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Course: MACM 101, Spring 2011
School: Simon Fraser
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Logic Introduction Discrete Propositional Mathematics Andrei Bulatov Discrete Mathematics Propositional Logic Use of Logic In mathematics and rhetoric: Give precise meaning to statements. Distinguish between valid and invalid arguments. Provide rules of `correct reasoning. Natural language can be very ambiguous `If you do your homework, then youll get to watch the game. `If you dont do your homework, then...

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Logic Introduction Discrete Propositional Mathematics Andrei Bulatov Discrete Mathematics Propositional Logic Use of Logic In mathematics and rhetoric: Give precise meaning to statements. Distinguish between valid and invalid arguments. Provide rules of `correct reasoning. Natural language can be very ambiguous `If you do your homework, then youll get to watch the game. `If you dont do your homework, then you will not get to watch ... `You do your homework, or youll fail the exam. = `If you dont do your homework, then youll fail the exam. 2-2 2-3 Discrete Mathematics Propositional Logic Use of Logic (cntd) In computing: Derive new data / knowledge from existing facts Design of computer circuits. Construction of computer programs. Verification of correctness of programs and circuit design. Specification What the customer really needed How the Programmer understood it What the customer got Discrete Mathematics Propositional Logic 2-4 Statements (propositions) Propositional logic deals with statements and their truth values A statement is a declarative sentence that can be true or false Truth values are TRUTH (T or 1) and FALSE (F or 0). Examples: - 1 + 1 = 2 (statement, T) - The moon is made of cheese (statement, F) - Go home! (not statement, imperative) - What a beautiful garden! (not statement, exclamation) - Alice said, `What a beautiful garden! (statement, depends on Alice) - y+1=2 - The God exists (not statement, uncertain) (statement, ?) Discrete Mathematics Propositional Logic Compound Statements Simplest statement are called primitive statement We cannot decide the truth value of a primitive statement. This is not what logic does. We shall use propositional variables to denote primitive statements, p, q, r, Instead we combine primitive statements by means of logic connectives into compound statements or formulas and look how the truth value of a compound statement depends on the truth values of the primitive statements it includes. We will denote compound statements by , , 2-5 Discrete Mathematics Propositional Logic 2-6 Logic Connectives negation (not, ) `It is not true that at least one politician was honest conjunction (and, ) `In this room there is a lady, and in the other room there is a tiger disjunction (or, ) `Margaret Mitchell wrote `Gone with the Wind or I am going home implication (if, then, ) `If there is a tiger in this room, then there is no lady there exclusive or (either , or , ) `There is either a tiger in this room, or a lady biconditional (equivalence) (if and only if, ) `There is a lady in this room if and only if there is a tiger in the other room Discrete Mathematics Propositional Logic Truth tables Truth table is a way to specify the exact dependence of the truth value of a compound statement through the values of primitive statements involved truth values of primitive statements (propositional variables) truth value of compound statements (formulas) p q 0 0 0 0 1 1 2-7 2-8 Discrete Mathematics Propositional Logic Truth Tables of Connectives (negation and conjunction) p p F (0) T (1) T (1) F (0) Negation Conjunction unary connective `Today is Friday `Today is not Friday p q pq 0 0 0 0 1 0 1 0 0 1 1 1 binary connective `Today is Friday `It is raining `Today is Friday and it is raining Discrete Mathematics Propositional Logic 2-9 Truth Tables of Connectives (disjunction) Disjunction `or p q pq 0 0 0 0 1 1 1 0 1 1 1 `Students inclusive who have taken calculus can take this course `Students who have taken computing can take this course 1 `Students who have taken calculus or computing can take this course Be careful with `or constructions in natural languages! `You do your homework, or youll fail the exam. `Today is Friday or Saturday Discrete Mathematics Propositional Logic Truth Tables of Connectives (exclusive or) Exclusive `or One of the statements is true but not both p q pq 0 0 0 0 1 1 1 0 1 1 1 0 `You can follow the rules or be disqualified. `Natalie will arrive today or Natalie will not arrive at all. 2-10 2-11 Discrete Mathematics Propositional Logic Truth Tables of Connectives (implication) Implication p q pq 0 0 1 0 1 1 1 0 0 1 1 1 Note that logical (material) implication does not assume any causal connection. `If black is white, then we live in Antarctic. If pigs fly, then Paris is in France. 2-12 Discrete Mathematics Propositional Logic Implication as a promise Implication can be thought of as a promise, and it is true if the promise is kept `If I am elected, then I will lower taxes - He is not elected and taxes are not lowered promise kept! - He is not elected and taxes are lowered promise kept! - He is elected, but (=and) taxes are not lowered promise broken! - He is elected and taxes are lowered promise kept! 2-13 Discrete Mathematics Propositional Logic Playing with Implication Parts of implication pq hypothesis antecedent premise conclusion consequence `if p, then q `if p, q `p is sufficient for q `q unless p `p implies q `p only if q `q if p `q when p `q whenever p `q follows from p `a sufficient condition for q is p `a necessary condition for p is q Discrete Mathematics Propositional Logic 2-14 Playing with Implication (cntd) Converse, contrapositive, and inverse pq Converse `The home team wins whenever it is raining (`If it is raining then the home team wins) qp `If the home team wins, then it is raining Contrapositive q p `If the home team does not win, then it is not raining Inverse p q `If it is not raining, then the home team does not win Discrete Mathematics Propositional Logic Truth Tables of Connectives (biconditional) Biconditional or Equivalence One of the statements is true if and only if the other is true p q pq 0 0 1 0 1 0 1 0 0 1 1 1 `You can take the flight if and only if you buy a ticket. 2-15 Discrete Mathematics Propositional Logic Example `You can access the Internet from campus if you are a computer science major or if you are not a freshman. p - `you can access the Internet from campus q - `you are a computer science major r - `you are a freshman 2-16 Discrete Mathematics Propositional Logic Tautologies Tautology is a compound statement (formula) that is true for all combinations of truth values of its propositional variables (p q) (q p) p q (p q) (q p) 0 0 1 0 1 1 1 0 1 1 1 1 2-17 Discrete Mathematics Propositional Logic Contradictions Contradiction is a compound statement (formula) that is false for all combinations of truth values of its propositional variables (p q) (p q) p q (p q) (p q) 0 0 0 0 1 0 1 0 0 1 1 0 2-18 Discrete Mathematics Propositional Logic Homework Exercises from the Book: No. 1 ,3, 4, 8a, 8c (page 54) 2-19
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Simon Fraser - MACM - 101
PropositionalIntroduction Logic IIDiscrete MathematicsAndrei BulatovDiscrete Mathematics Propositional Logic IIPrevious LectureStatements, primitive and compoundLogic connectives:negationconjunctiondisjunctionexclusive orimplicationbiconditio
Simon Fraser - MACM - 101
LawsIntroductionof LogicDiscrete MathematicsAndrei BulatovDiscrete Mathematics Laws of LogicPrevious LectureTruth tablesTautologies and contradictionsLogic equivalences4-2Discrete Mathematics Laws of LogicLogic EquivalencesCompound statements
Simon Fraser - MACM - 101
Rules ofIntroduction InferenceDiscrete MathematicsAndrei BulatovDiscrete Mathematics Rules of InferencePrevious LectureLogically equivalent statementsStatements and are equivalent iff is a tautologyMain logic equivalencesdouble negationDeMorgans
Simon Fraser - MACM - 101
Logic InferenceIntroductionDiscrete MathematicsAndrei BulatovDiscrete Mathematics - Logic InferencePrevious LectureValid and invalid argumentsArguments and tautologiesRules of inference6-26-3Discrete Mathematics Logic InferenceGeneral Definiti
Simon Fraser - MACM - 101
Problems to Week 2 Tutorial MACM101 (Fall 2011)1. State the converse, contrapositive, and inverse of each of these implications: If it snows today, I will ski tomorrow. I come to class whenever there is going to be a quiz. A positive integer is a prim
Simon Fraser - MACM - 101
Problems to Week 3 Tutorial MACM101 (Fall 2011)1. Verify that p (q (p q ) is a tautology. (If a similar problemwasnt solved in Tutorial 1.)2. Is (p q ) (q (p q ) a contradiction?3. Verify that(p q ) (q r) (r p) (p q ) (q r) (r p).4. Negate the follo
Simon Fraser - MACM - 101
Problems to Week 4 Tutorial MACM 101 (Fall 2011)1. Consider the universe of all polygons with three or four sides, anddene the following predicates for this universe:A(x):all interior angles of x are equal;E (x): x is a equilateral triangle;H (x):a
Simon Fraser - CHEM - 281
CHEMISTRY 281-4 (2011-3). Dr. Pete WilsonORGANIC CHEMISTRY I: PROBLEM SET 2.WEEK 3 LECTURE 7/8Selected Problems from Past Inter-term ExaminationsFor discussion in tutorials (Week 4) in addition to any other generalquestions you may have about the cou
Simon Fraser - CHEM - 281
CHEMISTRY 281-4 (2011-3). Dr. Pete WilsonORGANIC CHEMISTRY I: PROBLEM SET 3.WEEK 4 LECTURE 10/11 (Wednesday and Friday)Problems from Bruice (in part),6e Organic Chemistry Custom Volume 1For discussion in tutorials (Week 5) in addition to any other ge
Simon Fraser - CHEM - 281
CHEMISTRY 281-4 (2011-3). Dr. Pete WilsonORGANIC CHEMISTRY I: INTRODUCTORY ORGANIC CHEMISTRYLECTURE 1 - WEEK 1 (Wednesday, 7 September, 2011)An Introduction to Organic ChemistryPlease see the link from the course web-page An Introduction to Organic Ch
Simon Fraser - CHEM - 281
CHEMISTRY 281-4 (2011-3). Dr. Pete WilsonORGANIC CHEMISTRY I: INTRODUCTORY ORGANIC CHEMISTRYLECTURE 2 - WEEK 1 (Friday, 9 September, 2011)Course Information and OutlineIMPORTANT:Please see the link from the course web-page Course Information and Outl
Simon Fraser - CHEM - 281
Crash-course in Molecular Lego:Building Molecules with Carbon and Hydrogen AtomsHere Im using the Lewis structures of the atoms (i.e., just showing the valence shell electrons)to construct molecules and then Im showing various schematic / pictorial rep
Simon Fraser - CHEM - 281
CHEM 281 2011-3 Lecture 2 Class Problem Solution:Molecules / Compounds that have a Molecular Formula C5H10
Simon Fraser - CHEM - 281
PERIODIC TABLE OF THE ELEMENTSIIIIIIIVVVI VII VIII12HHe1.0084.002345678910LiBeBCNOFNe6.9419.01210.8112.0114.0116.0019.0020.181112131415161718NaMgAlSiPSClAr22.9924.3026.9828.0930.9732.0635.4539.95
Simon Fraser - CHEM - 281
CHEMISTRY 281-4 (2011-3). Dr. Pete WilsonORGANIC CHEMISTRY I: PROBLEM SET 1.WEEK 2 LECTURES 4/5 (Wednesday and Friday)Problems from Bruice,6e Organic Chemistry Custom Volume 1For discussion in tutorials (Week 3) in addition to any other generalquest
Simon Fraser - CHEM - 281
MACM 101 Discrete Mathematics IExercises on Propositional Logic. Due: Friday, September 30th (at the beginning of the class)Reminder: the work you submit must be your own. Any collaboration andconsulting outside resources must be explicitly mentioned o
Simon Fraser - CHEM - 282
282IntroductionLecture 1 January September 7th, 2011Welcome to Your Second OrganicChemistry Course !vPeggyPaduraru course instructorvofficeBLU9820vtelephone 778-782-5493 (no messages pls.)ve-mail mppadura@sfu.cavTeachingassistantsvKyleGreenw
Simon Fraser - CHEM - 282
282Aromaticity and BenzeneIntro to EASLecture 4 September 16, 2011Benzene C6H6v the six electrons are delocalized they roam freely within the doughnutshaped clouds that lie over and under the ring of carbon atomsv the C-C bonds in benzene have the s
Simon Fraser - CHEM - 282
282Reactions of Substituted BenzenesLecture 6 September 23, 2011Reactions of Alkyl Substituents (4)v reduction of a nitro substituent2Nomenclature of Disubstituted andPolysubstituted Benzenesv disubstituted benzenes can be named using o, m, p pref
Simon Fraser - CHEM - 282
282Delocalized ElectronsDienes and Their ReactionsLecture 2 September 9th , 2011Localized / Delocalized Electronsvlocalized electrons belong to a single atom or are confined to abond between two atomsl o ca l i zede l e c tr o n sH 3CNH2H 2CH
Simon Fraser - CHEM - 282
282Thermodynamic vs Kinetic ControlDiels-Alder ReactionLecture 3 September 14, 2011Thermodynamic vs Kinetic controlv the 1,4-addition product is not always the thermodynamicproductThe Diels-Alder Reactionvvvvcreates two new carbon-carbon bonds
Simon Fraser - CHEM - 282
282Electrophilic Aromatic SubstitutionReactionReactions of Alkyl SubstituentsLecture 5 September 21, 2011Electrophilic Aromatic Substitution Reactions- general mechanism -2Halogenation of Benzenev bromination or chlorination of benzene requires a
Simon Fraser - CHEM - 282
Problem Set 1CHEM 282Page1The questions below are part of a previous Chem 281 final exam. Go through this problem setin order to refresh the organic chemistry I material; these questions will not be discussed intutorial. If you need help with unders
Simon Fraser - CHEM - 282
Problem Set 2CHEM 282Page1Review - Delocalized Electrons Dienes1. Please give the product of the following sequences of reactions, and make sure that theseChem 281 alkenes reactions are understood:HBrH2SO4, H2OBr2, CH2Cl2ORBr2 , H2OmCPBAa. BH
Simon Fraser - CHEM - 282
Problem Set 3CHEM 282Page1Diels-Alder Reaction and Aromaticity1. What is the product of the following reaction?HCH2CCCH2O+H2CHCCheatCH32. Which diene and which dienophile could be used to prepare each of the followingcompounds?ClH3C
Simon Fraser - CHEM - 282
Problem Set 4CHEM 282PageElectrophilic Aromatic Substitution Reactions1. Write the reaction mechanism for the following reaction:CH3+AlCl3CH3CHCH2Cl2. Show the reactants and reagents required to prepare 1-phenylbutane from benzene.3. Provide the
Simon Fraser - CHEM - 121
CHEM 121Prof. Steven Holdcroft SSB 8102Prof. Gary LeachWed. Sept. 7, 2011Todays Lecture Announcements Overview1Address questions about getting into the course toCameron Ford, Chemistry Academic Advisor&lt;chemadv@sfu.ca&gt; NOT the labinstructors/lec
Simon Fraser - CHEM - 121
Fri. Sept. 9, 2011Todays Lecture 2.1 Early History of Chemistry 2.2 Fundamental Chemical Laws, etc 2.5 Characterizing the AtomLecture 21Atoms, Molecules, and Ions(Zumdahl, Chapter 2) History of Chemistry: Robert Boyle (1627-1691): First Chemist
Simon Fraser - CHEM - 121
Monday Sept. 12, 2011Todays Lecture2.6 Atomic Structure2.7 Molecules and Ions2.8 The Periodic table2.9 Naming Compounds3.1 Atomic MassesLecture 31 Modern View of Atomic Structure: Chemist perspective: an important feature of an atom is its elec
Simon Fraser - CHEM - 121
Wed. Sept. 14, 2011Todays Lecture3.23.33.43.53.63.73.83.9Lecture 4 StoichiometryThe MoleMolar massComposition of CompoundsDetermining FormulaeChemical EquationsBalancing Chemical EquationsStoichiometric CalculationsLimiting Reagents1
Simon Fraser - CHEM - 121
Friday. Sept. 16, 2011Todays Lecture Types of Chemical Reactions andSolution Stoichiometry4.1 Water4.2 Aqueous Solutions4.3 Solution Compositions4.4 -4.9 Chemical Reactionshttp:/www.chemistry.sfu.ca/courses/course/CHEM%20121Lecture 51 Water, th
Simon Fraser - CHEM - 121
Mon. Sept. 21, 2011Todays Lecture Types of Chemical Reactions andSolution Stoichiometry 4.9 Acid Base Reactions 4.10-4.11 Oxidation/ReductionReactionsLecture 612Oxidationof coppermetal bynitricacid.34Na/H2O Oxidation-reduction reactions
Simon Fraser - CHEM - 121
General Chemistry IIn- classPractice Midterm:Wed. October 5Midterm:Monday October24 6:30 8:30 pmAtoms, Molecules, IonsStoichiometryChemical ReactionsGasesThermochemistryAtomic TheoryPeriodic PropertiesChemical BondingLiquids/SolidsSolution
Simon Fraser - CHEM - 121
Friday Sept. 23, 2011Todays Lecture Gases 5.6 Kinetic Theory of Gases1 Kinetic Molecular Theory of Gases: simple model used to explain experimentalobservations about gas behavior Postulates: Due to large inter-particle distances, their volume may
Simon Fraser - CHEM - 121
Mon, Sept. 26, 2011Todays Lecture Gasesu rms =3RT3RT=N AmMu mp =2k B T2RT=mMuavg =8k B T8RT=mM 5.7 Effusion and DiffusionMidterm Monday Oct 5.8-5.9 Collisions24, 6:30-8:30 pm 5.10 Real Gases 5.11 Gases in the Atmosphere (omit) 1
Simon Fraser - CHEM - 121
Name: _ Date: _1. Which one of the following statements about atomic structure is false?A) The electrons occupy a very large volume compared to the nucleus.B) Almost all of the mass of the atom is concentrated in the nucleus.C) The protons and neutron
Aarhus Universitet, Handels- og IngeniørHøjskolen - ECON - 101
Aarhus Universitet, Handels- og IngeniørHøjskolen - ECON - 101
Ethics in Accounting9/23/11 11:40 AMPrint this pageEthics in AccountingOBJECTIVE 4This element provides expanded discussion ofethics and ethical issues as they relate to theaccounting and the financial reportingenvironment.Following the introduct
Aarhus Universitet, Handels- og IngeniørHøjskolen - ECON - 101
Ethics in Accounting9/23/11 11:42 AMPrint this pageEthics in AccountingOBJECTIVE 4This element provides expanded discussion ofethics and ethical issues as they relate to theaccounting and the financial reportingenvironment.Following the introduct
Aarhus Universitet, Handels- og IngeniørHøjskolen - ECON - 101
Ethics in Accounting9/23/11 11:42 AMPrint this pageEthics in AccountingOBJECTIVE 4This element provides expanded discussion ofethics and ethical issues as they relate to theaccounting and the financial reportingenvironment.Following the introduct
UNC Charlotte - ACCOUN - 1220
STAT 1220-090HOMEWORK 4Name: _Fall 2011Score_1. Given that the population mean = 68 and population standard deviation = 12, find thesample mean X and the sample standard deviation Xfor a random sample of size 50 (round offto two decimal places.)(a
Georgia Perimeter - PHYSICS 22 - physics 22
ABA10.0m/s3.83mIntroductory Physics3.00m20.0o PHYS 2211-1603.53mExam #1IChapters 1-4Name_Show all your work on these sheets, and make sure to box your answers!For problems 1 4 the position of an object is described by the equationx(t) = [3m
Georgia Perimeter - PHYSICS 22 - physics 22
7.5kg6.25x104kg25o0.25m/sFpush 10.0kg13o7.65x104kg s = 2.3005.00kg0 5 Introductory Physicsmg0.30m/s .0FFs 4 PHYS 2211-160 k =m kg0push.1506.25x10r7.65x104kgExam #2n 3.00kgfvfChapters 5-10W = mg = 73.5NIName_Solutions_Show all yo
Georgia Perimeter - PHYSICS 22 - physics 22
A19.0o o3.25kg0.500m22.03.00m3.00m o65.07.00kgPrinciples of PhysicsLn 25.0oOrigin68.0o PHYS 2211-160fk 13.0mWper o68.0Exam #3Wpl2.10Nmg o112.0IChapters 11 - 14Spring 2010Name_Solutions_You may use the constants G = 6.67x10-11N m2
Georgia Perimeter - PHYSICS 22 - physics 22
Principles of Physics IPHYS 2211-160Exam #4Chapter 15 and ThermodynamicsSpring 2010Name_Solutions_Make sure to show all of yourwork, and box your answers!You might like to know that 1.00atm = 1.013x105Pa , = 5.67x10-8J/(s m2 K4) , k = 1.38x10-23J/
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS2211 Assignment #11. Alex decides to go out for a walk. At time t1 = 3.00s he is at position x1 = 10.0m , at time t2= 10.0s he is at x2 = 25.0m , and at time t3 = 20.0s he is at x3 = 17.0m . (Remember that vectorsrequire a unit vector to
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS2211 Assignment #1 Solutions1. Alex decides to go out for a walk. At time t1 = 3.00s he is at position x1 = 10.0m , at time t2= 10.0s he is at x2 = 25.0m , and at time t3 = 20.0s he is at x3 = 17.0m . (Remember that vectorsrequire a unit
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS2211 Assignment #21. Starting at the initial position of +25m , Azadeh walks with a constant velocity of -3m/s .a) What is Azadehs position after 4sec?b) What is Azadehs position after 12sec?c) What is Azadehs velocity after 20sec?d) W
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS2211 Assignment #2 Solutions1. Starting at the initial position of +25m , Azadeh walks with a constant velocity of -3m/s .With xo = 25m and vo = -3m/s we have that x = xo + vot = 25m (3m/s)t , soa)What is Azadehs position after 4sec?x(
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS1111 Assignment #31. A ball is thrown vertically (you dont know up or down) out of a window in a tall building.It takes 3.30sec to pass a point 20 meters below where it started.a.) What was the balls initial velocity?b.) What is the vel
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS1111 Assignment #3 Solutions1. A ball is thrown vertically (you dont know up or down) out of a window in a tall building.It takes 3.30sec to pass a point 20 meters below where it started.a.) What was the balls initial velocity?20m in 3.
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS1111 Assignment #3 Solutions1. A ball is thrown vertically (you dont know up or down) out of a window in a tall building.It takes 3.30sec to pass a point 20 meters below where it started.a.) What was the balls initial velocity?20m in 3.
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS1111 Assignment #4Due Monday, Feb. 1, 20101. A vector A has a magnitude of 20.0m at points at an angle of 15.0o to the left of the y axis.What are its components?2. A vector B has components Bx = 15.0m, By = -5.00m . What are its magnit
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS1111 Assignment #4Due Monday, Feb. 1, 20101. A vector A has a magnitude of 20.0m at points at an angle of 15.0o to the left of the y axis.What are its components?2. A vector B has components Bx = 15.0m, By = -5.00m . What are its magnit
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS2211 Assignment #4 SolutionsDue Monday, Feb. 1, 20101. A vector A has a magnitude of 20.0m at points at an angle of 15.0o to the left of the y axis.What are its components?AWe see from the diagram that Ax will be negative andxA15.0o