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Course: MACM 101, Spring 2011
School: Simon Fraser
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Logic Discrete Laws Introductionof Mathematics Andrei Bulatov Discrete Mathematics Laws of Logic Previous Lecture Truth tables Tautologies and contradictions Logic equivalences 4-2 Discrete Mathematics Laws of Logic Logic Equivalences Compound statements and are said to be logically equivalent if the statement is true (false) if and only if is true (respectively, false) or The truth tables of and...

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Logic Discrete Laws Introductionof Mathematics Andrei Bulatov Discrete Mathematics Laws of Logic Previous Lecture Truth tables Tautologies and contradictions Logic equivalences 4-2 Discrete Mathematics Laws of Logic Logic Equivalences Compound statements and are said to be logically equivalent if the statement is true (false) if and only if is true (respectively, false) or The truth tables of and are equal or For any choice of truth values of the primitive statements (propositional variables) of and , formulas and have the same truth value If and are logically equivalent, we write 4-3 Discrete Mathematics Laws of Logic Why Logic Equivalences To simplify compound statements ``If you are a computer science major or a freshman and you are not a computer science major or you are granted access to the Internet, then you are a freshman or have access to the Internet To convert complicated compound statements to certain `normal form that is easier to handle Conjunctive Normal Form CNF 4-4 Discrete Mathematics Laws of Logic Example Equivalences Implication and its contrapositive p q pq q p 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 1 All tautologies are equivalent to T All contradictions are equivalent to F 4-5 Discrete Mathematics Laws of Logic Laws of Logic Double negation p p p p p 0 1 0 1 0 1 4-6 4-7 Discrete Mathematics Laws of Logic Laws of Logic (cntd) DeMorgans laws (p q) p q (p q) p q p q p q pq (p q) p q 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 Discrete Mathematics Laws of Logic Example Construct the negation of `Miguel has a cell phone and he has a laptop `Heather will go to the concert or Steve will go to the concert 4-8 Discrete Mathematics Laws of Logic `Algebraic Laws of Logic pq qp pq qp Commutative laws p (q r) (p q) r p (q r) (p q) r Associative laws p (q r) (p q) (p r) p (q r) (p q) (p r) pp p pp p Idempotent laws Distributive laws 4-9 Discrete Mathematics Laws of Logic 4-10 `Logic Laws of Logic pT p Identity p pF laws p p F p p T Inverse laws pF F pT T Domination laws p (p q) p p (p q) p Absorption laws the law of contradiction the law of excluded middle Discrete Mathematics Laws of Logic Example Simplify the statement (q r) (q p) r p 4-11 Discrete Mathematics Laws of Logic Expressing Connectives Some connectives can be expressed through others p q (p q) p q (p q) (q p) p q p q Theorem Every compound statement is logically equivalent to a statement that uses only conjunction, disjunction, and negation 4-12 Discrete Mathematics Laws of Logic Example ``If you are a computer science major or a freshman and you are not a computer science major or you are granted access to the Internet, then you are a freshman or have access to the Internet p - `you can access the Internet from campus q - `you are a computer science major r - `you are a freshman 4-13 Discrete Mathematics Laws of Logic Example Simplify the statement (p q) (p q) 4-14 Discrete Mathematics Laws of Logic 4-15 First Law of Substitution Suppose that the compound statement is a tautology. If p is a primitive statement that appears in and we replace each occurrence of p by the same statement q, then the resulting compound statement is also a tautology. Let = (p q) (q p), and we substitute p by p (s r) Therefore ((p (s r)) q) (q (p (s r)) is a tautology Discrete Mathematics Laws of Logic 4-16 Second Law of Substitution Let be a compound statement, p an arbitrary (not necessarily primitive!) statement that appears in , and let q be a statement such that p q. If we replace one or more occurrences of p by q, then for the resulting compound statement we have . Let = (p q) (q p), and we substitute the first occurrence of p by p (p q). Recall that p p (p q) by Absorption Law. Therefore (p q) (q p) ((p (p q)) q) (q p). Discrete Mathematics Laws of Logic Homework Exercises from the Book: No. 1ai, 2, 6a, 6b, 14a (page 66) - Express conjunction and disjunction through implication and negation (*) 4-17
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Simon Fraser - MACM - 101
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Simon Fraser - MACM - 101
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Simon Fraser - MACM - 101
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Simon Fraser - CHEM - 281
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Simon Fraser - CHEM - 281
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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
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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
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Simon Fraser - CHEM - 121
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Simon Fraser - CHEM - 121
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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
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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
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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
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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
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS2211 Assignment #5These problems are not to be turned in.They are for your practice for the exam.(Youre welcome!)1. While golfing, Haile hits a ball from the ground, at an angle of 12.0o above the horizontal andan initial speed of 55.0
Georgia Perimeter - PHYSICS 22 - physics 22
Jim Guinns PHYS2211 Assignment #5 SolutionsThese problems are not to be turned in.They are for your practice for the exam.(Youre welcome!)1. While golfing, Haile hits a ball from the ground, at an angle of 12.0o above the horizontal andan initial spe