Discrete Mathematics With Applications

1079 Pages·2003·12.811 MB·English

by  Koshy, Thomas

List of Symbols
Topic Symbol Meaning Page
LOGIC ∼p negationofp 8
p∨q disjunctionofpandq 7
p∧q conjunctionofpandq 5
pXORq exclusiveorofpandq 18
p→q pimpliesq 9
p(cid:6) q pifandonlyifq 14
p≡q pislogicallyequivalenttoq 20
F contradiction 21
T tautology 21
pNANDq notpandq 30
pNORq notporq 30
p|q Shefferstroke 31
P(x1,x2,...,xn) propositionalfunction 33
(∃x)P(x) ExistentialquantificationP(x) 32
(∀x)P(x) UniversalquantificationP(x) 32
∴ therefore 38
SETS {x1,x2,...,xn} setwithelementsx1,x2,...,andxn 68
{x|P(x)} set-buildernotation 69
Ø emptyset 70
U universalset 70
Z setofintegers 73
+
N,Z setofpositiveintegers 73
R setofrealnumbers 73
[a,b] closedinterval 73
[a,b) closed-openinterval 73
(a,b] open-closedinterval 73
(a,b) openinteval 73
A=B equalityofsetsAandB 70
x∈A xisanelementofA 68
x∈/ A xisnotanelementofA 68
A⊆B AisasubsetofB 69
P(A) powersetofA 72
|A| cardinalityofA 98
(a,b) orderedpair 86
(a1,a2,...,an) orderedset 86
A∪B unionofAandB 78
A∩B intersectionofAandB 78
A−B differenceofAandB 80
(cid:14)
A complementofA 81
A⊕B symmetricdifferenceofAandB 82
n
i=∪1Ai unionofsetsAi,i=1,2,...,n 86
n
i=∩1Ai intersectionofsetsAi,i=1,2,...,n 86
FUNCTIONS f(x) valueoffunctionfatx 119
f :A→B functionffromAtoB 118
f +g sumoffunctionsfandg 123
fg productoffunctionsfandg 123
Topic Symbol Meaning Page
iA identityfunctiononA 136
f−1 inverseoff 153
f ◦g compositionofgandf 151
amodb remainderwhenaisdividedbyb 132
adivb quotientwhenaisdividedbyb 132
(cid:17)x(cid:18) floorofx 126
(cid:19)x(cid:20) ceilingofx 127
sn nthtermthesequence{sn} 157
(cid:2)n
ai am+am+1+···+an 158
i=(cid:2)m
ai sumthetermsai,wherei∈S 160
i∈S
(cid:3)n
ai amam+1···an 163
i=m
n! nfactorial 241
f(x)=O(g(x)) f(x)isbig-ohofg(x) 237
f(x)=(cid:2)(g(x)) f(x)isbig-omegaofg(x) 243
f(x)=(cid:3)(g(x)) f(x)isbig-thetaofg(x) 245
min{x,y} minimumofxandy 252
max{x,y} maximumofxandy 251
MATRICES (aij)m×n m×nmatrixwithentriesaij 165
A+B sumofAandB 166
A−B differenceofAandB 167
kA scalarmultiplicationofAbyk 167
AB productofAandB 170
In identitymatrixofordern 166
AT transposeofA 174
INTEGERS a|b adividesb 189
a(cid:2)b aisnotafactorofb 189
gcd{a,b} greatestcommondivisoraandb 191
lcm{a,b} leastcommondivisorofaandb
(a1a2···an)b base-brepresentation 197
COMBINATORICS P(n,r) numberofr-permutationsofasetwithnelements 352
Dn numberofderangementsofnelements 361
C(cid:4)(n,r(cid:5)) numberofr-combinationsofasetwithnelements 366
n
binomialcoefficient:coefficientofxninexpansionof(1+x)n 366
r
N(Pi1Pi2...Pin) numberofelementshavingeachofthepropertiesPij, j=1,2,...,n 400
N(P(cid:14)i1P(cid:14)i2···P(cid:14)in) numberofelementsnothavinganyofthepropertiesPij, j=1,2,...,n 400
PROBABILITY P(E) probabilityofeventE 410
P(E|F) conditionalprobabilityofEgivenF 417
BOOLEAN A∨B joinofAandB 439
MATRICES A∧B meetofAandB 439
A(cid:21)B booleanproductofAandB 439
A[n] nthbooleanpowerofA 441
continued
Topic Symbol Meaning Page
RELATIONS S◦R compositeofrelationsSandR 463
Rn nthpowerofrelationR 466
R−1 inverserelation 469
a≡b(modm) aiscongruenttobmodulom 484
a(cid:22)≡b(modm) aisnotcongruenttobmodulom 484
[a]R equivalenceclassofawithrespecttoR 486
a≺b aislessthanb 495
a(cid:24)b aislessthanorequaltob 495
a(cid:25)b aisgreaterthanb
a(cid:26)b aisgreaterthanorequaltob
GRAPHS {u,v}or{u−v} undirectededge 517
G=(V,E) graphwithvertexsetVandedgesetE 517
deg(v) degreeofvertexv 520
Kn completegraphwithnvertices 524
Cn cyclewithnvertices 526
Wn wheelwithn+1vertices 526
Km,n completebipartitegraph 527
Qn n-cube 566
a-v1-···-vn−1-b pathfromatob 546
G1∪G2 unionofG1andG2 532
deg(R) degreeofregionR 581
BOOLEAN B {0,1} 807
ALGEBRA x,y booleanvariables 813
(cid:14)
x complementofx
x+y booleansumofxandy 814
x·y(orxy) booleanproductofxandy 814
x↑y xNANDy 820
x↓y xNORy 820
x x inverter 827
x
x+y ORgate 826
y
x
xy ANDgate 825
y
LANGUAGES xy concatenationofwordsxandy 76
λ emptystring 75
||x|| lengthofthestringx 75
wR reversalofw 113
AB concatenationoflanguagesAandB 736
∗
A Kleeneclosure 739
(cid:5) alphabet 75
FINITE-STATE σ startsymbol 744
MACHINES w→w(cid:14) production 744
w⇒w(cid:14) w(cid:14)isdirectlyderivablefromw 746
(cid:31)A nonterminalsymbol 748
(cid:31)A ::=(cid:31)B |x Backus–Naurform 748
(S,I,O,f,g,s0) finite-statemachinewithoutput 772
s0 startstate 760
(S,I,f,s0,F) finite-statemachinewithnooutput 761

Discrete Mathematics
with Applications

Discrete Mathematics
with Applications
Thomas Koshy
Framingham State College
Amsterdam Boston Heidelberg London NewYork Oxford Paris
SanDiego SanFrancisco Singapore Sydney Tokyo

Dedicated to
thememoryofaclosefriendandan
outstandingcolleagueKennethJ.
Preskenis(1940–2002)