slides for rosen, 5th edition - ai lab

Discrete Mathematics離散數學

Lecture 01 Lecture 01 September 13, 2006September 13, 2006

洪國寶洪國寶

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Outline

•• Course informationCourse information•• Overview of Discrete MathematicsOverview of Discrete Mathematics• Logic (chapter 1 ((§§1.1)1.1))

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Course information (1/3)

• Textbook Discrete Mathematics and its Applications, Discrete Mathematics and its Applications, 5th Edition,5th Edition, Kenneth H. Rosen, McGrawKenneth H. Rosen, McGraw--Hill, Inc., New York, 2003Hill, Inc., New York, 2003

http://wwwhttp://www.mhhe.mhhe.com/math.com/math/advmath/rosen/advmath/rosen/r5//r5/

歐亞書局

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•• Grading Grading (Tentative)(Tentative)Quizzes 30%Quizzes 30%Midterm exam 30%Midterm exam 30%Final exam 30%Final exam 30%Class participation 10%Class participation 10%

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•• The problems of the quizzes and exams are The problems of the quizzes and exams are allall from from examplesexamples and and exercisesexercises of the of the textbook.textbook.

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Outline


Overview of Discrete Mathematics

•• What is Discrete Mathematics? What is Discrete Mathematics? •• What is Mathematics, really? What is Mathematics, really?

–– ItIt’’s s notnot just about numbers!just about numbers!–– Mathematics is much more than that:Mathematics is much more than that:

Mathematics is the study of any and all absolutely certain truths about any and all perfectly well-defined concepts.


•• But, the concepts can relate to numbers, But, the concepts can relate to numbers, symbols, visual patterns, or symbols, visual patterns, or anythinganything!!

Mathemabsolutely certainperfectly well-defined

atics is the study of any and all truths about any and all

concepts.


•• What is Discrete Mathematics? What is Discrete Mathematics? –– The study of The study of discretediscrete, mathematical objects , mathematical objects

and and structuresstructures..


•• What are What are ““discrete structuresdiscrete structures”” anyway?anyway?–– ““DiscreteDiscrete”” -- Composed of distinct, separable Composed of distinct, separable

parts. (Opposite of parts. (Opposite of continuouscontinuous.).)discrete:continuous :: digital:analogdiscrete:continuous :: digital:analog

–– ““StructuresStructures”” -- objects built up from simpler objects built up from simpler objects according to a definite pattern.objects according to a definite pattern.

Discrete Structures We’ll Study

•• PropositionsPropositions•• PredicatesPredicates•• SetsSets•• (Discrete) Functions(Discrete) Functions•• Orders of GrowthOrders of Growth•• AlgorithmsAlgorithms•• IntegersIntegers

•• ProofsProofs•• SummationsSummations•• Permutations Permutations •• CombinationsCombinations•• RelationsRelations•• GraphsGraphs•• TreesTrees

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Relationships Between Structures

Sets

Sequencesn-tuples

MatricesNaturalnumbers

Integers

Relations

Functions

GraphsReal numbers

Complex numbers

Strings

Propositions

ProofsTreesOperatorsPrograms

Infiniteordinals Vectors

Groups

Bits

•• ““→→”” ::≡≡ ““Can be defined in terms ofCan be defined in terms of””

Some Notations We’ll Learn

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Why Study Discrete Math?

•• The basis of all of digital information The basis of all of digital information processing: processing: Discrete manipulations of Discrete manipulations of discrete structures represented in memory.discrete structures represented in memory.

•• ItIt’’s the basic language and conceptual s the basic language and conceptual foundation of all of computer science.foundation of all of computer science.

•• Discrete concepts are also widely used Discrete concepts are also widely used throughout math, science, engineering, throughout math, science, engineering, economics, biology, economics, biology, etc.etc., , ……

•• A generally useful tool for rational thought!A generally useful tool for rational thought!

Uses for Discrete Math in Computer Science

•• Algorithms & data Algorithms & data structuresstructures

•• Programming Programming language compilers & language compilers & interpreters.interpreters.

•• Computer networksComputer networks•• Operating systemsOperating systems•• Computer architectureComputer architecture

•• Database management Database management systemssystems

•• Computer securityComputer security•• Error correction codesError correction codes•• Graphics & animation Graphics & animation

algorithms, game algorithms, game enginesengines

•• Just about everything!Just about everything!

Goals of a Discrete Math Course

•• Teach students how to think mathematicallyTeach students how to think mathematically•• Five important themes Five important themes

-- mathematical reasoningmathematical reasoning-- combinatorial analysiscombinatorial analysis-- discrete structuresdiscrete structures-- algorithmic thinkingalgorithmic thinking-- applications and modelingapplications and modeling

Five Important Themes

-- mathematical reasoningmathematical reasoningto read, comprehend, and construct mathematical argumento read, comprehend, and construct mathematical argumentsts

-- combinatorial analysiscombinatorial analysisthe ability to count or enumerate objectsthe ability to count or enumerate objects

-- discrete structuresdiscrete structuresto work with discrete structures, which are the abstract to work with discrete structures, which are the abstract mathematical mathematical

structures used to represent discrete objects and relationships structures used to represent discrete objects and relationships between these between these objectsobjects

-- algorithmic thinkingalgorithmic thinkingthe specification of the algorithm, the verification thatthe specification of the algorithm, the verification that it works properly, and it works properly, and

the analysis of the computer memory and time required to performthe analysis of the computer memory and time required to perform itit

-- applications and modelingapplications and modeling

•• Upon completion of this course, the student Upon completion of this course, the student should be able to:should be able to:–– Check the validity of simple logical arguments.Check the validity of simple logical arguments.–– Creatively construct simple valid logical arguments.Creatively construct simple valid logical arguments.–– Describe the definitions and properties of a variety of Describe the definitions and properties of a variety of

specific types discrete structures.specific types discrete structures.–– Correctly read, write and analyze various types of Correctly read, write and analyze various types of

structures using standard notations.structures using standard notations.

Course Objectives

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Course Outline (1/11)

•• 11 The Foundations: Logic, Sets, and The Foundations: Logic, Sets, and FunctionsFunctions–– 1.11.1 LogicLogic–– 1.21.2 Propositional EquivalencesPropositional Equivalences–– 1.31.3 Predicates and QuantifiersPredicates and Quantifiers–– 1.41.4 Nested QuantifiersNested Quantifiers–– 1.51.5 Methods of proofMethods of proof–– 1.61.6 SetsSets–– 1.71.7 Set OperationsSet Operations–– 1.81.8 FunctionsFunctions

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•• 22 The Fundamentals: Algorithms, The Fundamentals: Algorithms, the Integers, and Matricesthe Integers, and Matrices–– 2.12.1 AlgorithmsAlgorithms–– 2.22.2 The Growth of FunctionsThe Growth of Functions–– 2.32.3 Complexity of AlgorithmsComplexity of Algorithms–– 2.42.4 The Integers and DivisionThe Integers and Division–– 2.52.5 Integers and AlgorithmsIntegers and Algorithms–– 2.62.6 Applications of Number TheoryApplications of Number Theory–– 2.72.7 MatricesMatrices

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•• 33 Mathematical Reasoning, Mathematical Reasoning, Induction, and RecursionInduction, and Recursion–– 3.13.1 Art and Strategy of ProofArt and Strategy of Proof–– 3.23.2 Sequences and SumsSequences and Sums–– 3.33.3 Mathematical InductionMathematical Induction–– 3.43.4 Recursive Definitions and Recursive Definitions and

Structural DefinitionStructural Definition–– 3.53.5 Recursive AlgorithmsRecursive Algorithms–– 3.63.6 Program CorrectnessProgram Correctness

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•• 44 CountingCounting–– 4.14.1 The Basics of CountingThe Basics of Counting–– 4.24.2 The Pigeonhole PrincipleThe Pigeonhole Principle–– 4.34.3 Permutations and CombinationsPermutations and Combinations–– 4.44.4 Binomial CoefficientsBinomial Coefficients–– 4.54.5 Generalized Permutations and Generalized Permutations and

CombinationsCombinations–– 4.64.6 Generating Permutations and Generating Permutations and

CombinationsCombinations

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•• 55 Discrete ProbabilityDiscrete Probability––5.15.1 An Introduction to Discrete An Introduction to Discrete

ProbabilityProbability––5.25.2 Probability TheoryProbability Theory––5.35.3 Expected Value and Expected Value and

VarianceVariance

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•• 66 Advanced Counting TechniquesAdvanced Counting Techniques–– 6.16.1 Recurrence RelationsRecurrence Relations–– 6.26.2 Solving Recurrence RelationsSolving Recurrence Relations–– 6.36.3 DivideDivide--andand--Conquer RelationsConquer Relations–– 6.46.4 Generating FunctionsGenerating Functions–– 6.56.5 InclusionInclusion--ExclusionExclusion–– 6.66.6 Applications of InclusionApplications of Inclusion--ExclusionExclusion

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•• 77 RelationsRelations–– 7.17.1 Relations and Their PropertiesRelations and Their Properties–– 7.27.2 nn--aryary Relations and Their Relations and Their

ApplicationsApplications–– 7.37.3 Representing RelationsRepresenting Relations–– 7.47.4 Closures of RelationsClosures of Relations–– 7.57.5 Equivalence RelationsEquivalence Relations–– 7.67.6 Partial OrderingsPartial Orderings

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•• 88 GraphsGraphs–– 8.18.1 Introduction to GraphsIntroduction to Graphs–– 8.28.2 Graph TerminologyGraph Terminology–– 8.38.3 Representing Graphs and Graph Representing Graphs and Graph

IsomorphismIsomorphism–– 8.48.4 ConnectivityConnectivity–– 8.58.5 Euler and Hamilton PathsEuler and Hamilton Paths–– 8.68.6 Shortest Path ProblemsShortest Path Problems–– 8.78.7 Planar GraphsPlanar Graphs–– 8.88.8 Graph ColoringGraph Coloring

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•• 99 TreesTrees–– 9.19.1 Introduction to TreesIntroduction to Trees–– 9.29.2 Applications of TreesApplications of Trees–– 9.39.3 Tree TraversalTree Traversal–– 9.49.4 Spanning TreesSpanning Trees–– 9.59.5 Minimum Spanning TreesMinimum Spanning Trees

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•• 1010 Boolean AlgebraBoolean Algebra–– 10.110.1 Boolean FunctionsBoolean Functions–– 10.210.2 Representing Boolean FunctionsRepresenting Boolean Functions–– 10.310.3 Logic GatesLogic Gates–– 10.410.4 Minimization of CircuitsMinimization of Circuits

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•• 1111 Modeling ComputationModeling Computation–– 11.111.1 Languages and GrammarsLanguages and Grammars–– 11.211.2 FiniteFinite--State Machines with OutputState Machines with Output–– 11.311.3 FiniteFinite--State Machines with No OutputState Machines with No Output–– 11.411.4 Language RecognitionLanguage Recognition–– 11.511.5 Turing MachinesTuring Machines

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Some Advice

•• Take noteTake note•• Read in advanceRead in advance•• Do exercisesDo exercises•• Take advantages of the web resourcesTake advantages of the web resources

–– http://wwwhttp://www.mhhe.mhhe.com/math.com/math/advmath/rosen/advmath/rosen/r5//r5/–– MIT 6.042 MIT 6.042 Mathematics for Computer ScienceMathematics for Computer Science

http://theoryhttp://theory.lcs.mit.edu.lcs.mit.edu/classes/6.042/fall06//classes/6.042/fall06/

http://www.mhhe.com/math/advmath/rosen/r5/

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Outline


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Logic (1/2)

Mathematical Logic is a tool for working with complicated compound statements. It includes:

• A language for expressing them.• A concise notation for writing them.• A methodology for reasoning about their truth or

falsity.• It is the foundation for expressing formal proofs

in all branches of mathematics.

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Logic (2/2)

•• Propositional logic (Propositional logic (§§1.11.1--1.2):1.2):–– Basic definitions. (Basic definitions. (§§1.1)1.1)–– Equivalence rules & derivations. (Equivalence rules & derivations. (§§1.2)1.2)

•• Predicate logic (Predicate logic (§§1.31.3--1.4)1.4)–– Predicates.Predicates.–– Quantified predicate expressions.Quantified predicate expressions.–– Equivalences & derivations.Equivalences & derivations.

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Topic #1 – Propositional Logic

Propositional Logic (§1.1)

Propositional LogicPropositional Logic is the logic of compound is the logic of compound statements built from simpler statements statements built from simpler statements using sousing so--called called BooleanBoolean connectives.connectives.

Some applications in computer scienceSome applications in computer science::•• Design of digital electronic circuits.Design of digital electronic circuits.•• Expressing conditions in programs.Expressing conditions in programs.•• Queries to databases & search engines.Queries to databases & search engines.

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Definition of a Proposition

A A propositionproposition ((pp, , qq, , rr, , ……) is simply a ) is simply a statement statement ((i.e.i.e., a declarative sentence), a declarative sentence) with with a definite meaninga definite meaning, having a , having a truth valuetruth valuethatthat’’s either s either truetrue (T) or (T) or falsefalse (F), but not (F), but not both.both.

(However, you might not (However, you might not knowknow the actual the actual truth value, and it might be situationtruth value, and it might be situation--dependent.)dependent.)

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Examples of Propositions

•• ““It is raining.It is raining.”” (In a given situation.)(In a given situation.)•• ““Toronto is the capital of Canada.Toronto is the capital of Canada.””•• ““2 + 2 = 32 + 2 = 3””But, the following are But, the following are NOTNOT propositions:propositions:•• ““What time is it?What time is it?”” (interrogative, question)(interrogative, question)•• ““La la la la la.La la la la la.”” (meaningless interjection)(meaningless interjection)•• ““Just do it!Just do it!”” (imperative, command)(imperative, command)•• ““x+1=2x+1=2”” (neither true nor false)(neither true nor false)•• ““1 + 21 + 2”” (expression with a non(expression with a non--true/false value)true/false value)

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Topic #1.0 – Propositional Logic: Operators

An An operatoroperator or or connectiveconnective combines one or combines one or more more operand operand expressions into a larger expressions into a larger expression. (expression. (E.g.E.g., , ““++”” in numeric in numeric exprsexprs.).)

UnaryUnary operators take 1 operand (operators take 1 operand (e.g.,e.g., −−3); 3); binary binary operators take 2 operands (operators take 2 operands (egeg 3 3 ×× 4).4).

PropositionalPropositional or or BooleanBoolean operators operate operators operate on propositions or truth values instead of on on propositions or truth values instead of on numbers.numbers.

Operators / Connectives

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Some Popular Boolean Operators

↔↔BinaryBinaryIFFIFFBiconditionalBiconditional operatoroperator→→BinaryBinaryIMPLIESIMPLIESImplication operatorImplication operator⊕⊕BinaryBinaryXORXORExclusiveExclusive--OR operatorOR operator

Disjunction operatorDisjunction operatorConjunction operatorConjunction operatorNegation operatorNegation operator

Formal NameFormal Name

∨∨BinaryBinaryOROR∧∧BinaryBinaryANDAND¬¬UnaryUnaryNOTNOT

SymbolSymbolArityArityNicknameNickname

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The Negation Operator

The unary The unary negation operatornegation operator “¬”“¬” ((NOTNOT) ) transforms a prop. into its logicaltransforms a prop. into its logical negationnegation..

E.g.E.g. If If pp = = ““Today is Friday.Today is Friday.””then then ¬¬pp = = ““Today is not Friday.Today is not Friday.””

Truth tableTruth table for NOT:for NOT: p ¬pT FF T

Operandcolumn

Resultcolumn

T :≡ True; F :≡ False“:≡” means “is defined as”

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The Conjunction Operator

•• The binary The binary conjunction operatorconjunction operator ““∧∧””((ANDAND) combines two propositions to form ) combines two propositions to form their logical their logical conjunctionconjunction..

•• Example 4. If Example 4. If pp==““Today is Friday.Today is Friday.”” and and q=q=““It is raining today.It is raining today.””, then , then pp∧∧qq==““Today Today is Friday is Friday andand it is raining today.it is raining today.”

∧NDND∧

”

Remember: “∧∧”” points up like an points up like an ““AA””, and it means , and it means ““∧∧NDND””

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•• Note that aNote that aconjunctionconjunctionpp11 ∧∧ pp2 2 ∧∧ …… ∧∧ ppnnof of nn propositionspropositionswill have 2will have 2nn rowsrowsin its truth table.in its truth table.

•• Also: Also: ¬¬ and and ∧∧ operations together are sufficient to express operations together are sufficient to express anyany Boolean truth table! (They are functionally complete.)Boolean truth table! (They are functionally complete.)

p q p∧qF F FF T FT F FT T T

Operand columns

Conjunction Truth Table

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The Disjunction Operator

•• The binary The binary disjunction operatordisjunction operator ““∨∨”” ((OROR) ) combines two propositions to form their combines two propositions to form their logical logical disjunctiondisjunction..

•• Example 5.Example 5.pp==““Today is Friday.Today is Friday.””q=q=““It is raining today.It is raining today.””pp∨∨qq==““Today is Friday Today is Friday oror iit is raining today.t is raining today.””

Meaning is like “and/or” in English.

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•• Note that Note that pp∨∨q q meansmeansthat that pp is true, or is true, or qq isistrue, true, or bothor both are true!are true!

•• So, this operation isSo, this operation isalso called also called inclusive or,inclusive or,because it because it includesincludes thethepossibility that both possibility that both pp and and qq are true.are true.

•• ““¬¬”” and and ““∨∨”” together are also universal.together are also universal.

p q p∨qF F FF T TT F TT T T

Notedifferencefrom AND

Disjunction Truth Table

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The Exclusive Or Operator

The binary The binary exclusiveexclusive--or operatoror operator ““⊕⊕”” ((XORXOR) ) combines two propositions to form their combines two propositions to form their logical logical ““exclusive orexclusive or””..

pp = = ““I will earn an A in this course,I will earn an A in this course,””qq == ““I will drop this course,I will drop this course,””pp ⊕⊕ q q = = ““I will either earn an A for this I will either earn an A for this

course, or I will drop it (but not both!)course, or I will drop it (but not both!)””

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•• Note that Note that pp⊕⊕q q meansmeansthat that pp is true, or is true, or qq isistrue, but true, but not bothnot both!!

•• This operation isThis operation iscalled called exclusive or,exclusive or,because it because it excludesexcludes thethepossibility that both possibility that both pp and and qq are true.are true.

•• ““¬¬”” and and ““⊕⊕”” together are together are notnot universal.universal.

p q p⊕qF F FF T TT F TT T F Note

differencefrom OR.

Exclusive-Or Truth Table

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The Implication Operator

The The implicationimplication p p →→ qq states that states that pp implies implies q.q.I.e.I.e., If , If pp is true, then is true, then qq is true; but if is true; but if pp is not is not

true, then true, then qq could be either true or false.could be either true or false.E.g.E.g., let , let p p = = ““You study hard.You study hard.””

q q = = ““You will get a good grade.You will get a good grade.””p p →→ q = q = ““If you study hard, then you will get If you study hard, then you will get

a good grade.a good grade.”” (else, it could go either way)(else, it could go either way)

hypothesis conclusion

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Implication Truth Table

•• p p →→ q q is is falsefalse onlyonly whenwhenpp is true but is true but qq is is notnot true.true.

•• p p →→ q q does does not not saysaythat that pp causescauses qq!!

•• p p →→ q q does does not not requirerequirethat that pp or or qq are ever trueare ever true!!

•• E.g.E.g. ““(1=0) (1=0) →→ pigs can flypigs can fly”” is TRUE!is TRUE!

p q p→qF F T F T T T F F T T T

The onlyFalsecase!

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English Phrases Meaning p → q

•• ““pp implies implies qq””•• ““if if pp, then , then qq””•• ““if if pp, , qq””•• ““when when pp, , qq””•• ““whenever whenever pp, , qq””•• ““q q if if pp””•• ““qq when when pp””•• ““qq whenever whenever pp””

•• ““p p only if only if qq””•• ““p p is sufficient for is sufficient for qq””•• ““qq is necessary for is necessary for pp””•• ““qq follows from follows from pp””•• ““q q is implied by is implied by pp””We will see some equivalent We will see some equivalent

logic expressions later.logic expressions later.

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Converse, Inverse, Contrapositive

Some terminology, for an implication Some terminology, for an implication p p →→ qq::•• Its Its converseconverse is: is: q q →→ pp..•• Its Its inverseinverse is: is: ¬¬pp →→ ¬¬qq..•• Its Its contrapositivecontrapositive:: ¬¬q q →→ ¬¬ p.p.•• One of these three has the One of these three has the same meaningsame meaning

(same truth table) as (same truth table) as pp →→ qq. Can you figure . Can you figure out which?out which?

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The biconditional operator

•• The The biconditionalbiconditional p p ↔ q q states that states that pp is true is true if and if and only ifonly if (IFF) q(IFF) q is true.is true.

•• Example 8. (p. 9)Example 8. (p. 9)p p = = ““You can take the flight.You can take the flight.””qq == ““You buy a ticket.You buy a ticket.””p p ↔ q = q = ““You can take the flight if and only if You can take the flight if and only if you buy a ticket.you buy a ticket.””

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Biconditional Truth Table

•• p p ↔ q q means that means that pp and and qqhave the have the samesame truth value.truth value.

•• Note this truth table is theNote this truth table is theexact exact oppositeopposite of of ⊕⊕’’s!s!–– p p ↔ q q means means ¬¬((p p ⊕⊕ qq))

•• p p ↔ q q does does not not implyimplypp and and qq are true, or cause each other.are true, or cause each other.

p q p ↔ qF F TF T FT F FT T T

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Nested Propositional Expressions

•• Use parentheses to Use parentheses to group subgroup sub--expressionsexpressions::““I just saw my old I just saw my old ffriendriend, and either , and either hehe’’s s ggrownrown or or II’’ve ve sshrunkhrunk..”” = = ff ∧∧ ((gg ∨∨ ss))–– ((ff ∧∧ gg) ) ∨∨ ss would mean something differentwould mean something different–– ff ∧∧ gg ∨∨ ss would be ambiguouswould be ambiguous

•• By convention, By convention, ““¬”¬” takes takes precedenceprecedence over over both both ““∧∧”” and and ““∨∨””..–– ¬¬s s ∧∧ ff means (means (¬¬ss)) ∧∧ f f , , not not ¬¬ ((s s ∧∧ ff))

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Precedence of Logical Operators

Operator PrecedenceOperator Precedence¬¬ 11∧∧ 22∨∨ 33→→ 44↔↔ 55

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Boolean Operations Summary

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

•• We have seen 1 unary operator (out of the 4 We have seen 1 unary operator (out of the 4 possible) and 5 binary operators (out of the possible) and 5 binary operators (out of the 16 possible). Their truth tables are below.16 possible). Their truth tables are below.

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Some Alternative Notations

Name: not and or xor implies iffPropositional logic: ¬ ∧ ∨ ⊕ → ↔Boolean algebra: p pq + ⊕C/C++/Java (wordwise): ! && || != ==C/C++/Java (bitwise): ~ & | ^Logic gates:

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Translating English Sentences

•• Example 9.Example 9.

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System Specifications

•• Propositional expressions are Propositional expressions are consistentconsistent if if there is an assignment of truth values to the there is an assignment of truth values to the variables in the expressions that makes all variables in the expressions that makes all the expressions true.the expressions true.


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Logic puzzles

•• Puzzles that can be solved using logical Puzzles that can be solved using logical reasoning are known as reasoning are known as logic puzzles..


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Topic #2 – Bits

Bits and Bit Operations

•• Why propositional logic can be applied in Why propositional logic can be applied in the design of digital circuit?the design of digital circuit?

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Topic #2 – Bits

Bits and Bit Operations

•• A A bitbit is a is a bibinary (base 2) dignary (base 2) digitit: 0 or 1.: 0 or 1.•• Bits may be used to represent truth values.Bits may be used to represent truth values.•• By convention: By convention:

0 represents 0 represents ““falsefalse””; 1 represents ; 1 represents ““truetrue””..•• Boolean algebraBoolean algebra is like ordinary algebra is like ordinary algebra

except that variables stand for bits, + means except that variables stand for bits, + means ““oror””, and multiplication means , and multiplication means ““andand””..–– See chapter 10 for more details.See chapter 10 for more details.

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Topic #2 – Bits

Bit Strings

•• AA Bit stringBit string is a sequence of zero or more bits. is a sequence of zero or more bits. The The length length of this string is the number of bits in of this string is the number of bits in the string.the string.–– More on sequences in More on sequences in §§3.2.3.2.

•• By convention, bit strings are written left to right: By convention, bit strings are written left to right: e.g.e.g. the first bit of the first bit of ““10011010101001101010”” is 1.is 1.

•• When a bit string represents a baseWhen a bit string represents a base--2 number, by 2 number, by convention the first bit is the convention the first bit is the most significantmost significant bit. bit. Ex. Ex. 1101110122=8+4+1=13.=8+4+1=13.

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Topic #2 – Bits

Bitwise Operations

•• Boolean operations can be extended to Boolean operations can be extended to operate on bit strings as well as single bits.operate on bit strings as well as single bits.

•• E.g.:E.g.:01 1011 011001 1011 011011 0001 110111 0001 110111 1011 1111 BitBit--wise ORwise OR01 0001 0100 BitBit--wise ANDwise AND10 1010 1011 BitBit--wise XORwise XOR

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End of §1.1

You have learned about:You have learned about:•• Propositions: What Propositions: What

they are.they are.•• Propositional logic Propositional logic

operatorsoperators’’–– Symbolic notations.Symbolic notations.–– English equivalents.English equivalents.–– Logical meaning.Logical meaning.–– Truth tables.Truth tables.

•• Atomic vs. compound Atomic vs. compound propositions.propositions.

•• Alternative notations.Alternative notations.•• Bits and bitBits and bit--strings.strings.•• Next section: Next section: §§1.21.2

–– Propositional Propositional equivalences.equivalences.

–– How to prove them.How to prove them.

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