Discrete mathematics.doc

7/24/2019 Discrete mathematics.doc

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Discrete mathematics

Discrete mathematics, also called finite mathematicsor decision mathematics, is the study of

mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the

notion of continuity. Objects studied in finite mathematics are largely countable sets such as integers,

finite graphs,and formal languages.

Discrete mathematics has become popular in recent decades because of its applications to computer

science. Concepts and notations from discrete mathematics are useful to study or describe objects or

problems in computer algorithmsand programming languages. In some mathematics curricula, finite

mathematics courses cover discrete mathematical concepts for business, while discrete mathematics

courses emphasize concepts for computer science majors.

Prepared by Mr. Romeo Balcita
http://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Countable_setshttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Graph_(mathematics)http://en.wikipedia.org/wiki/Formal_languagehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Programming_languagehttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Countable_setshttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Graph_(mathematics)http://en.wikipedia.org/wiki/Formal_languagehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Programming_language


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Discrete mathematics includes the following topics:

Prelims:

Logic a study of reasoning

!"ature of logico !.!Consistency, soundness, and completenesso !.#$ival conceptions of logico !.%Deductive and inductive reasoning

#&istory of logic %'opics in logic

o %.!(yllogistic logico %.#)redicate logico %.%*odal logico %.+Deduction and reasoningo %.*athematical logico %.-)hilosophical logico %./ogic and computation

o %.01rgumentation theory Set theory a study of collections of elements

o 1 History

o 2 Basic concepts

o 3 Interpretations

o 4 Axiomatic set theory

o 5 Applications

o

6 Areas of study 6.1 Cominatorial set theory

6.2 !escripti"e set theory

6.3 #u$$y set theory

6.4 Inner model theory

6.5 %ar&e cardinals

6.6 !eterminacy

6.' #orcin&

6.( Cardinal in"ariants

6.) *et+theoretic topolo&y

Combinatorics,o 2numerative combinatorics

)ermutations with repetitions )ermutations without repetitions Combinations without repetitions Combinations with repetitions 3ibonacci numbers
http://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Logic#Nature_of_logichttp://en.wikipedia.org/wiki/Logic#Nature_of_logichttp://en.wikipedia.org/wiki/Logic#Nature_of_logichttp://en.wikipedia.org/wiki/Logic#Consistency.2C_soundness.2C__and_completenesshttp://en.wikipedia.org/wiki/Logic#Consistency.2C_soundness.2C__and_completenesshttp://en.wikipedia.org/wiki/Logic#Consistency.2C_soundness.2C__and_completenesshttp://en.wikipedia.org/wiki/Logic#Rival_conceptions_of_logichttp://en.wikipedia.org/wiki/Logic#Rival_conceptions_of_logichttp://en.wikipedia.org/wiki/Logic#Rival_conceptions_of_logichttp://en.wikipedia.org/wiki/Logic#Deductive_and_inductive_reasoninghttp://en.wikipedia.org/wiki/Logic#Deductive_and_inductive_reasoninghttp://en.wikipedia.org/wiki/Logic#Deductive_and_inductive_reasoninghttp://en.wikipedia.org/wiki/Logic#History_of_logichttp://en.wikipedia.org/wiki/Logic#History_of_logichttp://en.wikipedia.org/wiki/Logic#History_of_logichttp://en.wikipedia.org/wiki/Logic#Topics_in_logichttp://en.wikipedia.org/wiki/Logic#Topics_in_logichttp://en.wikipedia.org/wiki/Logic#Topics_in_logichttp://en.wikipedia.org/wiki/Logic#Syllogistic_logichttp://en.wikipedia.org/wiki/Logic#Syllogistic_logichttp://en.wikipedia.org/wiki/Logic#Syllogistic_logichttp://en.wikipedia.org/wiki/Logic#Syllogistic_logichttp://en.wikipedia.org/wiki/Logic#Predicate_logichttp://en.wikipedia.org/wiki/Logic#Predicate_logichttp://en.wikipedia.org/wiki/Logic#Predicate_logichttp://en.wikipedia.org/wiki/Logic#Modal_logichttp://en.wikipedia.org/wiki/Logic#Modal_logichttp://en.wikipedia.org/wiki/Logic#Modal_logichttp://en.wikipedia.org/wiki/Logic#Deduction_and_reasoninghttp://en.wikipedia.org/wiki/Logic#Deduction_and_reasoninghttp://en.wikipedia.org/wiki/Logic#Deduction_and_reasoninghttp://en.wikipedia.org/wiki/Logic#Mathematical_logichttp://en.wikipedia.org/wiki/Logic#Mathematical_logichttp://en.wikipedia.org/wiki/Logic#Mathematical_logichttp://en.wikipedia.org/wiki/Logic#Philosophical_logichttp://en.wikipedia.org/wiki/Logic#Philosophical_logichttp://en.wikipedia.org/wiki/Logic#Philosophical_logichttp://en.wikipedia.org/wiki/Logic#Logic_and_computationhttp://en.wikipedia.org/wiki/Logic#Logic_and_computationhttp://en.wikipedia.org/wiki/Logic#Logic_and_computationhttp://en.wikipedia.org/wiki/Logic#Argumentation_theoryhttp://en.wikipedia.org/wiki/Logic#Argumentation_theoryhttp://en.wikipedia.org/wiki/Logic#Argumentation_theoryhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Historyhttp://en.wikipedia.org/wiki/Set_theory#Basic_conceptshttp://en.wikipedia.org/wiki/Set_theory#Interpretationshttp://en.wikipedia.org/wiki/Set_theory#Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Applicationshttp://en.wikipedia.org/wiki/Set_theory#Areas_of_studyhttp://en.wikipedia.org/wiki/Set_theory#Combinatorial_set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Descriptive_set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Fuzzy_set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Inner_model_theoryhttp://en.wikipedia.org/wiki/Set_theory#Large_cardinalshttp://en.wikipedia.org/wiki/Set_theory#Determinacyhttp://en.wikipedia.org/wiki/Set_theory#Forcinghttp://en.wikipedia.org/wiki/Set_theory#Cardinal_invariantshttp://en.wikipedia.org/wiki/Set_theory#Set-theoretic_topologyhttp://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Combinatorics#Permutations_with_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Permutations_without_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Combinations_without_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Combinations_with_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Fibonacci_numbershttp://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Logic#Nature_of_logichttp://en.wikipedia.org/wiki/Logic#Consistency.2C_soundness.2C__and_completenesshttp://en.wikipedia.org/wiki/Logic#Rival_conceptions_of_logichttp://en.wikipedia.org/wiki/Logic#Deductive_and_inductive_reasoninghttp://en.wikipedia.org/wiki/Logic#History_of_logichttp://en.wikipedia.org/wiki/Logic#Topics_in_logichttp://en.wikipedia.org/wiki/Logic#Syllogistic_logichttp://en.wikipedia.org/wiki/Logic#Predicate_logichttp://en.wikipedia.org/wiki/Logic#Modal_logichttp://en.wikipedia.org/wiki/Logic#Deduction_and_reasoninghttp://en.wikipedia.org/wiki/Logic#Mathematical_logichttp://en.wikipedia.org/wiki/Logic#Philosophical_logichttp://en.wikipedia.org/wiki/Logic#Logic_and_computationhttp://en.wikipedia.org/wiki/Logic#Argumentation_theoryhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Historyhttp://en.wikipedia.org/wiki/Set_theory#Basic_conceptshttp://en.wikipedia.org/wiki/Set_theory#Interpretationshttp://en.wikipedia.org/wiki/Set_theory#Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Applicationshttp://en.wikipedia.org/wiki/Set_theory#Areas_of_studyhttp://en.wikipedia.org/wiki/Set_theory#Combinatorial_set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Descriptive_set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Fuzzy_set_theoryhttp://en.wikipedia.org/wiki/Set_theory#Inner_model_theoryhttp://en.wikipedia.org/wiki/Set_theory#Large_cardinalshttp://en.wikipedia.org/wiki/Set_theory#Determinacyhttp://en.wikipedia.org/wiki/Set_theory#Forcinghttp://en.wikipedia.org/wiki/Set_theory#Cardinal_invariantshttp://en.wikipedia.org/wiki/Set_theory#Set-theoretic_topologyhttp://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Combinatorics#Permutations_with_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Permutations_without_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Combinations_without_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Combinations_with_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Fibonacci_numbers


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Midterms:

Algorithmics a study of methods of calculation

! 2tymology # 4hy algorithms are necessary5 an informal definition % 3ormalization of algorithms + 26ample

o +.! 1lgorithm analysis Classes

o - /egal issues &istory5 Development of the notion of 7algorithm7

Digital geometry Computabilityand complexitytheories dealing with theoretical and practical limitation

of algorithms Computability 'heory

! Introduction # 3ormal models of computation % )ower of automata

o %.! )ower of finite state machineso %.# )ower of pushdown automatao %.% )ower of 'uring machines

+ Concurrencybased models 8nreasonable models of computation

Comple6ity

! Disorganized comple6ity vs. organized comple6ity # (ources of comple6ity % (pecific meanings of comple6ity + (tudy of comple6ity Comple6ity topics

o .! Comple6 behaviouro .# Comple6 mechanismso .% Comple6 simulationso .+ Comple6 systemso . Comple6ity in datao .- Comple6 )roject *anagement

- 1pplications of comple6ity
http://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Algorithm#Etymologyhttp://en.wikipedia.org/wiki/Algorithm#Why_algorithms_are_necessary:_an_informal_definitionhttp://en.wikipedia.org/wiki/Algorithm#Formalization_of_algorithmshttp://en.wikipedia.org/wiki/Algorithm#Examplehttp://en.wikipedia.org/wiki/Algorithm#Algorithm_analysishttp://en.wikipedia.org/wiki/Algorithm#Classeshttp://en.wikipedia.org/wiki/Algorithm#Legal_issueshttp://en.wikipedia.org/wiki/Algorithm#History:_Development_of_the_notion_of_.22algorithm.22http://en.wikipedia.org/wiki/Digital_geometryhttp://en.wikipedia.org/wiki/Computabilityhttp://en.wikipedia.org/wiki/Complexityhttp://en.wikipedia.org/wiki/Computabilityhttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Introductionhttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Formal_models_of_computationhttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Power_of_automatahttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Power_of_finite_state_machineshttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Power_of_pushdown_automatahttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Power_of_Turing_machineshttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Concurrency-based_modelshttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Unreasonable_models_of_computationhttp://en.wikipedia.org/wiki/Complexityhttp://en.wikipedia.org/wiki/Complexity#Disorganized_complexity_vs._organized_complexityhttp://en.wikipedia.org/wiki/Complexity#Sources_of_complexityhttp://en.wikipedia.org/wiki/Complexity#Specific_meanings_of_complexityhttp://en.wikipedia.org/wiki/Complexity#Study_of_complexityhttp://en.wikipedia.org/wiki/Complexity#Complexity_topicshttp://en.wikipedia.org/wiki/Complexity#Complex_behaviourhttp://en.wikipedia.org/wiki/Complexity#Complex_mechanismshttp://en.wikipedia.org/wiki/Complexity#Complex_simulationshttp://en.wikipedia.org/wiki/Complexity#Complex_systemshttp://en.wikipedia.org/wiki/Complexity#Complexity_in_datahttp://en.wikipedia.org/wiki/Complexity#Complex_Project_Managementhttp://en.wikipedia.org/wiki/Complexity#Applications_of_complexityhttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Algorithm#Etymologyhttp://en.wikipedia.org/wiki/Algorithm#Why_algorithms_are_necessary:_an_informal_definitionhttp://en.wikipedia.org/wiki/Algorithm#Formalization_of_algorithmshttp://en.wikipedia.org/wiki/Algorithm#Examplehttp://en.wikipedia.org/wiki/Algorithm#Algorithm_analysishttp://en.wikipedia.org/wiki/Algorithm#Classeshttp://en.wikipedia.org/wiki/Algorithm#Legal_issueshttp://en.wikipedia.org/wiki/Algorithm#History:_Development_of_the_notion_of_.22algorithm.22http://en.wikipedia.org/wiki/Digital_geometryhttp://en.wikipedia.org/wiki/Computabilityhttp://en.wikipedia.org/wiki/Complexityhttp://en.wikipedia.org/wiki/Computabilityhttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Introductionhttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Formal_models_of_computationhttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Power_of_automatahttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Power_of_finite_state_machineshttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Power_of_pushdown_automatahttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Power_of_Turing_machineshttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Concurrency-based_modelshttp://en.wikipedia.org/wiki/Computability_theory_(computer_science)#Unreasonable_models_of_computationhttp://en.wikipedia.org/wiki/Complexityhttp://en.wikipedia.org/wiki/Complexity#Disorganized_complexity_vs._organized_complexityhttp://en.wikipedia.org/wiki/Complexity#Sources_of_complexityhttp://en.wikipedia.org/wiki/Complexity#Specific_meanings_of_complexityhttp://en.wikipedia.org/wiki/Complexity#Study_of_complexityhttp://en.wikipedia.org/wiki/Complexity#Complexity_topicshttp://en.wikipedia.org/wiki/Complexity#Complex_behaviourhttp://en.wikipedia.org/wiki/Complexity#Complex_mechanismshttp://en.wikipedia.org/wiki/Complexity#Complex_simulationshttp://en.wikipedia.org/wiki/Complexity#Complex_systemshttp://en.wikipedia.org/wiki/Complexity#Complexity_in_datahttp://en.wikipedia.org/wiki/Complexity#Complex_Project_Managementhttp://en.wikipedia.org/wiki/Complexity#Applications_of_complexity


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Finals:

Proofso ! *ethods of proof

!.! Direct proof !.# )roof by induction

!.% )roof by transposition !.+ )roof by contradiction !. )roof by construction !.- )roof by e6haustion !. )robabilistic proof !.0 Combinatorial proof !.9 "onconstructive proof !.!: )roof nor disproof !.!! 2lementary proof

o # 2nd of a proof Graph Theory

o . )rominent graphtheorists ! &istory

# Drawing graphs % ;raphtheoretic data structures

o %.! /ist structureso %.# *atri6 structures

+ )roblems in graph theoryo +.! 2numerationo +.# (ubgraphs, induced subgraphs,

and minorso +.% ;raph coloringo +.+ $oute problemso +. "etwor< flowo +.- =isibility graph problemso +. Covering problems

1pplications - $eferences (ee also

o .! $elated topicso .# 1lgorithmso .% (ubareaso .+ $elated areas of mathematics
http://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Methods_of_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Direct_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_inductionhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_transpositionhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_contradictionhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_constructionhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_exhaustionhttp://en.wikipedia.org/wiki/Mathematical_proof#Probabilistic_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Combinatorial_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Nonconstructive_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_nor_disproofhttp://en.wikipedia.org/wiki/Mathematical_proof#Elementary_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#End_of_a_proofhttp://en.wikipedia.org/wiki/Graph_theory#Prominent_graph_theoristshttp://en.wikipedia.org/wiki/Graph_theory#Prominent_graph_theoristshttp://en.wikipedia.org/wiki/Graph_theory#Historyhttp://en.wikipedia.org/wiki/Graph_theory#Drawing_graphshttp://en.wikipedia.org/wiki/Graph_theory#Graph-theoretic_data_structureshttp://en.wikipedia.org/wiki/Graph_theory#List_structureshttp://en.wikipedia.org/wiki/Graph_theory#Matrix_structureshttp://en.wikipedia.org/wiki/Graph_theory#Problems_in_graph_theoryhttp://en.wikipedia.org/wiki/Graph_theory#Enumerationhttp://en.wikipedia.org/wiki/Graph_theory#Subgraphs.2C_induced_subgraphs.2C_and_minorshttp://en.wikipedia.org/wiki/Graph_theory#Subgraphs.2C_induced_subgraphs.2C_and_minorshttp://en.wikipedia.org/wiki/Graph_theory#Graph_coloringhttp://en.wikipedia.org/wiki/Graph_theory#Route_problemshttp://en.wikipedia.org/wiki/Graph_theory#Network_flowhttp://en.wikipedia.org/wiki/Graph_theory#Visibility_graph_problemshttp://en.wikipedia.org/wiki/Graph_theory#Covering_problemshttp://en.wikipedia.org/wiki/Graph_theory#Applicationshttp://en.wikipedia.org/wiki/Graph_theory#Referenceshttp://en.wikipedia.org/wiki/Graph_theory#See_alsohttp://en.wikipedia.org/wiki/Graph_theory#Related_topicshttp://en.wikipedia.org/wiki/Graph_theory#Algorithmshttp://en.wikipedia.org/wiki/Graph_theory#Subareashttp://en.wikipedia.org/wiki/Graph_theory#Related_areas_of_mathematicshttp://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Methods_of_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Direct_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_inductionhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_transpositionhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_contradictionhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_constructionhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_exhaustionhttp://en.wikipedia.org/wiki/Mathematical_proof#Probabilistic_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Combinatorial_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Nonconstructive_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#Proof_nor_disproofhttp://en.wikipedia.org/wiki/Mathematical_proof#Elementary_proofhttp://en.wikipedia.org/wiki/Mathematical_proof#End_of_a_proofhttp://en.wikipedia.org/wiki/Graph_theory#Historyhttp://en.wikipedia.org/wiki/Graph_theory#Drawing_graphshttp://en.wikipedia.org/wiki/Graph_theory#Graph-theoretic_data_structureshttp://en.wikipedia.org/wiki/Graph_theory#List_structureshttp://en.wikipedia.org/wiki/Graph_theory#Matrix_structureshttp://en.wikipedia.org/wiki/Graph_theory#Problems_in_graph_theoryhttp://en.wikipedia.org/wiki/Graph_theory#Enumerationhttp://en.wikipedia.org/wiki/Graph_theory#Subgraphs.2C_induced_subgraphs.2C_and_minorshttp://en.wikipedia.org/wiki/Graph_theory#Subgraphs.2C_induced_subgraphs.2C_and_minorshttp://en.wikipedia.org/wiki/Graph_theory#Graph_coloringhttp://en.wikipedia.org/wiki/Graph_theory#Route_problemshttp://en.wikipedia.org/wiki/Graph_theory#Network_flowhttp://en.wikipedia.org/wiki/Graph_theory#Visibility_graph_problemshttp://en.wikipedia.org/wiki/Graph_theory#Covering_problemshttp://en.wikipedia.org/wiki/Graph_theory#Applicationshttp://en.wikipedia.org/wiki/Graph_theory#Referenceshttp://en.wikipedia.org/wiki/Graph_theory#See_alsohttp://en.wikipedia.org/wiki/Graph_theory#Related_topicshttp://en.wikipedia.org/wiki/Graph_theory#Algorithmshttp://en.wikipedia.org/wiki/Graph_theory#Subareashttp://en.wikipedia.org/wiki/Graph_theory#Related_areas_of_mathematicshttp://en.wikipedia.org/wiki/Graph_theory#Prominent_graph_theoristshttp://en.wikipedia.org/wiki/Graph_theory#Prominent_graph_theorists


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References

Donald 2. >nuth,The Art of Computer Programming

>enneth &. $osen, Handbook of Discrete and Combinatorial MathematicsC$C )ress. I(?" :

0+9%:!+9!.

>enneth &. $osen, Discrete Mathematics and Its Applicationsth ed. *c;raw &ill. I(?" ::

#9%:%%:. Companion 4eb site5 http5@@www.mhhe.com@math@advmath@rosen@

$ichard Aohnsonbaugh, Discrete Mathematics-th ed. *acmillan. I(?" :!%:+0:%!.

Companion 4eb site5 http5@@wps.prenhall.com@esmBjohnsonbauBdiscrtmathB-@

$alph ). ;rimaldi, Discrete and Combinatorial Mathematics: An Applied Introductionth ed.

1ddison 4esley. I(?" :#:!#-%+%

"orman /. ?iggs, Discrete Mathematics#nd ed. O6ford 8niversity )ress. I(?" :!90:!0.

Companion 4eb site5 http5@@www.oup.co.u


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Prelims

Logic

Logicis the study of the principles of valid inference and demonstration. 'he word derives from ;ree

"#$%&'logike, fem. of"#$%&()logikos, 7possessed of reason, intellectual, dialectical, argumentativefrom"($#)logos, 7word, thought, idea, argument, account, reason, or principle7.

1s a formal science,logic investigates and classifies the structure of statementsand arguments, bothrough the study of formal systems of inferenceand through the study of arguments in naturalanguage. 'he field of logic ranges from core topics such as the study of validity, fallacies anparado6es, to specialized analysis of reasoning using probabilityand to arguments involving causalit/ogic is also commonly used today in argumentation theory.E%F

'raditionally, logic was considered a branch of philosophy, a part of the classical triviumof grammalogic, and rhetoric. (ince the midnineteenth century formal logichas been studied in the conte6t o

foundations of mathematics, where it was often called symbolic logic. In !09 3rege publishe?egriffsschrift 5 1 formula language or pure thought modelled on that of arithemetic which inauguratemodern logic with the invention of quantifiernotation. In !9:% 1lfred "orth 4hitehead and ?ertran$ussellattempted to establish logic formally as the cornerstone of mathematics with the publication o)rincipia *athematica.E+F&owever, e6cept for the elementary part, the system of )rincipia is no longemuch used, having been largely superseded by set theory. 1t the same time the developments in thfield of /ogic since 3rege, $usselland 4ittgensteinhad a profound influence on both the practice ophilosophy and the ideas concerning the nature of philosophical problems especially in the 2nglisspea


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Contents

!"ature of logico !.!Consistency, soundness, and completenesso !.#$ival conceptions of logico !.%Deductive and inductive reasoning

#&istory of logic %'opics in logic

o %.!(yllogistic logico %.#)redicate logico %.%*odal logico %.+Deduction and reasoningo %.*athematical logico %.-)hilosophical logico %./ogic and computationo %.01rgumentation theory

Nature of logic

3orm is central to logic. It complicates e6position that GformalG in 7formal logic7 is commonly used in anambiguous manner. (ymbolic logic is just one


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Consistency,which means that none of the theorems of the system contradict oneanother.

Soundness, which means that the systemGs rules of proof will never allow a falseinference from a true premise. If a system is sound and its a6ioms are true then itstheorems are also guaranteed to be true.

Completeness, which means that there are no true sentences in the system that cannot,at least in principle, be proved in the system.

"ot all systems achieve all three virtues. 'he wor< of >urt ;delhas shown that no useful system ofarithmetic can be both consistent and complete5

Ri$al conceptions of logic

/ogic arose from a concern with correctness of argumentation. *odern logicians usually wish to ensurethat logic studies just those arguments that arise from appropriately general forms of inferenceH so fore6ample the (tanford 2ncyclopedia of )hilosophysays of logic that it 7does not, however, cover goodreasoning as a whole. 'hat is the job of the theory of rationality. $ather it deals with inferences whosevalidity can be traced bac< to the formal features of the representations that are involved in thatinference, be they linguistic, mental, or other representations7 &ofweber #::+.

?y contrast, Immanuel >antargued that logic should be conceived as the science of judgment, an ideata


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'he first sustained wor< on the subject of logic which has survived was that of 1ristotle.E!:F'he formallysophisticated treatmentEcitation neededFof modern logic descends from the ;ree< tradition, the latter mainlybeing informed from the transmission of1ristotelian logic.

/ogic in Islamic philosophyalso contributed to the development of modern logic, which included thedevelopment of 71vicennian logic7 as an alternative to 1ristotelian logic.1vicennaGssystem of logic wasresponsible for the introduction of hypothetical syllogism,E!!Ftemporalmodal logic,E!#FE!%Fand inductivelogic.E!+FE!F'he rise of the1shariteschool, however, limited original wor< on logic in Islamic philosophy,

though it did continue into the !th century and had a significant influence on 2uropean logic during the$enaissance.

In India, innovations in the scholastic school, called "yaya, continued from ancient times into the early!0th century, though it did not survive long into the colonial period. In the #:th century, westernphilosophers lilaus ;lashoff have tried to e6plore certain aspects of theIndian tradition of logic. 1ccording to &ermann 4eyl!9#95

Occidental mathematics has in past centuries bro


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Predicate logic

/ogic as it is studied today is a very different subject to that studied before, and the principal differenceis the innovation of predicate logic. 4hereas 1ristotelian syllogistic logic specified the forms that therelevant part of the involved judgements too


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'his motivation is still alive, although it no longer ta


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#athematical logic

*athematical logic really refers to two distinct areas of research5 the first is the application of thetechniques of formal logic to mathematics and mathematical reasoning, and the second, in the otherdirection, the application of mathematical techniques to the representation and analysis of formal logic.

'he earliest use of mathematics and geometryin relation to logic and philosophy goes bac< to the

ancient ;reerip


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/ogic and the philosophy of language are closely related. )hilosophy of language has to do with thestudy of how our language engages and interacts with our thinnowledge representation formalisms and methods,&orn clausesin logic programming, and description logic.

3urthermore, computers can be used as tools for logicians. 3or e6ample, in symbolic logic andmathematical logic, proofs by humans can be computerassisted. 8sing automated theorem provingthemachines can find and chec< proofs, as well as wor< with proofs too lengthy to be written out by hand.

Argumentation theory

1rgumentation theoryis the study and research of informal logic, fallacies, and critical questions as the

relate to every day and practical situations. (pecific types of dialogue can be analyzed and questionedto reveal premises, conclusions, and fallacies. 1rgumentation theory is now applied in artificialintelligenceand law.
http://en.wikipedia.org/wiki/Alan_Turinghttp://en.wikipedia.org/wiki/Entscheidungsproblemhttp://en.wikipedia.org/wiki/Kurt_G%C3%83%C2%B6delhttp://en.wikipedia.org/wiki/Incompleteness_theoremshttp://en.wikipedia.org/wiki/Mathematical_notationhttp://en.wikipedia.org/wiki/Logic_programminghttp://en.wikipedia.org/wiki/Prologhttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Argumentation_theoryhttp://en.wikipedia.org/wiki/ACM_Computing_Classification_Systemhttp://en.wikipedia.org/wiki/ACM_Computing_Classification_Systemhttp://en.wikipedia.org/w/index.php?title=Logics_and_meanings_of_programs&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mathematical_logic_and_formal_languages&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mathematical_logic_and_formal_languages&action=edit&redlink=1http://en.wikipedia.org/wiki/Formal_semantics_of_programming_languageshttp://en.wikipedia.org/wiki/Formal_semantics_of_programming_languageshttp://en.wikipedia.org/wiki/Formal_methodshttp://en.wikipedia.org/wiki/Hoare_logichttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/w/index.php?title=Arithmetic_and_logic_structures&action=edit&redlink=1http://en.wikipedia.org/wiki/Modal_logichttp://en.wikipedia.org/wiki/Default_logichttp://en.wikipedia.org/wiki/Knowledge_representation_formalisms_and_methodshttp://en.wikipedia.org/wiki/Horn_clausehttp://en.wikipedia.org/wiki/Logic_programminghttp://en.wikipedia.org/wiki/Description_logichttp://en.wikipedia.org/wiki/Automated_theorem_provinghttp://en.wikipedia.org/wiki/Argumentation_theoryhttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Lawhttp://en.wikipedia.org/wiki/Alan_Turinghttp://en.wikipedia.org/wiki/Entscheidungsproblemhttp://en.wikipedia.org/wiki/Kurt_G%C3%83%C2%B6delhttp://en.wikipedia.org/wiki/Incompleteness_theoremshttp://en.wikipedia.org/wiki/Mathematical_notationhttp://en.wikipedia.org/wiki/Logic_programminghttp://en.wikipedia.org/wiki/Prologhttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Argumentation_theoryhttp://en.wikipedia.org/wiki/ACM_Computing_Classification_Systemhttp://en.wikipedia.org/wiki/ACM_Computing_Classification_Systemhttp://en.wikipedia.org/w/index.php?title=Logics_and_meanings_of_programs&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mathematical_logic_and_formal_languages&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mathematical_logic_and_formal_languages&action=edit&redlink=1http://en.wikipedia.org/wiki/Formal_semantics_of_programming_languageshttp://en.wikipedia.org/wiki/Formal_semantics_of_programming_languageshttp://en.wikipedia.org/wiki/Formal_methodshttp://en.wikipedia.org/wiki/Hoare_logichttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/w/index.php?title=Arithmetic_and_logic_structures&action=edit&redlink=1http://en.wikipedia.org/wiki/Modal_logichttp://en.wikipedia.org/wiki/Default_logichttp://en.wikipedia.org/wiki/Knowledge_representation_formalisms_and_methodshttp://en.wikipedia.org/wiki/Horn_clausehttp://en.wikipedia.org/wiki/Logic_programminghttp://en.wikipedia.org/wiki/Description_logichttp://en.wikipedia.org/wiki/Automated_theorem_provinghttp://en.wikipedia.org/wiki/Argumentation_theoryhttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Law


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Set theory

Set theoryis the branch of mathematicsthat studies sets, which are collections of objects. 1lthough antype of objects can be collected into a set, set theory is applied most often to objects that are relevant tomathematics.

'he modern study of set theory was initiated by Cantorand Dede


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?eginning with the wor< of Nenoaround +: ?C, mathematicians had been struggling with the conceptof infinity.2specially notable is the wor< of ?ernard ?olzanoin the first half of the !9th century. 'hemodern understanding of infinity began !0-!, with ;eorg CantorGs wor< on number theory. 1n !0#meeting between Cantor and Dede


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Interpretations

1


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(et theory is also a promising foundational system for much of mathematics. (ince the publication of thfirst volume of Principia Mathematica, it has been claimed that most or even all mathematical theoremscan be derived using an aptly designed set of a6ioms for set theory, augmented with many definitions,using first or second order logic (ee *etamath. 3or e6ample, properties of the naturaland realnumberscan be derived within set theory, as each number system can be identified with a set ofequivalence classes under a suitable equivalence relationwhose field is some infinite set.

(et theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete

mathematicsis li


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!nner model theory

1n inner modelof NermeloK3raen


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Set)theoretic topology

Set)theoretic topologystudies questions of general topologythat are settheoretic in nature orthat require advanced methods of set theory for their solution. *any of these theorems are independentof N3C, requiring stronger a6ioms for their proof. 1 famous problem is the normal *oore space questiona question in general topology that was the subject of intense research. 'he answer to the normal

*oore space question was eventually proved to be independent of N3C.

Set

'his article is about mathematical sets. 3or other uses, see (et disambiguation.This article gi,es an introduction to what mathematicians call 5intuiti,e5 or 5nai,e5 set theor!6 for more detailed account see 7ai,e set theor!8 9or a rigorous modern aiomatictreatment of sets;seeAiomatic set theor!8

1 setis a collection of distinct objects considered as a whole. (ets are one of the most fundamentalconceptsin mathematics. 'he study of the structure of sets, set theory, is rich and ongoing. &aving onlybeen invented at the end of the !9th century, set theory is now a ubiquitous part of mathematicseducation, being introduced from primary schoolin many countries.Ecitation neededF(et theory can be viewedas a foundation from which nearly all of mathematics can be derived.

In philosophy, sets are ordinarily considered to be abstract objectsE!FE#F

E%F

E+F

the physical to


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Contents

EhideF ! Definition # Describing sets % *embership + Cardinality (ubsets

o .! )ower set - (pecial sets ?asic operations

o .! 8nionso .# Intersectionso .% Complementso .+ Cartesian product

Definition

1t the beginning of his 1eitr


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P-, !!Q U P!!, -Q U P!!, !!, -, !!Q

are equivalent, because the e6tensional specification means merely that each of the elements listed is amember of the set.

3or sets with many elements, the enumeration of members can be abbreviated. 3or instance, the set ofthe first thousand positive whole numbers may be specified e6tensionally as5

P!, #, %, ..., !:::Q,where the ellipsis7...7 indicates that the list continues in the obvious way. 2llipses may also be usedwhere sets have infinitely many members. 'hus the set of positive even numbers can be written as P#, +-, 0, ... Q.

'he notation with braces may also be used in an intensional specification of a set. In this usage, thebraces have the meaning 7the set of all ...7 (o EU Pplayingcard suitsQ is the set whose four membersare V, W, X, and Y. 1 more general form of this is setbuilder notation, through which, for instance, the se9of the twenty smallest integers that are four less than perfect squarescan be denoted5

9U P n#

Z + nis an integerH and : [ n[ !9QIn this notation, the colon757 means 7such that7, and the description can be interpreted as 79is the setof all numbers of the form n#Z +, such that nis a whole number in the range from : to !9 inclusive.7(ometimes the vertical bar7\7 is used instead of the colon.

One often has the choice of specifying a set intensionally or e6tensionally. In the e6amples above, forinstance,AU Cand 1U D.

Membership

If something is or is not an element of a particular set then this is symbolised by

and respectively.(o, with respect to the sets defined above5

+ Aand #0 9since #0 U !] Z +H but 9 9and green 1.

Cardinality

'he cardinality \>\ of a set >is 7the number of members of >.7 3or e6ample, since the 3rench flag hasthree colors, \1\ U %.

'here is a set with no members and zero cardinality, which is called the empt! setor the null set and denoted by the symbol ^. 3or e6ample, the setAof all threesided squares has zero members \A\ U :and thusAU ^. 'hough, li


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Subsets

If every member of setAis also a member of set 1, thenAis said to be a subsetof 1, writtenalso pronouncedA is contained in 1. 2quivalently, we can write , read as 1 is a superset of A,1 includes A, or 1 contains A. 'he relationshipbetween sets established by is called inclusionorcontainment.

IfAis a subset of, but not equal to, 1, thenAis called aproper subsetof 1, written A is a

proper subset of 1 or 1 is proper superset of A.

"ote that the e6pressions and are used differently by different authorsH some authorsuse them to mean the same as respectively , whereas other use them to mean the

same as respectively .

Ais a subsetof 1

26ample5

'he set of all men is a proper subsetof the set of all people.

'he empty set is a subset of every set and every set is a subset of itself5

Po%er set

'he power set of a set >can be defined as the set of all subsets of >. 'his includes the subsets formedfrom the members of >and the empty set. If a finite set >has cardinality nthen the power set of >hascardinality #n. If >is an infinite either countableor uncountable set then the power set of >is alwaysuncountable. 'he power set can be written as #>.

1s an e6ample, the power set #P!, #, %Qof P!, #, %Q is equal to the set PP!, #, %Q, P!, #Q, P!, %Q, P#, %Q, P!Q, P#Q,P%Q, ^Q. 'he cardinality of the original set is %, and the cardinality of the power set is # %, or 0. 'hisrelationship is one of the reasons for the terminology power set. (imilarly, its notation is an e6ample of ageneral conventionproviding notations for sets based on their cardinalities.
http://en.wikipedia.org/wiki/Relation_(mathematics)http://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Countablehttp://en.wikipedia.org/wiki/Uncountablehttp://en.wikipedia.org/wiki/Combinatorics#set_sizes_motivate_a_naming_conventionhttp://en.wikipedia.org/wiki/Image:Venn_A_subset_B.svghttp://en.wikipedia.org/wiki/Relation_(mathematics)http://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Countablehttp://en.wikipedia.org/wiki/Uncountablehttp://en.wikipedia.org/wiki/Combinatorics#set_sizes_motivate_a_naming_convention


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Special sets

'here are some sets which hold great mathematical importance and are referred to with such regularitythat they have acquired special names and notational conventions to identify them. One of these is theempty set. *any of these sets are represented using ?lac


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!ntersections

1 new set can also be constructed by determining which members two sets have 7in common7. 'heintersectionofAand 1, denoted byA 1, is the set of all things which are members of bothAand 1. I

A 1 U ^, thenAand 1are said to be dis?oint.

'he intersectionofAand 1

26amples5

P!, #Q Pred, whiteQ U ^

P!, #, greenQ Pred, white, greenQ U PgreenQ P!, #Q P!, #Q U P!, #Q.

(ome basic properties of intersections5

A 1 U 1A A 1 8 A AA U A A ^ U ^ A 8 1if and only ifA 1UA.

Complements

'wo sets can also be 7subtracted7. 'he relati,e complementofAin 1also called the set theoreticdifferenceof 1andA, denoted by 1A, or 1ZA is the set of all elements which are members of 1,but not members ofA. "ote that it is valid to 7subtract7 members of a set that are not in the set, such asremoving greenfrom P!,#,%QH doing so has no effect.

In certain settings all sets under discussion are considered to be subsets of a given universal set*. Insuch cases, *A, is called the absolute complementor simply complementofA, and is denoted byA.

'he relati$e complementofAin 1

'he complementofAin *

26amples5

P!, #Q Pred, whiteQ U P!, #Q
http://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Universe_(mathematics)http://en.wikipedia.org/wiki/Image:Venn1100.svghttp://en.wikipedia.org/wiki/Image:Venn0100.svghttp://en.wikipedia.org/wiki/Image:Venn0001.svghttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Universe_(mathematics)


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P!, #, greenQ Pred, white, greenQ U P!, #Q P!, #Q P!, #Q U If *is the set of integers, Eis the set of even integers, and -is the set of odd integers,

then the complement of Ein *is -, or equivalently, E U -.

(ome basic properties of complements5

A8A@U *

AA@U A@ UA AAU A 1UA 1@.

Cartesian product

1 new set can be constructed by associating every element of one set with every element of another se'he Cartesian productof two setsAand 1, denoted byA 1is the set of all ordered pairsa, b such

that ais a member ofAand bis a member of 1.

26amples5

P!, #Q Pred, whiteQ U P!,red, !,white, #,red, #,whiteQ P!, #, greenQ Pred, white, greenQ U P!,red, !,white, !,green, #,red, #,white, #,green,

green,red, green,white, green,greenQ P!, #Q P!, #Q U P!,!, !,#, #,!, #,#Q

(ome basic properties of cartesian products5

A U AU A 18 C U A 1 8 A C A8 1 CU A C 8 1 C

/etAand 1be finite setsH

\A 1\ U \1A\ U \A\ \1\
http://en.wikipedia.org/wiki/Ordered_pairshttp://en.wikipedia.org/wiki/Ordered_pairs


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Combinatorics

Combinatoricsis a branch of pure mathematicsconcerning the study of discreteand usually finiteobjects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic

theoryand geometry, as well as to applied subjects in computer scienceand statistical physics. 1spectof combinatorics include 7counting7 the objects satisfying certain criteria enumerati,e combinatorics,deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria as incombinatorial designsandmatroidtheor!, finding 7largest7, 7smallest7, or 7optimal7 objects etremalcombinatoricsand combinatorial optimi/ation, and finding algebraicstructures these objects may havealgebraic combinatorics.

Combinatorics is as much about problem solving as theory building, though it has developed powerfultheoretical methods, especially since the later twentieth century see the page /ist of combinatoricstopicsfor details of the more recent development of the subject. One of the oldest and most accessibleparts of combinatorics is graph theory, which also has numerous natural connections to other areas.

'here are many combinatorial patterns and theoremsrelated to the structure of combinatoric sets.'hese often focus on a partitionor ordered partitionof a set. (ee the /ist of partition topicsfor ane6panded list of related topics or the /ist of combinatorics topicsfor a more general listing. (ome of themore notable results are highlighted below.

1n e6ample of a simple combinatorial question is the following5 4hat is the number of possibleorderings of a dec< of # distinct playing cardsM 'he answer is # # factorial, which is equal to abou0.:-0 !:-.

1nother e6ample of a more difficult problem5 ;iven a certain number nof people, is it possible to assignthem to sets so that each person is in at least one set, each pair of people is in e6actly one set togetherevery two sets have e6actly one person in common, and no set contains everyone, all but one person,or e6actly one personM 'he answer depends on n. (ee 7Design theory7 below.

Combinatorics is used frequently in computer scienceto obtain estimates on the number of elements ocertain sets. 1 mathematician who studies combinatorics is often referred to as a combinatorialistorcombinatorist.

Contents

2numerative combinatorics

o )ermutations with repetitionso )ermutations without repetitionso Combinations without repetitionso Combinations with repetitions

o 3ibonacci numbers
http://en.wikipedia.org/wiki/Pure_mathematicshttp://en.wikipedia.org/wiki/Countable_sethttp://en.wikipedia.org/wiki/Finite_sethttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Ergodic_theoryhttp://en.wikipedia.org/wiki/Ergodic_theoryhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Statistical_physicshttp://en.wikipedia.org/wiki/Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Combinatorial_designhttp://en.wikipedia.org/wiki/Matroidhttp://en.wikipedia.org/wiki/Extremal_combinatoricshttp://en.wikipedia.org/wiki/Extremal_combinatoricshttp://en.wikipedia.org/wiki/Combinatorial_optimizationhttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Algebraic_combinatoricshttp://en.wikipedia.org/wiki/List_of_combinatorics_topicshttp://en.wikipedia.org/wiki/List_of_combinatorics_topicshttp://en.wikipedia.org/wiki/Graph_theoryhttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Partition_of_a_sethttp://en.wikipedia.org/wiki/Ordered_partitionhttp://en.wikipedia.org/wiki/List_of_partition_topicshttp://en.wikipedia.org/wiki/List_of_combinatorics_topicshttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Combinatorics#Design_theoryhttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Combinatorics#Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Combinatorics#Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Combinatorics#Permutations_with_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Permutations_without_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Combinations_without_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Combinations_with_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Fibonacci_numbershttp://en.wikipedia.org/wiki/Pure_mathematicshttp://en.wikipedia.org/wiki/Countable_sethttp://en.wikipedia.org/wiki/Finite_sethttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Ergodic_theoryhttp://en.wikipedia.org/wiki/Ergodic_theoryhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Statistical_physicshttp://en.wikipedia.org/wiki/Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Combinatorial_designhttp://en.wikipedia.org/wiki/Matroidhttp://en.wikipedia.org/wiki/Extremal_combinatoricshttp://en.wikipedia.org/wiki/Extremal_combinatoricshttp://en.wikipedia.org/wiki/Combinatorial_optimizationhttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Algebraic_combinatoricshttp://en.wikipedia.org/wiki/List_of_combinatorics_topicshttp://en.wikipedia.org/wiki/List_of_combinatorics_topicshttp://en.wikipedia.org/wiki/Graph_theoryhttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Partition_of_a_sethttp://en.wikipedia.org/wiki/Ordered_partitionhttp://en.wikipedia.org/wiki/List_of_partition_topicshttp://en.wikipedia.org/wiki/List_of_combinatorics_topicshttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Combinatorics#Design_theoryhttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Combinatorics#Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Combinatorics#Permutations_with_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Permutations_without_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Combinations_without_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Combinations_with_repetitionshttp://en.wikipedia.org/wiki/Combinatorics#Fibonacci_numbers


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numerati!e combinatorics

Counting the number of ways that certain patterns can be formed is the central problem of enumerativecombinatorics. 'wo e6amples of this type of problem are counting combinations and countingpermutations as discussed in the previous section. *ore generally, given an infinite collection of finitesets P>iQ inde6ed by the natural numbers,enumerative combinatorics seenfor each n. 1lthough counting the number of elements

in a set is a rather broad mathematical problem, many of the problems that arise in applications have arelatively simple combinatorial description.

'he simplest such functions are closed formulas, which can be e6pressed as a composition ofelementary functions such as factorials, powers, and so on. 3or instance, as shown below, the numberof different possible orderings of a dec< of ncards is fn U n. Often, no closed form is initially availableIn these cases, we frequently first derive a recurrence relation, then solve the recurrence to arrive at thedesired closed form.

3inally, fn may be e6pressed by a formal power series, called its generating function, which is mostcommonly either the ordinary generating function

or the e6ponential generating function

Often, a complicated closed formula yields little insight into the behavior of the counting function as thenumber of counted objects grows. In these cases, a simple asymptoticappro6imation may be

preferable. 1 function gn is an asymptotic appro6imation to fn if as infinity. In

this case, we write .

Once determined, the generating function may allow one to e6tract all the information given by theprevious approaches. In addition, the various natural operations on generating functions such asaddition, multiplication, differentiation, etc., have a combinatorial significanceH this allows one to e6tendresults from one combinatorial problem in order to solve others.

Permutations%ith repetitions

4hen the order matters, and an object can be chosen more than once, the number of permutations is

where nis the number of objects from which you can choose and ris the number to be chosen.

3or e6ample, if you have the letters 1, ?, C, and D and you wish to discover the number of ways toarrange them in three letter patterns trigrams

!. order matters e.g., 1? is different from ?1, both are included as possibilities
http://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Mathematical_problemhttp://en.wikipedia.org/wiki/Closed_formulahttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Formal_power_serieshttp://en.wikipedia.org/wiki/Generating_functionhttp://en.wikipedia.org/wiki/Ordinary_generating_functionhttp://en.wikipedia.org/wiki/Exponential_generating_functionhttp://en.wikipedia.org/wiki/Asymptotichttp://en.wikipedia.org/wiki/Extended_real_number_linehttp://en.wikipedia.org/wiki/Permutationshttp://en.wikipedia.org/wiki/Trigramhttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Mathematical_problemhttp://en.wikipedia.org/wiki/Closed_formulahttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Formal_power_serieshttp://en.wikipedia.org/wiki/Generating_functionhttp://en.wikipedia.org/wiki/Ordinary_generating_functionhttp://en.wikipedia.org/wiki/Exponential_generating_functionhttp://en.wikipedia.org/wiki/Asymptotichttp://en.wikipedia.org/wiki/Extended_real_number_linehttp://en.wikipedia.org/wiki/Permutationshttp://en.wikipedia.org/wiki/Trigram


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#. an object can be chosen more than once 11 possible

you find that there are +%or -+ ways. 'his is because for the first slot you can choose any of the fourvalues, for the second slot you can choose any of the four, and for the final slot you can choose any ofthe four letters. *ultiplying them together gives the total.

Permutations %ithout repetitions

4hen the order matters and each object can be chosen only once, then the number of permutations is

where nis the number of objects from which you can choose, ris the number be chosen and 77 is the standard symbol meaning factorial.

3or e6ample, if you have five people and are going to choose three out of these, you will have @ Z %U -: permutations.

"ote that if nU rmeaning the number of chosen elements is equal to the number of elements to choosfromH five people and pic< all five then the formula becomes

where : U !.

3or e6ample, if you have the same five people and you want to find out how many ways you mayarrange them, it would be or + % # ! U !#: ways. 'he reason for this is that you can choosefrom for the initial slot, then you are left with only + to choose from for the second slot etc. *ultiplyingthem together gives the total of !#:.

Combinations%ithout repetitions

4hen the order does not matter and each object can be chosen only once, the number of combinationsis the binomial coefficient5

where nis the number of objects from which you can choose and kis the number to be chosen.

3or e6ample, if you have ten numbers and wish to choose you would have !:@!: Z U ##ways to choose. 'he binomial coefficient is also used to calculate the number of permutations in alottery.

Combinations %ith repetitions

Main articles: MultisetMultiset coefficientsand >tars and bars Bprobabilit!
http://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Combinationshttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Multiset#Multiset_coefficientshttp://en.wikipedia.org/wiki/Stars_and_bars_(probability)http://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Combinationshttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Multiset#Multiset_coefficientshttp://en.wikipedia.org/wiki/Stars_and_bars_(probability)


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4hen the order does not matter and an object can be chosen more than once, then the number ofcombinations is

where nis the number of objects from which you can choose and kis the number to be chosen.

3or e6ample, if you have ten types of donuts n on a menu to choose from and you want three donutsk there are !: S % Z ! @ %!: Z ! U ##: ways to choose see also multiset.

"ibonacci numbers

/et fn be the number of distinct subsets of the set that do not contain twoconsecutive integers. 4hen nU +, we have the sets PQ, P!Q, P#Q, P%Q, P+Q, P!,%Q, P!,+Q, P#,+Q, so f+ U 0. 4count the desired subsets of >n by separately counting those subsets that contain element nandthose that do not. If a subset contains n, then it does not contain element nZ !. (o there are e6actly fnZ # of the desired subsets that contain element n. 'he number of subsets that do not contain nis simp

fnZ !. 1dding these numbers together, we get the recurrence relation5

where f! U # and f# U %.

1s early as !#:#, /eonardo 3ibonaccistudied these numbers. 'hey are now called 3ibonacci numbersin particular, fn is


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MidtermsAlgorithm

3lowchartsare often used to graphically represent algorithms.

In mathematics,computing, linguisticsand related disciplines, an algorithmis a sequence ofinstructions, often used for calculation, data processing. It is formally a type of effective methodin whica list of welldefined instructions for completing a tas< will, when given an initial state, proceed through awelldefined series of successive states, eventually terminating in an endstate. 'he transition from onestate to the ne6t is not necessarily deterministicH some algorithms, leene !9+%5#+ or 7effective method7 $osser !9%95##H thoseformalizations included the ;del&erbrand>leene recursive functionsof !9%:, !9%+ and !9%,1lonzoChurchGslambda calculus of !9%-, 2mil )ostGs73ormulation I7 of !9%-, and1lan 'uringGs'uringmachines of !9%- and !9%9.
http://en.wikipedia.org/wiki/Flowcharthttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Computinghttp://en.wikipedia.org/wiki/Linguisticshttp://en.wikipedia.org/wiki/Calculationhttp://en.wikipedia.org/wiki/Data_processinghttp://en.wikipedia.org/wiki/Effective_methodhttp://en.wikipedia.org/wiki/Deterministichttp://en.wikipedia.org/wiki/Probabilistic_algorithmshttp://en.wikipedia.org/wiki/Entscheidungsproblemhttp://en.wikipedia.org/wiki/David_Hilberthttp://en.wikipedia.org/wiki/Effective_calculabilityhttp://en.wikipedia.org/wiki/Recursion_(computer_science)http://en.wikipedia.org/wiki/Alonzo_Churchhttp://en.wikipedia.org/wiki/Alonzo_Churchhttp://en.wikipedia.org/wiki/Lambda_calculushttp://en.wikipedia.org/wiki/Emil_Posthttp://en.wikipedia.org/wiki/Alan_Turinghttp://en.wikipedia.org/wiki/Turing_machineshttp://en.wikipedia.org/wiki/Turing_machineshttp://en.wikipedia.org/wiki/Image:LampFlowchart.svghttp://en.wikipedia.org/wiki/Image:LampFlowchart.svghttp://en.wikipedia.org/wiki/Flowcharthttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Computinghttp://en.wikipedia.org/wiki/Linguisticshttp://en.wikipedia.org/wiki/Calculationhttp://en.wikipedia.org/wiki/Data_processinghttp://en.wikipedia.org/wiki/Effective_methodhttp://en.wikipedia.org/wiki/Deterministichttp://en.wikipedia.org/wiki/Probabilistic_algorithmshttp://en.wikipedia.org/wiki/Entscheidungsproblemhttp://en.wikipedia.org/wiki/David_Hilberthttp://en.wikipedia.org/wiki/Effective_calculabilityhttp://en.wikipedia.org/wiki/Recursion_(computer_science)http://en.wikipedia.org/wiki/Alonzo_Churchhttp://en.wikipedia.org/wiki/Alonzo_Churchhttp://en.wikipedia.org/wiki/Lambda_calculushttp://en.wikipedia.org/wiki/Emil_Posthttp://en.wikipedia.org/wiki/Alan_Turinghttp://en.wikipedia.org/wiki/Turing_machineshttp://en.wikipedia.org/wiki/Turing_machines


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human %ho is capable of carrying out only $ery elementary operations on symbols B1oolos effre! FG; FGGG;p8 FG

'he words 7enumerably infinite7 mean 7countable using integers perhaps e6tending to infinity7. 'hus?oolos and Aeffrey are saying that an algorithm impliesinstructions for a process that 7creates7 outputintegers from an arbitrar!7input7 integer or integers that, in theory, can be chosen from : to infinity. 'huwe might e6pect an algorithm to be an algebraic equation such as y + m nJ two arbitrary 7inputvariables7 mand nthat produce an output y. 1s we see in1lgorithm characterizationsJ the word

algorithm implies much more than this, something on the order of for our addition e6ample5)recise instructions in language understood by 7the computer7 for a 7fast, efficient, good7

processthat specifies the 7moves7 of 7the computer7 machine or human, equipped with thenecessary internallycontained information and capabilities to find, decode, and then muncharbitrary input integers@symbols mand n, symbols and +... and reliably, correctly, 7effectively7produce, in a 7reasonable7 time, outputinteger yat a specified place and in a specified format.

'he concept of algorithmis also used to define the notion of decidability. 'hat notion is central fore6plaining how formal systemscome into being starting from a small set of a6iomsand rules. In logic,the time that an algorithm requires to complete cannot be measured, as it is not apparently related withour customary physical dimension. 3rom such uncertainties, that characterize ongoing wor


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described as starting 7from the top7 and going 7down to the bottom7, an idea that is described moreformally by flow of control.

(o far, this discussion of the formalization of an algorithm has assumed the premises of imperativeprogramming. 'his is the most common conception, and it attempts to describe a tas< in discrete,7mechanical7 means. 8nique to this conception of formalized algorithms is the assignment operation,setting the value of a variable. It derives from the intuition of 7memory7 as a scratchpad. 'here is ane6ample below of such an assignment.

3or some alternate conceptions of what constitutes an algorithm see functional programmingand logicprogramming.

Termination

(ome writers restrict the definition of algorithmto procedures that eventually finish. In such a category>leene places the 7decision procedureor decision methodor algorithmfor the question7 >leene!9#5!%-. Others, including >leene, include procedures that could run forever without stoppingH such aprocedure has been called a 7computational method7 >nuth !995 or 7calculation procedureoralgorithm7 >leene !9#5!%H however, >leene notes that such a method must eventually e6hibit 7some

object7 >leene !9#5!%.*ins


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-xpressing algorithms

1lgorithms can be e6pressed in many


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/igh)le$el description

!. 1ssume the first item is largest.#. /oo< at each of the remaining items in the list and if it is larger than the largest item so far, ma


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Classes

'here are various ways to classify algorithms, each with its own merits.

Classification by implementation

One way to classify algorithms is by implementation means.

Recursionor iteration5 1 recursive algorithmis one that invo


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Di$ide and con5uer. 1 divide and conquer algorithmrepeatedly reduces an instance of aproblem to one or more smaller instances of the same problem usually recursively, until theinstances are small enough to solve easily. One such e6ample of divide and conquer is mergesorting. (orting can be done on each segment of data after dividing data into segments andsorting of entire data can be obtained in conquer phase by merging them. 1 simpler variant ofdivide and conquer is called decrease and con5uer algorithm, that solves an identicalsubproblem and uses the solution of this subproblem to solve the bigger problem. Divide andconquer divides the problem into multiple subproblems and so conquer stage will be more

comple6 than decrease and conquer algorithms. 1n e6ample of decrease and conquer algorithmis binary search algorithm. Dynamic programming. 4hen a problem shows optimal substructure, meaning the optimal

solution to a problem can be constructed from optimal solutions to subproblems, and overlappingsubproblems, meaning the same subproblems are used to solve many different probleminstances, a quic


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1. )robabilistic algorithms are those that maee also: 4ist of algorithms

2very field of science has its own problems and needs efficient algorithms. $elated problems in one fielare often studied together. (ome e6ample classes are search algorithms, sorting algorithms, mergealgorithms,numerical algorithms, graph algorithms, string algorithms, computational geometricalgorithms,combinatorial algorithms,machine learning, cryptography, data compressionalgorithms andparsing techniques.

3ields tend to overlap with each other, and algorithm advances in one field may improve those of other,sometimes completely unrelated, fields. 3or e6ample, dynamic programming was originally invented foroptimization of resource consumption in industry, but is now used in solving a broad range of problemsin many fields.

Classification by complexity

>ee also: Compleit! class

1lgorithms can be classified by the amount of time they need to complete compared to their input size.'here is a wide variety5 some algorithms complete in linear time relative to input size, some do so in ane6ponential amount of time or even worse, and some never halt. 1dditionally, some problems may havemultiple algorithms of differing comple6ity, while other problems might have no algorithms or no


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?urgin #::, p. #+ uses a generalized definition of algorithms that rela6es the common requirementthat the output of the algorithm that computes a function must be determined after a finite number ofsteps. &e defines a superrecursive class of algorithms as 7a class of algorithms in which it is possible tcompute functions not computable by any 'uring machine7 ?urgin #::, p. !:. 'his is closely relatedto the study of methods of hypercomputation.

'egal issues

>ee also: >oftware patents for a general o,er,iew of the patentabilit! of software; including

computerLimplemented algorithms8

1lgorithms, by themselves, are not usually patentable. In the 8nited (tates, a claim consisting solely ofsimple manipulations of abstract concepts, numbers, or signals do not constitute 7processes7 8()'O#::- and hence algorithms are not patentable as in ;ottschal< v. ?enson. &owever, practicalapplications of algorithms are sometimes patentable. 3or e6ample, in Diamond v. Diehr, the applicationof a simple feedbachwarizmiwhose wor


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#echanical contri$ances %ith discrete states

The cloc'5 ?olter credits the invention of the weightdriven cloc


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calculation7 or 7effective calculability7 i.e., a calculation that would succeed. In rapid succession thefollowing appeared51lonzo Church,(tephen >leeneand A.?. $osserGscalculus, cf footnote in1lonzChurch!9%-a59:, !9%-b5!!: a finelyhoned definition of 7general recursion7 from the wor< of ;delacting on suggestions of Aacques &erbrandcf ;delGs )rinceton lectures of !9%+ and subsequentsimplifications by >leene !9%-5#%ff, !9+%5#ff. ChurchGs proof !9%-500ff that the2ntscheidungsproblemwas unsolvable, 2mil )ostGsdefinition of effective calculability as a wor


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out on onedimensional paper, i.e., on a tape divided into squares. I shall also suppose that thenumber of symbols which may be printed is finite....7'he behavior of the computer at any moment is determined by the symbols which he isobserving, and his 7state of mind7 at that moment. 4e may suppose that there is a bound ? to thnumber of symbols or squares which the computer can observe at one moment. If he wishes toobserve more, he must use successive observations. 4e will also suppose that the number ofstates of mind which need be taleene, Church, 'uring and )ostF . . . 4e may ta


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equivalent, so it doesnGt matter which one is used. *oreover, the fact that all three are equivalentis a very strong argument for the correctness of any one.7 $osser !9%95##K-

$osserGs footnote references the wor< of ! Church and >leene and their definition of definability,in particular ChurchGs use of it in hisAn *nsol,able Problem of Elementar! 7umber Theor!!9%-H #&erbrand and ;del and their use of recursion in particular ;delGs use in his famous paper -n9ormall! *ndecidable Propositions of Principia Mathematica and +elated >!stems I!9%!H and %)ost !9%- and 'uring !9%- in their mechanismmodels of computation.

Stephen C? leenedefined as his nowfamous 7'hesis I7 leene !9+%5#%

/istory after .


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Digital geometry

Digital geometrydeals with discretesets usually discrete pointsets considered to be digitizedmodelor imagesof objects of the #D or %D 2uclidean space.

(imply put, digiti(ingis replacing an object by a discrete set of its points. 'he images we see on the '=

screen, the raster display of a computer, or in newspapers are in fact digitalimages.Its main application areas are computer graphicsand image analysis.

*ain aspects of study are5

Constructing digitized representations of objects, with the emphasis on precision and efficiencyeither by means of synthesis, see, for e6ample, ?resenhamGs line algorithmor digital dis


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Computability theory 2computer science4

3or the branch of mathematical logic called computability theory, see $ecursion theory.

In computer science,computability theoryis the branch of the theory of computationthat studieswhich problems are computationally solvable using different models of computation.

Computability theory differs from the related discipline of computational comple6ity theory, which dealswith the question of how efficiently a problem can be solved, rather than whether it is solvable at all.

Contents

! Introduction # 3ormal models of computation % )ower of automata

o %.! )ower of finite state machineso %.# )ower of pushdown automatao %.% )ower of 'uring machines

%.%.! 'he halting problem %.%.# ?eyond recursive languages

+ Concurrencybased models 8nreasonable models of computation

o .! Infinite e6ecutiono .# Oracle machineso .% /imits of hypercomputation

Introduction

1 central question of computer science is to address the limits of computing devices. One approach toaddressing this question is understanding the problems we can use computers to solve. *oderncomputing devices often seem to possess infinite capacity for calculation, and itGs easy to imagine that,given enough time, we might use computers to solve any problem. &owever, it is possible to show clearlimits to the ability of computers, even given arbitrarily vast computational resources, to solve evenseemingly simple problems. )roblems are formally e6pressed as a decision problem which is toconstruct a mathematical function that for each input returns either :or !. If the value of the function onthe input is :then the answer is 7no7 and otherwise the answer is 7yes7.

'o e6plore this area, computer scientists invented automata theorywhich addresses problems such asthe following5 ;iven a formal language, and a string, is the string a member of that languageM 'his is asomewhat esoteric way of as


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?ut in what real sense is this observation trueM Can we define a formal sense in which we canunderstand how hard a particular problem is to solve on a computerM It is the goal of computabilitytheory of automata to answer just this question.

%ormal models of computation

In order to begin to answer the central question of automata theory, it is necessary to define in a formalway what an automaton is. 'here are a number of useful models of automata. (ome widely


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we can see that to find a language that is not regular, we must construct a language that would requirean infinite number of states.

1n e6ample of such a language is the set of all strings consisting of the letters GaG and GbG which containan equal number of the letter GaG and GbG. 'o see why this language cannot be correctly recognized by afinite state machine, assume first that such a machine Me6ists. Mmust have some number of states n"ow consider the stringconsisting of nS ! GaGs followed by nS ! GbGs.

1s Mreads in, there must be some state in the machine that is repeated as it reads in the first seriesof GaGs, since there are nS ! GaGs and only nstates by the pigeonhole principle. Call this state >, andfurther let dbe the number of GaGs that our machine read in order to get from the first occurrence of >tosome subsequent occurrence during the GaG sequence. 4e , we can add in an additional dwhere d : GaGs and we will be again at state >. 'his means that we


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string is not in a language, since it may run forever in such a case. 1 language which is accepted bysuch a 'uring machine is called a recursi$ely enumerable language.

'he 'uring machine, it turns out, is an e6ceedingly powerful model of automata. 1ttempts to amend thedefinition of a 'uring machine to produce a more powerful machine are surprisingly met with failure. 3ore6ample, adding an e6tra tape to the 'uring machine, giving it a #dimensional or % or anydimensionainfinite surface to wor< with can all be simulated by a 'uring machine with the basic !dimensional tape'hese models are thus not more powerful. In fact, a consequence of the Church'uring thesisis that

there is no reasonable model of computation which can decide languages that cannot be decided by a'uring machine.

'he question to as< then is5 do there e6ist languages which are recursively enumerable, but notrecursiveM 1nd, furthermore, are there languages which are not even recursively enumerableM

The halting problem

Main article: Halting problem

'he halting problem is one of the most famous problems in computer science, because it has profoundimplications on the theory of computability and on how we use computers in everyday practice. 'he

problem can be phrased5

i,en a description of a Turing machine and its initial input; determine whether the program;when eecuted on this input; e,er halts Bcompletes8 The alternati,e is that it runs fore,er withouhalting8

&ere we are as


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machine given in the input as well, by interleaving the e6ecution of the two programs. (ince the directsimulation will eventually halt if the program it is simulating halts, and since by assumption thesimulation of Mwill eventually halt if the input program would never halt, we


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History of computability theory

'he lambda calculus, an important precursor to formal computability theory, was developed by1lonzoChurchand (tephen Cole >leene.1lan 'uringis most often considered the father of modern computerscience, and laid many of the important foundations of computability and comple6ity theory, including thfirst description of the 'uring machinein E!F, !9%- as well as many of the important early results.

Complexity

3or other uses, see Comple6ity disambiguation.

In general usage, complexityoften tends to be used to characterize something with many parts inintricate arrangement. In science there are at this time a number of approaches to characterizingcomple6ity, many of which are reflected in this article. (eth /loydof *.I.'.writes that he once gave apresentation which set out %# definitions of comple6ity. E!F

Definitions are often tied to the concept of a systemK a set of parts or elements which haverelationships among them differentiated from relationships with other elements outside the relationalregime. *any definitions tend to postulate or assume that comple6ity e6presses a condition of numerouelements in a system and numerous forms of relationships among the elements.

(ome definitions


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Contents

! Disorganized comple6ity vs. organized comple6ity # (ources of comple6ity % (pecific meanings of comple6ity + (tudy of comple6ity Comple6ity topics

o .! Comple6 behaviouro .# Comple6 mechanismso .% Comple6 simulationso .+ Comple6 systemso . Comple6ity in datao .- Comple6 )roject *anagement

- 1pplications of comple6ity (ee also 0 $eferences 9 3urther reading

!: 26ternal lin


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organized comple6ity is a city neighborhood as a living mechanism, with the neighborhood peopleamong the systems parts. E+F

Sources of complexity

'he source of disorganized comple6ity is the large number of parts in the system of interest, and thelac< of correlation between elements in the system.

'here is no consensus at present on general rules regarding the sources of organized comple6ity,though the lac< of randomness implies correlations between elements. (ee e.g. $obert 8lanowi6zGstreatment of ecosystems. EFConsistent with prior statements here, the number of parts and types ofparts in the system and the number of relations between the parts would have to be nontrivial Khowever, there is no general rule to separate trivial from nontrivial.

Specific meanings of complexity

In several scientific fields, 7comple6ity7 has a specific meaning 5

In computational comple6ity theory, the amounts of resourcesrequired for the e6ecution of

algorithms is studied. 'he most popular types of computational comple6ity are the timecomple6ity of a problem equal to the number of steps that it tarohn$hodes comple6ityis an important topic in the study of finite semigroupsand automata.

'here are different specific forms of comple6ity5

In the sense of how complicated a problem is from the perspective of the person trying to solve itlimits of comple6ity are measured using a term from cognitive psychology, namely the hrair limit.
http://en.wikipedia.org/wiki/Complexity#cite_note-3http://en.wikipedia.org/wiki/Complexity#cite_note-4http://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_resourcehttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Problem_sizehttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Computer_storagehttp://en.wikipedia.org/wiki/Problem_sizehttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Complexity_classhttp://en.wikipedia.org/wiki/P_(complexity)http://en.wikipedia.org/wiki/NP_(complexity)http://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Algorithmic_information_theoryhttp://en.wikipedia.org/wiki/Kolmogorov_complexityhttp://en.wikipedia.org/wiki/String_(computer_science)http://en.wikipedia.org/wiki/Computer_programhttp://en.wikipedia.org/wiki/Information_processinghttp://en.wikipedia.org/wiki/Propertyhttp://en.wikipedia.org/wiki/Observationhttp://en.wikipedia.org/wiki/State_(computer_science)http://en.wikipedia.org/wiki/Physical_systemshttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Systemhttp://en.wikipedia.org/wiki/Entropy_(disambiguation)http://en.wikipedia.org/wiki/Statistical_mechanicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Krohn-Rhodes_complexityhttp://en.wikipedia.org/wiki/Semigrouphttp://en.wikipedia.org/wiki/Automata_theoryhttp://en.wikipedia.org/wiki/Cognitive_psychologyhttp://en.wikipedia.org/wiki/Hrair_limithttp://en.wikipedia.org/wiki/Complexity#cite_note-3http://en.wikipedia.org/wiki/Complexity#cite_note-4http://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_resourcehttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Problem_sizehttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Computer_storagehttp://en.wikipedia.org/wiki/Problem_sizehttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Complexity_classhttp://en.wikipedia.org/wiki/P_(complexity)http://en.wikipedia.org/wiki/NP_(complexity)http://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Algorithmic_information_theoryhttp://en.wikipedia.org/wiki/Kolmogorov_complexityhttp://en.wikipedia.org/wiki/String_(computer_science)http://en.wikipedia.org/wiki/Computer_programhttp://en.wikipedia.org/wiki/Information_processinghttp://en.wikipedia.org/wiki/Propertyhttp://en.wikipedia.org/wiki/Observationhttp://en.wikipedia.org/wiki/State_(computer_science)http://en.wikipedia.org/wiki/Physical_systemshttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Systemhttp://en.wikipedia.org/wiki/Entropy_(disambiguation)http://en.wikipedia.org/wiki/Statistical_mechanicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Krohn-Rhodes_complexityhttp://en.wikipedia.org/wiki/Semigrouphttp://en.wikipedia.org/wiki/Automata_theoryhttp://en.wikipedia.org/wiki/Cognitive_psychologyhttp://en.wikipedia.org/wiki/Hrair_limit


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8nruly comple6itydenotes situations that do not have clearly defined boundaries, coherentinternal dynamics, or simply mediated relations with their e6ternal conte6t, as coined by )eter'aylor.

Comple6 adaptive systemdenotes systems which have some or all of the following attributes E-F

o 'he number of parts and types of parts in the system and the number of relationsbetween the parts is nontrivial K however, there is no general rule to separate trivial fromnontrivialH

o

'he system has memory or includes feedbac


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Complex systems

Main article: Comple s!stem

(ystems theoryhas long been concerned with the study of comple6 systemsIn recent times,compleit! theor!and comple s!stemshave also been used as names of the field. 'hese systemscabe biological, economic, technological, etc. $ecently, comple6ity is a natural domain of interest of the

real world sociocognitive systems and emerging systemicsresearch. Comple6 systems tend to be highdimensional, nonlinearand hard to model. In specific circumstances they may e6hibit low dimensionalbehaviour.

Complexity in data

In information theory, algorithmic information theoryis concerned with the comple6ity of stringsof data.

Comple6 strings are harder to compress. 4hile intuition tells us that this may depend on the codec useto compress a string a codec could be theoretically created in any arbitrary language, including one inwhich the very small command 7x7 could cause the computer to output a very complicated string li


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Applications of complexity

Computational comple6ity theoryis the study of the comple6ity of problems that is, the difficulty ofsolvingthem. )roblems can be classified by comple6ity classaccording to the time it ta


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#athematical proof

In mathematics,a proofis a convincing demonstration that some mathematical statementis necessaritrue, within the accepted standards of the field. 1 proof is a logically deducedargument, not an empiricaone. 'hat is, the proof must demonstrate that a proposition is true in all cases to which it applies, withoua single e6ception. 1n unproven proposition believed or strongly suspected to be true is


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Methods of proof

Direct proof

Main article: Direct proof

In direct proof, the conclusion is established by logically combining the a6ioms, definitions, and earlier

theorems. 3or e6ample, direct proof can be used to establish that the sum of two evenintegersisalways even5

3or any two even integersand !we can writeU #aand !U #bfor some integers aand b,since bothand !are multiples of #. ?ut the sumS !U #aS #bU #aS b is also a multiple of#, so it is therefore even by definition.

'his proof uses definition of even integers, as well as distribution law.

Proof by induction

Main article: Mathematical induction

In proof by induction, first a 7base case7 is proved, and then an 7induction rule7 is used to prove aoften infinite series of other cases. (ince the base case is true, the infinity of other cases must also betrue, even if all of them cannot be proved directly because of their infinite number. 1 subset of inductionis Infinite descent. Infinite descent can be used to prove the irrationality of the square root of two.

'he principle of mathematical induction states that5 /et 7U P !, #, %, +, ... Q be the set of natural numberand ).n/be a mathematical statement involving the natural number nbelonging to &such that .i/P! true, ie, Pn is true for nU ! .ii/PmS ! is true whenever Pm is true, ie, Pm is true implies thatPmS ! is true. Then )2n4 is true for all natural numbers n?

Proof by transposition

Main article: Transposition Blogic

Proof by Transpositionestablishes the conclusion 7ifpthen 07 by proving the equivalent contrapositivstatement 7if not 0then not p7.

Proof by contradiction

Main article: +eductio ad absurdum

In proof by contradictionalso


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integers. (o a#is even, which implies that amust also be even. (o we can write aU #c, where cis also an integer. (ubstitution into the original equation yields #b#U #c#U +c#. Dividing bothsides by # yields b#U #c#. ?ut then, by the same argument as before, # divides b#, so bmust beeven. &owever, if aand bare both even, they share a factor, namely #. 'his contradicts our

assumption, so we are forced to conclude that is irrational.

Proof by construction

Main article: Proof b! construction

Proof by construction, or proof by e6ample, is the construction of a concrete e6ample with a propertyto show that something having that property e6ists. Aoseph /iouville, for instance, proved the e6istenceof transcendental numbersby constructing an e6plicit e6ample.

Proof by exhaustion

Main article: Proof b! ehaustion

In Proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and

proving each one separately. 'he number of cases sometimes can become very large. 3or e6ample, thfirst proof of the four colour theoremwas a proof by e6haustion with !,9%- cases. 'his proof wascontroversial because the majority of the cases were chec


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e6ample of a nonconstructive proof shows that there e6ist two irrational numbersaand bsuch that abisa rational number5

2ither is a rational number and we are done ta


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Graph theory

1 drawingof a graph.

In mathematics and computer science, graph theoryis the study of graphs5 mathematicalstructures used to model pairwise relations between objects from a certain collection. 1 7graph7 in thisconte6t refers to a collection of verticesor GnodesG and a collection of edgesthat connect pairs ofvertices. 1 graph may be undirected, meaning that there is no distinction between the two verticesassociated with each edge, or its edges may be directedfrom one verte6 to anotherH see graphmathematicsfor more detailed definitions and for other variations in the types of graphs that arecommonly considered. 'he graphs studied in graph theory should not be confused with 7 graphs offunctions7 and other


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History

'he >nigsberg ?ridge problem.

'he paper written by /eonhard 2uleron the >e,en 1ridges of Onigsbergand published in !%-isregarded as the first paper in the history of graph theory. E!F'his paper, as well as the one written by=andermondeon the knight problem;carried on with the anal!sis situsinitiated by /eibniz. 2ulerGsformula relating the number of edges, vertices, and faces of a conve6 polyhedron was studied andgeneralized by CauchyE#Fand /G&uillier,E%Fand is at the origin of topology.

*ore than one century after 2ulerGs paper on the bridges of >nigsbergand while /istingintroducedtopology, Cayleywas led by the study of particular analytical forms arising from differential calculustostudy a particular class of graphs, the trees. 'his study had many implications in theoretical chemistry.'he involved techniques mainly concerned the enumeration of graphshaving particular properties.2numerative graph theory then rose from the results of Cayley and the fundamental results published b)lyabetween !9%and !9%and the generalization of these by De ?ruijnin !99. Cayley linenneth 1ppeland 4olfgang &aempe, and others. 'he study and the generalization of this problem by 'ait&eawood, $amsey and &adwigerhas in particular led to the study of the colorings of the graphs

embedded on surfaces with arbitrary genus. 'aitGs reformulation generated a new class of problems, thefactori/ation problems, particularly studied by )etersenand >{nig. 'he wor


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'he introduction of probabilistic methods in graph theory, especially in the study of 2rd{sand$}nyiofthe asymptotic probability of graph connectivity,

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