For instances ( n n ) = 1

    ,s ( n n ; 1 ) =

    ;

    n

    2

    ,s ( n 1 ) = ( n ; 1 ) !

    ,s ( n 0 ) =

    s ( 0 k ) = 0 for all k n 2 N but s ( 0 0 ) = 1 .

    Lemma 3.1.

    s ( n k ) = s ( n ; 1 k ; 1 ) + ( n ; 1 ) s ( n ; 1 k )

    Proof. Either the pointn

    is in a cycleon its own (s ( n ; 1 k ; 1 )

    such) or it is not. In this

    case, n can be inserted into any of n ; 1 places in any of the s ( n ; 1 k ) permutations

    off 1 : : : n ; 1 g

    .

    We can use this recurrence to prove this proposition. (Proof left as exercise.)

    Proposition.

    x

    n

    =

    X

    k

    s ( n k ) x

    k

    3.3.2 Transpositions and shuffles

    A transposition is a permutation which swaps two points and fixes the rest.

    Theorem 3.2. Every permutation is the product of transpositions.

    Proof. Since every permutations is the product of cycles we only need to check for

    cycles. This is easy: ( i1

    i

    2

    : : : i

    k

    ) = ( i

    1

    i

    2

    ) ( i

    2

    i

    3

    ) : : : ( i

    k ; 1

    i

    k

    ) .

    Theorem 3.3. For a given permutation , the number of transpositions used to write

    as their product is either always even or always odd.

    1See the Algebra and Geometry course for more details.

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    28 CHAPTER 3. SETS, FUNCTIONS AND RELATIONS

    We writes i g n =

    (

    + 1 if always even

    ;1 if always odd

    . We say that

    is aneven

    oddpermutation.

    Letc ( )

    be the number of cycles in the disjoint cycle representation of

    (including

    fixed points).

    Lemma 3.2. If = ( a b ) is a transposition thatc ( ) = c ( ) 1 .

    Proof. If a and b are in the same cycle of then has two cycles, so c ( ) =

    c ( ) + 1

    . Ifa

    andb

    are in different cycles then they contract them together andc ( ) =

    c ( ) ; 1 .

    Proof of theorem 3.3. Assume = 1

    : : :

    k

    =

    1

    : : :

    l

    . Then c ( ) = c ( ) + k

    c ( ) + l ( m o d 2 )

    . Hencek l ( m o d 2 )

    as required.

    We note that s i g n = ( ; 1 ) n ; c ( ) , thus s i g n ( 1

    2

    ) = s i g n

    1

    s i g n

    2

    and thus

    s i g n

    is a homomorphism fromS

    n

    tof 1 g

    .

    Ak

    -cycle is an even permutation iffk

    is odd. A permutation is aneven

    oddper-

    mutation iff the number of even length cycles in the disjoint cycle representation iseven

    odd.

    3.3.3 Order of a permutation

    If is a permutation then the order of is the least natural number n such that n = .

    The order of the permutation

    is the lcm of the lengths of the cycles in the disjoint

    cycle decomposition of .

    In card shuffling we need to maximise the order of the relevant permutation

    . One

    can show (see) that for of maximal length we can take all the cycles in the disjoint

    cycle representation to have prime power length. For instance with3 0

    cards we can get

    a 2 S

    3 0

    with an order of4 6 2 0

    (cycle type3 4 5 7 1 1

    ).

    3.3.4 Conjugacy classes in Sn

    Two permutations 2 Sn

    are conjugate iff 9 2 Sn

    such that = ; 1 .

    Theorem 3.4. Two permutations are conjugate iff they have the same cycle type.

    This theorem is proved in the Algebra and Geometry course. We note the corollary

    that the number of conjugacy classes inS

    n

    equals the number of partitions ofn

    .

    3.3.5 Determinants of an n n matrix

    In the Linear Maths course you will prove that ifA = ( a

    i j

    )

    is ann n

    matrix then

    d e t A =

    X

    2 S

    n

    s i g n

    n

    Y

    j = 1

    a

    j ( j )

    :

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    3.4. BINARY RELATIONS 29

    3.4 Binary Relations

    A binary relation on a set S is a property that any pair of elements of S may or may

    not have. More precisely:

    WriteS S

    , the Cartesian square ofS

    for the set of pairs of elements ofS

    ,S S =

    f ( a b ) : a b 2 S g. A binary relation

    Ron

    S

    is a subset ofS S

    . We writea R b

    iff

    ( a b ) 2 R . We can think of R as a directed graph with an edge from a to b iff a R b .

    A relation R is:

    reflexive iff a

    Ra

    8a

    2S ,

    symmetric iff

    a R b ) b R a 8 a b 2 S,

    transitive iff

    a R b b R c ) a R c 8 a b c 2 S,

    antisymmetric iff

    a R b b R a ) a = b 8 a b 2 S.

    The relationR

    onS

    is an equivalence relation if it is reflexive, symmetric and

    transitive. These are nice properties designed to make R behave something like = .

    Definition. If R is a relation on S , then

    a ]

    R

    = a ] = f b 2 S : a R b g :

    IfR

    is an equivalence then these are the equivalence classes.

    Theorem 3.5. If R is an equivalence relation then the equivalence classes form a par-

    tition ofS .

    Proof. Ifa 2 S

    thena 2 a ]

    , so the classes cover all ofS

    . If a ] \ b ] 6=

    then

    9 c 2 a ] \ b ] . Now a R c and b R c ) c R b . Thus a R b and b 2 a ] . If d 2 b ]

    thenb R d

    soa R d

    and thus b ] a ]

    . We can similarly show that a ] b ]

    and thus

    a ] = b ]

    .

    The converse of this is true: if we have a partition ofS

    we can define an equivalence

    relation onS

    bya R b

    iffa

    andb

    are in the same part.

    An application of this is the proof of Lagranges Theorem. The idea is to show that

    being in the same (left/right) coset is an equivalence relation.

    Given an equivalence class on S the quotient set is S = R , the set of all equivalence

    classes. For instance ifS = R

    anda R b

    iffa ; b 2 Z

    thenS = R

    is (topologically) a

    circle. If S = R 2 and ( a1

    b

    1

    ) R ( a

    2

    b

    2

    ) iff a1

    ; a

    2

    2 Z and b1

    ; b

    2

    2 Z the quotient

    set is a torus.

    Returning to a general relationR

    , for eachk 2 N

    we define

    R

    ( k )

    = f ( a b ) : there is a path of length at k from a to b g :

    R

    ( 1 )

    = Rand

    R

    ( 1 )

    = t ( R )

    , the transitive closure ofR

    .R

    ( 1 ) is defined asS

    i 1

    R

    ( i ) .

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    30 CHAPTER 3. SETS, FUNCTIONS AND RELATIONS

    3.5 Posets

    Ris a (partial) order on

    S

    if it is reflexive, anti-symmetric and transitive. The setS

    is

    a poset (partially ordered set) if there is an orderR

    on S .

    We generally writea b

    iff( a b ) 2 R

    , anda < b

    iffa b

    anda 6= b

    .

    ConsiderD

    n

    , the set of divisors ofn

    .D

    n

    is partially ordered by division,a b

    if

    a

    jb . We have the Hasse diagram, in this case for D 3 6 :

    36

    18 12

    9

    3

    46

    2

    1

    A descending chain is a sequence a1

    > a

    2

    > a

    3

    > : : : . An antichain is a subset of

    S

    with no two elements directly comparable, for instancef 4 6 9 g

    inD

    3 6

    .

    Proposition. If S is a poset with no chains of length > n then S can be covered by at

    most n antichains.

    Proof. Induction on n . Take n > 1 and let M be the set of all maximal elements in S .

    NowS n M

    has no chains of length> n ; 1

    andM

    is an antichain.

    3.5.1 Products of posets

    Suppose A and B are posets. Then A B has various orders; two of them being

    product order: ( a1

    b

    1

    ) ( a

    2

    b

    2

    ) iff a1

    a

    2

    and b1

    b

    2

    ,

    lexicographic order:

    ( a

    1

    b

    1

    ) ( a

    2

    b

    2

    )

    if eithera

    1

    a

    2

    or ifa

    1

    = a

    2

    then

    b

    1

    b

    2

    .

    Exercise: check that these are orders.

    Note that there are no infinite descending chains inN N

    under lexicographic

    order. Such posets are said to be well ordered. The principle of induction follows from

    well-ordering as discussed earlier.

    3.5.2 Eulerian Digraphs

    A digraph is Eulerian if there is a closed path covering all the edges. A necessary

    condition is: the graph is connected and even (each vertex has an equal number of in

    and out edges). This is in fact sufficient.

    Proposition. The set of such digraphs is well-ordered under containment.

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    3.6. COUNTABILITY 31

    Proof. Assume proposition is false and letG

    be a minimal counterexample. LetT

    be

    a non-trivial closed path in G , for instance the longest closed path. Now T must be

    even, soG n T

    is even. Hence each connected component ofG n T

    is Eulerian asG

    is minimal. But then G is Eulerian: you can walk along T and include all edges of

    connected components ofG n T

    when encountered giving a contradiction. Hence

    there are no minimal counterexamples.

    3.6 Countability

    Definition. A setS

    is countable if eitherj S j

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    32 CHAPTER 3. SETS, FUNCTIONS AND RELATIONS

    Theorem 3.6. A countable union of countable sets is countable. That is, if I is a

    countable indexing set andAi

    is countable8

    i

    2I then

    S

    i 2 I

    A

    i

    is countable.

    Proof. Identify firstI

    with the subsetf ( I ) N

    . DefineF : A 7! N

    bya 7! 2

    n

    3

    m

    where n is the smallest index i with a2

    A

    i

    , and m = fn

    ( a ) . This is well-defined and

    injective (stop to think about it for a bit).

    Theorem 3.7. The set of all algebraic numbers is countable.

    Proof. LetP

    n

    be the set of all polynomials of degree at mostn

    with integral coeffi-

    cients. Then the mapc

    n

    x

    n

    + + c

    1

    x + c

    0

    7! ( c

    n

    : : : c

    1

    c

    0

    )

    is an injection fromP

    n

    toZ

    n + 1 . Hence eachP

    n

    is countable. It follows that the set of all polynomials with

    integral coefficients is countable. Each polynomial has finitely many roots, so the set

    of algebraic numbers is countable.

    Theorem 3.8 (Cantors diagonal argument). R is uncountable.

    Proof. Assume R is countable, then the elements can be listed as

    r

    1

    = n

    1

    : d

    1 1

    d

    1 2

    d

    1 3

    : : :

    r

    2

    = n

    2

    : d

    2 1

    d

    2 2

    d

    1 3

    : : :

    r

    3

    = n

    3

    : d

    3 1

    d

    3 2

    d

    3 3

    : : :

    (in decimal notation). Now define the real r = 0 : d1

    d

    2

    d

    3

    : : : by di

    = 0 if di i

    6= 0 and

    d

    i

    = 1

    ifd

    i i

    = 0

    . This is real, but it differs fromr

    i

    in thei

    th decimal place. So the list

    is incomplete and the reals are uncountable.

    Exercise: use a similiar proof to show that P ( N ) is uncountable.

    Theorem 3.9. The set of all transcendental numbers is uncountable. (And therefore at

    least non-empty!)

    Proof. LetA

    be the set of algebraic numbers andT

    the set of transcendentals. ThenR = A T , so if T was countable then so would R be. Thus T is uncountable.

    3.7 Bigger sets

    The material from now on is starred.

    Two sets S and T have the same cardinality ( j S j = j T j ) if there is a bijection

    betweenS

    andT

    . One can show (the Schroder-Bernstein theorem) that if there is an

    injection from S to T and an injection from T to S then there is a bijection between S

    andT

    .

    For any setS

    , there is an injection fromS

    toP ( S )

    , simplyx 7! f x g

    . However

    there is never a surjection S 7! P ( S ) , so j S j

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    3.7. BIGGER SETS 33

    Proof. Letf : S 7! P ( S )

    be a surjection and considerX 2 P ( S )

    defined byf x 2

    S : x =

    2f ( x )

    g. Now

    9x

    0

    2S such that f ( x 0 ) = X . If x 0

    2X then x 0 =

    2f ( x

    0

    ) but

    f ( x

    0

    ) = X

    a contradiction. But ifx

    0

    =2 Xthen

    x

    0

    =2 f ( x

    0

    )

    andx

    0

    2 X giving a

    contradiction either way.

    If there is aninjection

    surjection

    f : A 7! Bthen there exists a

    surjection

    injection

    g : B 7! A.

    Moreover we can ensure thatg f =

    A

    f g =

    B

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    34 CHAPTER 3. SETS, FUNCTIONS AND RELATIONS

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    References

    Hardy & Wright, An Introduction to the Theory of Numbers, Fifth ed., OUP, 1988.

    This book is relevant to quite a bit of the course, and I quite enjoyed (parts of!) it.

    H. Davenport, The Higher Arithmetic, Sixth ed., CUP, 1992.

    A very good book for this course. Its also worth a read just for interests sake.

    Ive also heard good things about Biggs book, but havent read it.

    35


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