Discrete Mathematics

41 Pages·1995·0.205 MB·English

by  Saxl Dr.J.

Discrete Mathematics
Dr. J. Saxl
Michælmas1995
ThesenotesaremaintainedbyPaulMetcalfe.
[email protected].
Revision: 2.3
Date: 1999/10/21 11:21:05
Thefollowingpeoplehavemaintainedthesenotes.
–date PaulMetcalfe
Contents
Introduction v
1 Integers 1
1.1 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thedivisionalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 TheEuclideanalgorithm . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 ApplicationsoftheEuclideanalgorithm . . . . . . . . . . . . . . . . 4
1.4.1 ContinuedFractions . . . . . . . . . . . . . . . . . . . . . . 5
1.5 ComplexityofEuclideanAlgorithm . . . . . . . . . . . . . . . . . . 6
1.6 PrimeNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6.1 Uniquenessofprimefactorisation . . . . . . . . . . . . . . . 7
1.7 Applicationsofprimefactorisation . . . . . . . . . . . . . . . . . . . 7
1.8 ModularArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.9 SolvingCongruences . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.9.1 Systemsofcongruences . . . . . . . . . . . . . . . . . . . . 9
1.10 Euler’sPhiFunction . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.10.1 PublicKeyCryptography. . . . . . . . . . . . . . . . . . . . 10
2 InductionandCounting 11
2.1 ThePigeonholePrinciple . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 StrongPrincipleofMathematicalInduction . . . . . . . . . . . . . . 12
2.4 RecursiveDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 SelectionandBinomialCoefficients . . . . . . . . . . . . . . . . . . 13
2.5.1 Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.2 Somemoreidentities . . . . . . . . . . . . . . . . . . . . . . 14
2.6 SpecialSequencesofIntegers . . . . . . . . . . . . . . . . . . . . . 16
2.6.1 Stirlingnumbersofthesecondkind . . . . . . . . . . . . . . 16
2.6.2 GeneratingFunctions . . . . . . . . . . . . . . . . . . . . . . 16
2.6.3 Catalannumbers . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.4 Bellnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6.5 PartitionsofnumbersandYoungdiagrams . . . . . . . . . . 18
2.6.6 Generatingfunctionforself-conjugatepartitions . . . . . . . 20
3 Sets,FunctionsandRelations 23
3.1 Setsandindicatorfunctions . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 DeMorgan’sLaws . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Inclusion-ExclusionPrinciple . . . . . . . . . . . . . . . . . 24
iii
iv CONTENTS
3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Stirlingnumbersofthefirstkind . . . . . . . . . . . . . . . . 27
3.3.2 Transpositionsandshuffles . . . . . . . . . . . . . . . . . . . 27
3.3.3 Orderofapermutation . . . . . . . . . . . . . . . . . . . . . 28
3.3.4 ConjugacyclassesinS . . . . . . . . . . . . . . . . . . . . 28
n
3.3.5 Determinantsofann×nmatrix. . . . . . . . . . . . . . . . 28
3.4 BinaryRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.1 Productsofposets . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.2 EulerianDigraphs . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 Biggersets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Introduction
These notes are based on the course “Discrete Mathematics” given by Dr. J. Saxl in
CambridgeintheMichælmasTerm1995. Thesetypesetnotesaretotallyunconnected
withDr.Saxl.
Othersetsofnotesareavailablefordifferentcourses. Atthetimeoftypingthese
courseswere:
Probability DiscreteMathematics
Analysis FurtherAnalysis
Methods QuantumMechanics
FluidDynamics1 QuadraticMathematics
Geometry DynamicsofD.E.’s
FoundationsofQM Electrodynamics
MethodsofMath.Phys FluidDynamics2
Waves(etc.) StatisticalPhysics
GeneralRelativity DynamicalSystems
PhysiologicalFluidDynamics BifurcationsinNonlinearConvection
SlowViscousFlows TurbulenceandSelf-Similarity
Acoustics Non-NewtonianFluids
SeismicWaves
Theymaybedownloadedfrom
http://www.istari.ucam.org/maths/ or
http://www.cam.ac.uk/CambUniv/Societies/archim/notes.htm
[email protected]
setsyourequire.
v
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Chapter 1
Integers
Notation. The“naturalnumbers”,whichwewilldenotebyN,are
{1,2,3,...}.
TheintegersZare
{...,−2,−1,0,1,2,...}.
Wewillalsousethenon-negativeintegers,denotedeitherbyN orZ ,whichisN∪
0 +
{0}. TherearealsotherationalnumbersQandtherealnumbersR.
GivenasetS,wewritex∈S ifxbelongstoS,andx∈/ S otherwise.
Thereareoperations+and·onZ. Theyhavecertain“nice”propertieswhichwe
willtakeforgranted. Thereisalso“ordering”. Nissaidtobe“well-ordered”,which
meansthateverynon-emptysubsetofNhasaleastelement.Theprincipleofinduction
followsfromwell-ordering.
Proposition(PrincipleofInduction). LetP(n)beastatementaboutnforeachn∈
N. SupposeP(1)istrueandP(k)trueimpliesthatP(k+1)istrueforeachk ∈ N.
ThenP istrueforalln.
Proof. SupposeP isnottrueforalln. ThenconsiderthesubsetSofNofallnumbers
k for which P is false. Then S has a least element l. We know that P(l−1) is true
(sincel>1),sothatP(l)mustalsobetrue. ThisisacontradictionandP holdsforall
n.
1.1 Division
Given two integers a, b ∈ Z, we say that a divides b (and write a | b) if a (cid:54)= 0 and
b = a·q forsomeq ∈ Z(aisadivisorofb). aisaproperdivisorofbifaisnot±1
or±b.
Note. If a | b and b | c then a | c, for if b = q a and c = q b for q , q ∈ Z then
1 2 1 2
c=(q ·q )a.Ifd|aandd|bthend|ax+by.Theproofofthisisleftasanexercise.
1 2
1
2 CHAPTER1. INTEGERS
1.2 The division algorithm
Lemma1.1. Given a, b ∈ N there exist unique integers q, r ∈ N with a = qb+r,
0≤r <b.
Proof. Takeqthelargestpossiblesuchthatqb≤aandputr =a−qb.Then0≤r <b
sincea−qb≥0but(q+1)b≥a. Nowsupposethata=q b+rwithq ,r ∈Nand
1 1 1
0 ≤ r < b. Then0 = (q−q )b+(r−r )andb | r−r . But−b < r−r < bso
1 1 1 1 1
thatr =r andhenceq =q .
1 1
Itisclearthatb|aiffr =0intheabove.
Definition. Givena,b∈Nthend∈Nisthehighestcommonfactor(greatestcommon
divisor)ofaandbif:
1. d|aandd|b,
2. ifd(cid:48) |aandd(cid:48) |bthend(cid:48) |d(d(cid:48) ∈N).
Thehighestcommonfactor(henceforthhcf)ofaandbiswritten(a,b)orhcf(a,b).
Thehcfisobviouslyunique—ifcandc(cid:48)arebothhcf’sthentheybothdivideeach
otherandarethereforeequal.
Theorem1.1(Existanceofhcf). Fora, b ∈ Nhcf(a,b)exists. Moreoverthereexist
integersxandysuchthat(a,b)=ax+by.
Proof. ConsiderthesetI ={ax+by :x,y ∈Zandax+by >0}. ThenI (cid:54)=∅solet
dbetheleastmemberofI. Now∃x ,y suchthatd=ax +by ,sothatifd(cid:48) |aand
0 0 0 0
d(cid:48) |bthend(cid:48) |d.
Now write a = qd+r with q, r ∈ N , 0 ≤ r < d. We have r = a−qd =
0
a(1−qx )+b(−qy ).Sor =0,asotherwiser ∈I:contrarytodminimal.Similiarly,
0 0
d|bandthusdisthehcfofaandb.
Lemma1.2. If a, b ∈ N and a = qb + r with q, r ∈ N and 0 ≤ r < b then
0
(a,b)=(b,r).
Proof. Ifc|aandc|bthenc|randthusc|(b,r). Inparticular,(a,b)|(b,r). Now
notethatifc | bandc | r thenc | aandthusc | (a,b). Therefore(b,r) | (a,b)and
hence(b,r)=(a,b).
1.3 The Euclidean algorithm
Supposewewanttofind(525,231). Weuselemmas(1.1)and(1.2)toobtain:
525=2×231+63
231=3×63+42
63=1×42+21
42=2×21+0
So(525,231)=(231,63)=(63,42)=(42,21)=21. Ingeneral,tofind(a,b):
1.3. THEEUCLIDEANALGORITHM 3
a=q b+r with0<r <b
1 1 1
b=q r +r with0<r <r
2 1 2 2 1
r =q r +r with0<r <r
1 3 2 3 3 2
.
.
.
r =q r +r with0<r <r
i−2 i i−1 i i i−1
.
.
.
r =q r +r with0<r <r
n−3 n−1 n−2 n−1 n n−1
r =q r +0.
n−2 n n−1
This process must terminate as b > r > r > ··· > r > 0. Using Lemma
1 2 n−1
(1.2), (a,b) = (b,r ) = ··· = (r ,r ) = r . So (a,b) is the last non-zero
1 n−2 n−1 n−1
remainderinthisprocess.
Wenowwishtofindx andy ∈ Zwith(a,b) = ax +by . Wecandothisby
0 0 0 0
backsubstitution.
21=63−1×42
=63−(231−3×63)
=4×63−231
=4×(525−2×231)−231
=4×525−9×231.
This works in general but can be confusing and wasteful. These numbers can be
calculatedatthesametimeas(a,b)ifweknowweshallneedthem.
We introduce A and B . We put A = B = 0 and A = B = 1. We
i i −1 0 0 −1
iterativelydefine
A =q A +A
i i i−1 i−2
B =q B +B .
i i i−1 i−2
NowconsideraB −bA .
j j
Lemma1.3.
aB −bA =(−1)j+1r .
j j j
Proof. Weshalldothisusingstronginduction. Wecaneasilyseethat(1.3)holdsfor
j = 1 and j = 2. Now assume we are at i ≥ 2 and we have already checked that
r =(−1)i−1(aB −bA )andr =(−1)i(aB −bA ). Now
i−2 i−2 i−2 i−i i−1 i−1
r =r −q r
i i−2 i i−1
=(−1)i−1(aB −bA )−q (−1)i(aB −bA )
i−2 i−2 i i−1 i−1
=(−1)i+1(aB −bA ),usingthedefinitionofA andB .
i i i i
4 CHAPTER1. INTEGERS
Lemma1.4.
A B −A B =(−1)i
i i+1 i+1 i
Proof. ThisisdonebybacksubstitutionandusingthedefinitionofA andB .
i i
Animmediatecorollaryofthisisthat(A ,B )=1.
i i
Lemma1.5.
a b
A = B = .
n (a,b) n (a,b)
Proof. (1.3)fori = ngivesaB = bA . Therefore a B = b A . Now a
n n (a,b) n (a,b) n (a,b)
and b are coprime. A and B are coprime and thus this lemma is therefore an
(a,b) n n
immediateconsequenceofthefollowingtheorem.
Theorem1.2. Ifd|ceand(c,d)=1thend|e.
Proof. Since (c,d) = 1 we can write 1 = cx+dy for some x, y ∈ Z. Then e =
ecx+edyandd|e.
Definition. Theleastcommonmultiple(lcm)ofaandb(written[a,b])istheintegerl
suchthat
1. a|landb|l,
2. ifa|l(cid:48)andb|l(cid:48)thenl|l(cid:48).
Itiseasytoshowthat[a,b]= ab .
(a,b)
1.4 Applications of the Euclidean algorithm
Takea,bandc∈Z.Supposewewanttofindallthesolutionsx,y ∈Zofax+by =c.
Anecessaryconditionforasolutiontoexististhat(a,b)|c,soassumethis.
Lemma1.6. If(a,b)|cthenax+by =chassolutionsinZ.
Proof. Take x(cid:48) and y(cid:48) ∈ Z such that ax(cid:48) +by(cid:48) = (a,b). Then if c = q(a,b) then if
x =qx(cid:48)andy =qy(cid:48),ax +by =c.
0 0 0 0
Lemma1.7. Any other solution is of the form x = x + bk , y = y − ak for
0 (a,b) 0 (a,b)
k ∈Z.
Proof. These certainly work as solutions. Now suppose x and y is also a solution.
1 1
Then a (x −x ) = − b (y −y ). Since a and b are coprime we have
(a,b) 0 1 (a,b) 0 1 (a,b) (a,b)
a | (y − y ) and b | (x − x ). Say that y = y − ak , k ∈ Z. Then
(a,b) 0 1 (a,b) 0 1 1 0 (a,b)
x =x + bk .
1 0 (a,b)