Discrete Mathematics for Computer Science Prof. Raissa...
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B P M /92 Module #1 - Logic B P M /92 Module #1 - Logic UC Davis ECS20, Winter 2017 Discrete Mathematics for Computer Science Prof. Raissa D’Souza (slides adopted from Michael Frank and Haluk Bingöl)
Transcript of 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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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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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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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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Some Notations We’ll
Learn
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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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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.
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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).
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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
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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
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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
