discrete mathematics for computer science prof. raissa...
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UCDavisECS20,Winter2017DiscreteMathematicsforComputerScience
Prof.RaissaD’Souza(slidesadoptedfromMichaelFrankandHalukBingöl)
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Calculusisforcontinuoussystems • Calculusf(x)versusx,considersacon/nuousvariablex.• Con/nuousnumbers:xisanumberwhichcanhaveanynumberofdecimalplaces,• E.g.3.1415• Transcendentalnumbers:pi=3.1415926…..
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Examplesofdiscretesystems?• Discrete=consis2ngdis2nctorindividualunits;o:encomesinintegerquan22es.
• .….• ……• …..
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2
What is Mathematics,
really?
• It’s not just about numbers!
• Mathematics is much more than that:
• But, these concepts can be about numbers, symbols,
objects, images, sounds, anything!
Mathematics is, most generally, the study of any and all absolutely certain truths about
any and all perfectly well-defined concepts.
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5
So, what’s this class
about?
What are “discrete structures” anyway?
• “Discrete” (≠ “discreet”!) - Composed of distinct,
separable parts. (Opposite of continuous.)
discrete:continuous :: digital:analog
• “Structures” - Objects built up from simpler objects
according to some definite pattern.
• “Discrete Mathematics” - The study of discrete,
mathematical objects and structures.
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6
Discrete Structures
We’ll Study• Propositions
• Predicates
• Proofs
• Sets
• Functions
• Orders of Growth
• Algorithms
• Integers
• Summations
• Sequences
• Strings
• Permutations
• Combinations
• Relations
• Graphs
• Trees
• Logic Circuits
• Automata
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Exampleproblems• Given8leIers,howmanypossiblepasswords?• Encryp/on(mappingonesymboltoanother)• Sor/ngalist• Howlikelyisyourpokerhand?
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7
Some Notations We’ll
Learn
9
Uses for Discrete Math in Computer
Science
• Advanced algorithms & data structures
• Programming language compilers & interpreters.
• Computer networks
• Operating systems
• Computer architecture
• Database management
systems
• Cryptography
• Error correction codes
• Graphics & animation
algorithms, game
engines, etc.…
• I.e., the whole field!
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12
Course Objectives
• Upon completion of this course, the student should be able to:
• Check validity of simple logical arguments (proofs).
• Check the correctness of simple algorithms.
• Creatively construct simple instances of valid logical arguments and correct algorithms.
• Describe the definitions and properties of a variety of specific types of discrete structures.
• Correctly read, represent and analyze various types of discrete structures using standard notations.
Think!
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Module#1:FoundationsofLogic(§§1.1-1.3,~3lectures)• Mathema2calLogicisatoolforworkingwithelaboratecompoundstatements.
• Itincludes:• Aformallanguageforexpressingthem.• Aconcisenota/onforwri/ngthem.• Amethodologyforobjec/velyreasoningabouttheirtruthorfalsity.
• Itisthefounda/onforexpressingformalproofsinallbranchesofmathema/cs.
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Examples:Englishisambiguous
• Areyounotgoingtothepartytonight?• Youmustbeatleast5feettallorovertheageof14toridetherollercoaster.
• Thelawniswet,soeitheritrainedlastnightorthesprinklerswentonthismorning,butnotbothhappened.
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FoundationsofLogic:Overview• Proposi/onallogic(§1.1-1.2):
– Basicdefini/ons.(§1.1)– Equivalencerules&deriva/ons.(§1.2)
• Predicatelogic(§1.3-1.4)– Predicates.– Quan/fiedpredicateexpressions.– Equivalences&deriva/ons.
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PropositionalLogic(§1.1)• Proposi2onalLogicisthelogicofcompoundstatementsbuiltfromsimplerstatementsusingso-calledBooleanconnec2ves.
• Someapplica/onsincomputerscience• Designofdigitalelectroniccircuits.• Expressingcondi/onsinprograms.• Queriestodatabases&searchengines.
Topic #1 – Propositional Logic
George Boole (1815-1864)
Chrysippus of Soli (ca. 281 B.C. – 205 B.C.)
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DeYinitionofaProposition• Defini&on:Aproposi2on(denotedp,q,r,…)issimply:• Adeclara2vestatementwithsomedefinitemeaning,(notvagueorambiguous)
• havingatruthvaluethatiseithertrue(T)orfalse(F)• itisneverboth,neither,orsomewhere“inbetween”
• However,youmightnotknowtheactualtruthvalue,• and,thetruthvaluemightdependonthesitua/onorcontext.
• Later,wewillstudyprobabilitytheory,inwhichweassigndegreesofcertainty(“between” TandF)toproposi/ons.• Butfornow:thinkTrue/Falseonly!
Topic #1 – Propositional Logic
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ExamplesofPropositions• “BeijingisthecapitalofChina.”• “1+2=5” • “Itisraining.”(T/Fassessedgivensitua/on.)
• But,thefollowingareNOTproposi/ons:• “Who’sthere?”(interroga/ve,ques/on)• “Lalalalala.”(meaninglessinterjec/on)• “Justdoit!”(impera/ve,command)• “1+2”(expressionwithanon-true/falsevalue;doesnotassertanything)
Topic #1 – Propositional Logic
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Absurdstatementscanbepropositions
• “Pigscanfly”
• “Themoonismadeofgreencheese”
• “1+1=10”
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Example–Arethesestatementspropositions?
• “Thisstatementistrue”• “Thisstatementisfalse”
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Example–Arethesestatementspropositions?
• “Thisstatementistrue”• Yes,andthetruthvalueisT
• “Thisstatementisfalse”
• No,notaproposi2on;cannotassignTorF
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Operators/Connectives• Anoperatororconnec2vecombinesoneormoreoperandexpressionsintoalargerexpression.(E.g.,“+”innumericexpressions.)– Unaryoperatorstake1operand(e.g.,−3);– binaryoperatorstake2operands(eg3×4).
• Proposi2onalorBooleanoperatorsoperateonproposi/ons(ortheirtruthvalues)insteadofonnumbers.
Topic #1.0 – Propositional Logic: Operators
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SomePopularBooleanOperators
Formal Name Nickname Arity Symbol
Negation operator NOT Unary ¬
Conjunction operator AND Binary ∧
Disjunction operator OR Binary ∨
Exclusive-OR operator XOR Binary ⊕
Implication operator IMPLIES Binary →
Biconditional operator IFF Binary ↔
Topic #1.0 – Propositional Logic: Operators
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Truthtables
• To evaluate the T or F status of a compound proposition
• Enumerate over all T and F combinations of all the propositions
• If an outcome of a compound proposition is F is this violates logic.
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TheNegationOperator• Theunarynega2onoperator“¬”(NOT)transformsaprop.intoitslogicalnega2on.
• E.g.Ifp=“Ihavebrownhair.”• then¬p=“Idonothavebrownhair.” • ThetruthtableforNOT:
p ¬p T F F T
T :≡ True; F :≡ False “:≡” means “is defined as”
Operand column
Result column
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TheConjunctionOperator• Thebinaryconjunc2onoperator“∧”(AND)combinestwoproposi/onstoformtheirlogicalconjunc2on.
• E.g.Ifp=“Iwillhavesaladforlunch.”andq=“Iwillhavesteakfordinner.”,
• thenp∧q=“IwillhavesaladforlunchandIwillhavesteakfordinner.”
Remember: “∧” points up like an “A”, and it means “∧ND”
∧ND
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ConjunctionTruthTable• In-classexerciseontheboard:Truthtablefor“p^q”
• Comparelengthofthe“not”(unary)to“and”(binary)truthtables.
• Howmanyrowsinthetruthtableifthecompoundproposi/oninvolvesthreebasicproposi/ons,p,q,r?
• Howmanyrowsinthetruthtableifthecompoundproposi/oninvolvesNbasicproposi/ons?
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ConjunctionTruthTable• Notethataconjunc/onp1∧p2∧…∧pnofnproposi/onswillhave2nrowsinitstruthtable.
• Remark.¬and∧opera/onstogetheraresufficienttoexpressanyBooleantruthtable!
p q p∧q F F F F T F T F F T T T
Operand columns
Topic #1.0 – Propositional Logic: Operators
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TheDisjunctionOperator• Thebinarydisjunc2onoperator“∨”(OR)combinestwoproposi/onstoformtheirlogicaldisjunc2on.
• p=“Mycarhasabadengine.”• q=“Mycarhasabadcarburetor.”• p∨q=“Eithermycarhasabadengine,ormycarhasabadcarburetor.”
After the downward- pointing “axe” of “∨” splits the wood, you can take 1 piece OR the other, or both.
∨
Topic #1.0 – Propositional Logic: Operators
Meaning is like “and/or” in English.
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DisjunctionTruthTable
• In-classexerciseontheboard:truthtablefor“pvq”
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DisjunctionTruthTable
• Notethatp∨qmeansthatpistrue,orqistrue,orbotharetrue!
• So,thisopera/onisalsocalledinclusiveor,becauseitincludesthepossibilitythatbothpandqaretrue.
• Remark.“¬”and“∨”togetherarealsouniversal.(WecanwriteanyBooleanlogicfunc/onintermsofthoseoperators.)
p q p∨qF F FF T TT F TT T T
Note difference from AND
Topic #1.0 – Propositional Logic: Operators
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NestedPropositionalExpressions• Useparenthesestogroupsub-expressions:“Ijustsawmyoldfriend,andeitherhe’sgrownorI’veshrunk.”
• First break it down into propositions: • f = “Ijustsawmyoldfriend”• g = “he’s grown” • s = “I’ve shrunk”
• =f∧(g∨s)• (f∧g)∨swouldmeansomethingdifferent• f∧g∨swouldbeambiguous
• Byconven/on,“¬”takesprecedenceoverboth“∧”and“∨”.• ¬s∧fmeans(¬s)∧f,not¬(s∧f)
Topic #1.0 – Propositional Logic: Operators
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ASimpleExercise• Letp=“Itrainedlastnight”,q=“Thesprinklerscameonlastnight,”r=“Thelawnwaswetthismorning.”
• TranslateeachofthefollowingintoEnglish:• ¬p=• r∧¬p=• ¬r∨p∨q=
Topic #1.0 – Propositional Logic: Operators
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TheExclusiveOrOperator• Thebinaryexclusive-oroperator“⊕”(XOR)combinestwoproposi/onstoformtheirlogical“exclusiveor”(exclusivedisjunc/on)
• p=“IwillearnanAinthiscourse,”• q=“Iwilldropthiscourse,”• p⊕q=“IwilleitherearnanAinthiscourse,orIwilldropit(butnotboth!)”
• A more common phrase: “Your entrée comes with either soup of salad”
Topic #1.0 – Propositional Logic: Operators
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Exclusive-OrTruthTable
• Notethatp⊕qmeansthatpistrue,orqistrue,butnotboth!
• Thisopera/oniscalledexclusiveor,becauseitexcludesthepossibilitythatbothpandqaretrue.
• Remark.“¬”and“⊕”togetherarenotuniversal.
p q p⊕qF F FF T TT F TT T F Note
difference from OR.
Topic #1.0 – Propositional Logic: Operators
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NaturalLanguageisAmbiguous
• NotethatEnglish“or”canbeambiguousregardingthe“both”case!
• “PatisasingerorPatisawriter.”-
• “PatisaliveorPatisdeceased.”-
• Needcontexttodisambiguatethemeaning!• Forthisclass,assume“or”meansinclusive.
p q p "or" qF F FF T TT F TT T ?
∨
⊕
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TheImplicationOperator
• Theimplica2onp→qstatesthatpimpliesq.
• i.e.,Ifpistrue,thenqistrue;butifpisnottrue,thenqcouldbeeithertrueorfalse.
• E.g.,letp=“YoumasterECS20.”q=“Youwillgetagoodjob.”
• p→q=“IfyoumasterECS20,thenyouwillgetagoodjob.”
(else,itcouldgoeitherway;somegreatjobsdonotrequirediscretemath)
Topic #1.0 – Propositional Logic: Operators
antecedent consequent
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ImplicationTruthTable• p→qisfalseonlywhenpistruebutqisnottrue.
• p→qdoesnotsaythatpcausesq!
• p→qdoesnotrequirethatporqareevertrue!
• E.g.“(1=0)→pigscanfly”isTRUE!
p q p→q F F T F T T T F F T T T
The only False case!
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ExamplesofImplications• “Ifthislectureeverends,thenthesunwillrisetomorrow.” TrueorFalse?
• “IfTuesdayisadayoftheweek,thenIamapenguin.” TrueorFalse?
• “If1+1=6,thenBushispresident.”TrueorFalse?
• “Ifthemoonismadeofgreencheese,thenIamricherthanBillGates.” TrueorFalse?
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ExamplesofImplications• “Ifthislectureeverends,thenthesunwillrisetomorrow.” TrueorFalse?
• “IfTuesdayisadayoftheweek,thenIamapenguin.” TrueorFalse?
• “If1+1=6,thenBushispresident.”TrueorFalse?
• “Ifthemoonismadeofgreencheese,thenIamricherthanBillGates.” TrueorFalse?
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Whydoesthisseemwrong?• Considerasentencelike,
• “IfIweararedshirttomorrow,thenglobalpeacewillprevail”
• Inlogic,weconsiderthesentenceTruesolongaseitherIdon’tweararedshirt,orglobalpeaceisnotachieved.
• But,innormalEnglishconversa/on,ifIweretomakethisclaim,youwouldthinkthatIwaslying.• Whythisdiscrepancybetweenlogic&language?
• Logicisaboutconsistency.
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EnglishPhrasesMeaningp→q• “pimpliesq”• “ifp,thenq”• “ifp,q”• “whenp,q”• “wheneverp,q”• “qifp”• “qwhenp”• “qwheneverp”
• “ponlyifq”• “pissufficientforq”• “qisnecessaryforp”• “qfollowsfromp”• “qisimpliedbyp”
• Wewillseesomeequivalentlogicexpressionslater.
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Converse,Inverse,Contrapositive
• Someterminology,foranimplica/onp→q:• Itsconverseis: q→p.• Itsinverseis: ¬p→¬q.• Itscontraposi2ve: ¬q→¬p.• Oneofthesethreehasthesamemeaning(sametruthtable)asp→q.Canyoufigureoutwhich?
(Let’sworkitoutontheboard).
Topic #1.0 – Propositional Logic: Operators
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• Provingtheequivalenceofp→qanditscontraposi/veusingtruthtables:
p q ¬q ¬p p→q ¬q →¬pF F T T T TF T F T T TT F T F F FT T F F T T
Topic #1.0 – Propositional Logic: Operators
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Thebiconditionaloperator
• Thebicondi2onalp↔qstatesthatp→qandq→p• Inotherwords,pistrueifandonlyif(IFF)qistrue.• p=“Clintonwinsthe2016elec/on.”• q=“Clintonwillbepresidentforallof2017.”• p↔q=“If,andonlyif,Clintonwinsthe2016elec/on,Clintonwillbepresidentforallof2017.”
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BiconditionalTruthTable
• p↔qmeansthatpandqhavethesametruthvalue.
• Remark.Thistruthtableistheexactoppositeof⊕’s!• Thus,p↔qmeans¬(p⊕q)
• p↔qdoesnotimplythatpandqaretrue,orthateitherofthemcausestheother,orthattheyhaveacommoncause.
p q p ↔ qF F TF T FT F FT T T
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BooleanOperationsSummary• Wehaveseen1unaryoperator(outofthe4possible)and5binaryoperators(outofthe16possible).Theirtruthtablesarebelow.
p q ¬p p∧q p∨q p⊕q p→q p↔qF F T F F F T TF T T F T T T FT F F F T T F FT T F T T F T T
Topic #1.0 – Propositional Logic: Operators
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SomeAlternativeNotations
Name: not and or xor implies iffPropositional logic: ¬ ∧ ∨ ⊕ → ↔Boolean algebra: p pq + ⊕C/C++/Java (wordwise): ! && || != ==C/C++/Java (bitwise): ~ & | ^Logic gates:
Topic #1.0 – Propositional Logic: Operators
End of lecture 1