andtheCellTheory.**CellStructure**:Explorationofthestructureoftypicaleukaryoticcells,including:**CellMembrane**:Structure,composition(lipidbilayer,proteins),functionsincelltransport,signaling,andrecognition.**Cy
m**:Composition,organellessuspendedwithin(e.g.,cytoskeleton,ribosomes).**Nucleus**:Structure,functions(DNAstorage,transcription,regulationofcellactivities).**Organelles**:Detailedexaminationoforganellessuchas
hondria,endoplasmicreticulum,Golgiapparatus,lysosomes,andtheirrolesincellularprocesses.**ProkaryoticCells**:Comparisonofprokaryoticandeukaryoticcellstructures,emphasizingdifferencesinorganelles,geneticmateria
cleusinprokaryotes),andcellularprocesses.**CellularOrganizationandFunction**:Integrationofcellularcomponentsandorganellestosupportcellularfunctionssuchasmetabolism,proteinsynthesis,energyproduction(e.g.,ATPsy
is),andmaintenanceofhomeostasis.**CellularTransport**:Mechanismsofcellulartransport,including:**PassiveTransport**:Diffusion,osmosis,facilitateddiffusion.**ActiveTransport**:Ionpumps,carrierproteins,endocyto
ocytosis.**BulkTransport**:Phagocytosis,pinocytosis.**CellCommunication**:Overviewofcellularcommunicationprocesses,including:**ReceptorProteins**:Recognitionofsignalingmolecules(ligands).**SignalTransd
**:Transmissionofsignalswithincells(e.g.,secondmessengers).**CellSignalingPathways**:Examplesofsignalingpathways(e.g.,hormonesignaling,neurotransmission).**CellCycleandCellDivision**:Phasesofthecellcycle(
hase,mitosis,cytokinesis)andtheirregulation.Importanceofcelldivisioningrowth,repair,andreproduction.**CancerandCellRegulation**:Introductiontocancerasuncontrolledcellgrowthanddivision.Causesofcancer(mutations
onmentalfactors),mechanismsoftumorformation,andapproachestocancertreatment.**StemCellsandRegenerativeMedicine**:Overviewofstemcells,theirproperties(pluripotent,multipotent),andpotentialapplicationsinregene
medicineandresearch.**MicroscopyandCellVisualization**:Techniquesandtypesofmicroscopyusedtostudycells,includinglightmicroscopy,electronmicroscopy,andfluorescencemicroscopy.**SummaryandKeyConcepts**
hapterconcludeswithasummaryofkeyconceptsrelatedtocellstructure,function,andorganization,reinforcingunderstandingofcellularbiologyfundamentals.Chapter4providesacomprehensiveexplorationofthestructureandfuncti
ells,highlightingtheircomplexityandorganizationatthemicroscopiclevel.Itservesasafoundationalchapterinunderstandinghowcellularstructuresandprocessescontributetothefunctioningoforganismsandecosystems.Ifyouhaves
cquestionsaboutanyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter4,feelfreetoask!Chapter5Chapter5of"EssentialsofBiology"bySylviaMaderandMichaelWindelspecht,titled"TheDynamic
ypicallyexploresthedynamicprocessesandactivitiesthatoccurwithincells.Here‘sanoverviewofwhatyoumightfindinthischapter:**IntroductiontoCellularDynamics**:Definitionofcellulardynamicsandanoverviewofthedynam
essesthatoccurwithincellstomaintainlifeandrespondtoenvironmentalchanges.**EnergyTransformations**:Explorationofcellularenergyprocesses,including:**CellularRespiration**:Overviewofaerobicandanaerobicrespira
ncludingglycolysis,Krebscycle,andoxidativephosphorylation(electrontransportchain).**Photosynthesis**:Overviewofphotosyntheticprocessesinplantsandalgae,includinglightreactionsandCalvincycle.**MetabolicPathwa
Detailedexaminationofmetabolicpathwayswithincells,including:**CatabolicPathways**:Breakdownofmolecules(e.g.,glucose)toreleaseenergy.**AnabolicPathways**:Synthesisofmolecules(e.g.,proteins,lipids)usingener
Cell
Mathematics for Computer Science
revised Wednesday 6th June, 2018, 13:43
Eric Lehman
Google Inc.
F Thomson Leighton
Department of Mathematics
and the Computer Science and AI Laboratory,
Massachussetts Institute of Technology;
Akamai Technologies
Albert R Meyer
Department of Electrical Engineering and Computer Science
and the Computer Science and AI Laboratory,
Massachussetts Institute of Technology
,Contents
I Proofs
Introduction 3
0.1 References 4
1 What is a Proof? 5
1.1 Propositions 5
1.2 Predicates 8
1.3 The Axiomatic Method 8
1.4 Our Axioms 9
1.5 Proving an Implication 11
1.6 Proving an ―If and Only If‖ 13
1.7 Proof by Cases 15
1.8 Proof by Contradiction 16
1.9 Good Proofs in Practice 17
1.10 References 19
2 The Well Ordering Principle 29
2.1 Well Ordering Proofs 29
2.2 Template for WOP Proofs 30
2.3 Factoring into Primes 32
2.4 Well Ordered Sets 33
3 Logical Formulas 47
3.1 Propositions from Propositions 48
3.2 Propositional Logic in Computer Programs 52
3.3 Equivalence and Validity 54
3.4 The Algebra of Propositions 57
3.5 The SAT Problem 62
3.6 Predicate Formulas 63
3.7 References 68
4 Mathematical Data Types 103
4.1 Sets 103
4.2 Sequences 108
4.3 Functions 109
4.4 Binary Relations 111
4.5 Finite Cardinality 115
,iv Contents
5 Induction 137
5.1 Ordinary Induction 137
5.2 Strong Induction 146
5.3 Strong Induction vs. Induction vs. Well Ordering 153
6 State Machines 173
6.1 States and Transitions 173
6.2 The Invariant Principle 174
6.3 Partial Correctness & Termination 182
6.4 The Stable Marriage Problem 187
7 Recursive Data Types 217
7.1 Recursive Definitions and Structural Induction 217
7.2 Strings of Matched Brackets 221
7.3 Recursive Functions on Nonnegative Integers 225
7.4 Arithmetic Expressions 227
7.5 Games as a Recursive Data Type 232
7.6 Search Trees 238
7.7 Induction in Computer Science 257
8 Infinite Sets 295
8.1 Infinite Cardinality 296
8.2 The Halting Problem 305
8.3 The Logic of Sets 309
8.4 Does All This Really Work? 314
II Structures
Introduction 339
9 Number Theory 341
9.1 Divisibility 341
9.2 The Greatest Common Divisor 346
9.3 Prime Mysteries 353
9.4 The Fundamental Theorem of Arithmetic 356
9.5 Alan Turing 358
9.6 Modular Arithmetic 362
9.7 Remainder Arithmetic 364
9.8 Turing‘s Code (Version 2.0) 367
9.9 Multiplicative Inverses and Cancelling 369
9.10 Euler‘s Theorem 373
, ―mcs‖ — 2018/6/6 — 13:43 — page v — #5
v Contents
9.11 RSA Public Key Encryption 378
9.12 What has SAT got to do with it? 381
9.13 References 382
10 Directed graphs & Partial Orders 421
10.1 Vertex Degrees 423
10.2 Walks and Paths 424
10.3 Adjacency Matrices 427
10.4 Walk Relations 430
10.5 Directed Acyclic Graphs & Scheduling 431
10.6 Partial Orders 439
10.7 Representing Partial Orders by Set Containment 442
10.8 Linear Orders 444
10.9 Product Orders 444
10.10 Equivalence Relations 445
10.11 Summary of Relational Properties 447
10.12 References 449
11 Communication Networks 481
11.1 Routing 481
11.2 Routing Measures 482
11.3 Network Designs 485
12 Simple Graphs 501
12.1 Vertex Adjacency and Degrees 501
12.2 Sexual Demographics in America 503
12.3 Some Common Graphs 505
12.4 Isomorphism 507
12.5 Bipartite Graphs & Matchings 509
12.6 Coloring 514
12.7 Walks in Simple Graphs 519
12.8 Connectivity 521
12.9 Special Walks and Tours 523
12.10 k-connected Graphs 525
12.11 Forests & Trees 527
12.12 References 535
13 Planar Graphs 575
13.1 Drawing Graphs in the Plane 575
13.2 Definitions of Planar Graphs 575
13.3 Euler‘s Formula 586
13.4 Bounding the Number of Edges in a Planar Graph 587
m**:Composition,organellessuspendedwithin(e.g.,cytoskeleton,ribosomes).**Nucleus**:Structure,functions(DNAstorage,transcription,regulationofcellactivities).**Organelles**:Detailedexaminationoforganellessuchas
hondria,endoplasmicreticulum,Golgiapparatus,lysosomes,andtheirrolesincellularprocesses.**ProkaryoticCells**:Comparisonofprokaryoticandeukaryoticcellstructures,emphasizingdifferencesinorganelles,geneticmateria
cleusinprokaryotes),andcellularprocesses.**CellularOrganizationandFunction**:Integrationofcellularcomponentsandorganellestosupportcellularfunctionssuchasmetabolism,proteinsynthesis,energyproduction(e.g.,ATPsy
is),andmaintenanceofhomeostasis.**CellularTransport**:Mechanismsofcellulartransport,including:**PassiveTransport**:Diffusion,osmosis,facilitateddiffusion.**ActiveTransport**:Ionpumps,carrierproteins,endocyto
ocytosis.**BulkTransport**:Phagocytosis,pinocytosis.**CellCommunication**:Overviewofcellularcommunicationprocesses,including:**ReceptorProteins**:Recognitionofsignalingmolecules(ligands).**SignalTransd
**:Transmissionofsignalswithincells(e.g.,secondmessengers).**CellSignalingPathways**:Examplesofsignalingpathways(e.g.,hormonesignaling,neurotransmission).**CellCycleandCellDivision**:Phasesofthecellcycle(
hase,mitosis,cytokinesis)andtheirregulation.Importanceofcelldivisioningrowth,repair,andreproduction.**CancerandCellRegulation**:Introductiontocancerasuncontrolledcellgrowthanddivision.Causesofcancer(mutations
onmentalfactors),mechanismsoftumorformation,andapproachestocancertreatment.**StemCellsandRegenerativeMedicine**:Overviewofstemcells,theirproperties(pluripotent,multipotent),andpotentialapplicationsinregene
medicineandresearch.**MicroscopyandCellVisualization**:Techniquesandtypesofmicroscopyusedtostudycells,includinglightmicroscopy,electronmicroscopy,andfluorescencemicroscopy.**SummaryandKeyConcepts**
hapterconcludeswithasummaryofkeyconceptsrelatedtocellstructure,function,andorganization,reinforcingunderstandingofcellularbiologyfundamentals.Chapter4providesacomprehensiveexplorationofthestructureandfuncti
ells,highlightingtheircomplexityandorganizationatthemicroscopiclevel.Itservesasafoundationalchapterinunderstandinghowcellularstructuresandprocessescontributetothefunctioningoforganismsandecosystems.Ifyouhaves
cquestionsaboutanyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter4,feelfreetoask!Chapter5Chapter5of"EssentialsofBiology"bySylviaMaderandMichaelWindelspecht,titled"TheDynamic
ypicallyexploresthedynamicprocessesandactivitiesthatoccurwithincells.Here‘sanoverviewofwhatyoumightfindinthischapter:**IntroductiontoCellularDynamics**:Definitionofcellulardynamicsandanoverviewofthedynam
essesthatoccurwithincellstomaintainlifeandrespondtoenvironmentalchanges.**EnergyTransformations**:Explorationofcellularenergyprocesses,including:**CellularRespiration**:Overviewofaerobicandanaerobicrespira
ncludingglycolysis,Krebscycle,andoxidativephosphorylation(electrontransportchain).**Photosynthesis**:Overviewofphotosyntheticprocessesinplantsandalgae,includinglightreactionsandCalvincycle.**MetabolicPathwa
Detailedexaminationofmetabolicpathwayswithincells,including:**CatabolicPathways**:Breakdownofmolecules(e.g.,glucose)toreleaseenergy.**AnabolicPathways**:Synthesisofmolecules(e.g.,proteins,lipids)usingener
Cell
Mathematics for Computer Science
revised Wednesday 6th June, 2018, 13:43
Eric Lehman
Google Inc.
F Thomson Leighton
Department of Mathematics
and the Computer Science and AI Laboratory,
Massachussetts Institute of Technology;
Akamai Technologies
Albert R Meyer
Department of Electrical Engineering and Computer Science
and the Computer Science and AI Laboratory,
Massachussetts Institute of Technology
,Contents
I Proofs
Introduction 3
0.1 References 4
1 What is a Proof? 5
1.1 Propositions 5
1.2 Predicates 8
1.3 The Axiomatic Method 8
1.4 Our Axioms 9
1.5 Proving an Implication 11
1.6 Proving an ―If and Only If‖ 13
1.7 Proof by Cases 15
1.8 Proof by Contradiction 16
1.9 Good Proofs in Practice 17
1.10 References 19
2 The Well Ordering Principle 29
2.1 Well Ordering Proofs 29
2.2 Template for WOP Proofs 30
2.3 Factoring into Primes 32
2.4 Well Ordered Sets 33
3 Logical Formulas 47
3.1 Propositions from Propositions 48
3.2 Propositional Logic in Computer Programs 52
3.3 Equivalence and Validity 54
3.4 The Algebra of Propositions 57
3.5 The SAT Problem 62
3.6 Predicate Formulas 63
3.7 References 68
4 Mathematical Data Types 103
4.1 Sets 103
4.2 Sequences 108
4.3 Functions 109
4.4 Binary Relations 111
4.5 Finite Cardinality 115
,iv Contents
5 Induction 137
5.1 Ordinary Induction 137
5.2 Strong Induction 146
5.3 Strong Induction vs. Induction vs. Well Ordering 153
6 State Machines 173
6.1 States and Transitions 173
6.2 The Invariant Principle 174
6.3 Partial Correctness & Termination 182
6.4 The Stable Marriage Problem 187
7 Recursive Data Types 217
7.1 Recursive Definitions and Structural Induction 217
7.2 Strings of Matched Brackets 221
7.3 Recursive Functions on Nonnegative Integers 225
7.4 Arithmetic Expressions 227
7.5 Games as a Recursive Data Type 232
7.6 Search Trees 238
7.7 Induction in Computer Science 257
8 Infinite Sets 295
8.1 Infinite Cardinality 296
8.2 The Halting Problem 305
8.3 The Logic of Sets 309
8.4 Does All This Really Work? 314
II Structures
Introduction 339
9 Number Theory 341
9.1 Divisibility 341
9.2 The Greatest Common Divisor 346
9.3 Prime Mysteries 353
9.4 The Fundamental Theorem of Arithmetic 356
9.5 Alan Turing 358
9.6 Modular Arithmetic 362
9.7 Remainder Arithmetic 364
9.8 Turing‘s Code (Version 2.0) 367
9.9 Multiplicative Inverses and Cancelling 369
9.10 Euler‘s Theorem 373
, ―mcs‖ — 2018/6/6 — 13:43 — page v — #5
v Contents
9.11 RSA Public Key Encryption 378
9.12 What has SAT got to do with it? 381
9.13 References 382
10 Directed graphs & Partial Orders 421
10.1 Vertex Degrees 423
10.2 Walks and Paths 424
10.3 Adjacency Matrices 427
10.4 Walk Relations 430
10.5 Directed Acyclic Graphs & Scheduling 431
10.6 Partial Orders 439
10.7 Representing Partial Orders by Set Containment 442
10.8 Linear Orders 444
10.9 Product Orders 444
10.10 Equivalence Relations 445
10.11 Summary of Relational Properties 447
10.12 References 449
11 Communication Networks 481
11.1 Routing 481
11.2 Routing Measures 482
11.3 Network Designs 485
12 Simple Graphs 501
12.1 Vertex Adjacency and Degrees 501
12.2 Sexual Demographics in America 503
12.3 Some Common Graphs 505
12.4 Isomorphism 507
12.5 Bipartite Graphs & Matchings 509
12.6 Coloring 514
12.7 Walks in Simple Graphs 519
12.8 Connectivity 521
12.9 Special Walks and Tours 523
12.10 k-connected Graphs 525
12.11 Forests & Trees 527
12.12 References 535
13 Planar Graphs 575
13.1 Drawing Graphs in the Plane 575
13.2 Definitions of Planar Graphs 575
13.3 Euler‘s Formula 586
13.4 Bounding the Number of Edges in a Planar Graph 587
