CSC 125 :: Final Exam May 3 & 5, 2010
|
|
- Piers Nash
- 5 years ago
- Views:
Transcription
1 CSC 125 :: Final Exam May 3 & 5, 2010 Name KEY (1 5) Complete the truth tables below: p Q p q p q p q p q p q T T T T F T T T F F T T F F F T F T T T F F F F F F T T Match the following logical equivalences with the answers found in the Answer Bank. Write the correct letter just to the right of the symbol. Note: Some answers will be used more than once: 6. Identity Laws 7. Domination Laws 8. Idempotent Laws 9. Double Negation Law 10. Commutative Laws p T p p F p p T T p F F p p p p p p ( p) p p q q p p q q p 1
2 11. Associative Laws 12. Distributive Laws 13. DeMorgan's Laws 14. Absorption Laws 15. Negation Laws (p q) r p (q r) (p q) r p (q r) p (q r) (p q) (p r) p (q r) (p q) (p r) (p q) p q (p q) p q p (p q) p p (p q) p p p T p p F Fill in the missing portion of each of the rules of inference named below: 16. Modus ponens 17. Modus tollens p q p. q p q q. p 18. Hypothetical syllogism p q q r p r 19. Disjunctive syllogism p q p. q 2
3 20. Resolution p q p r q r (21 26) Let B(x,y) be the statement student x attended basketball game y, where the domain for x consists of all students in this class and the domain for y consists of all basketball games this season. Match each of the quantified expressions below with the sentences found in the Answer Bank. Write the letter of the matching sentence just to the right of the expression. Please note: Some answers may be used more than once. 21. x y B(x,y) A student from this class attended a basketball game this season. 22. x y B(x,y) x y B(x,y) x y B(x,y) Note: Equivalent to #38. No student from this class attended a basketball game this season. {For all students and all basketball games, it not true that any student attended any game} 23. x y B(x,y) A student from this class attended every basketball game this season. 24. x y B(x,y) x y B(x,y) x y B(x,y) Note: Equivalent to #37. No student from this class attended every basketball game this season. 25. x y B(x,y) Every student from this class attended a basketball game this season. 26. x y B(x,y) Every student from this class attended every basketball game this season. (27 29) Let F(x,y) be the statement x can fool y, where the domain for both x and y consists of all people in the world. Use quantifiers to express these sentences. 27. Everyone can fool Fred. x F(x,Fred) 3
4 28. Evelyn can fool everybody. y F(Evelyn,y) 29. Someone can fool everybody. x y F(x,y) (30 32) Let the domain of P(x) consist of the integers 1, 2 and 3 Write out each proposition using disjunctions, conjunctions and negations. 30. x P(x) P(1) P(2) P(3) 31. x P(x) P(1) P(2) P(3) 32. x P(x) P(1) P(2) P(3) (33 37) Determine whether the arguments below are valid or invalid, then circle either Valid or Invalid. If valid, using the list of valid argument forms found in the Answer Bank, write the letter of the rule of inference; if invalid write the letter of the logical fallacy, again using the list of invalid argument forms found in the Answer Bank. 33. If we get 10" of snow, school will be cancelled. We got 8" of snow. School will not be cancelled. Invalid denying the hypothesis 34. If we pigs can fly, Santa Claus is real. Pigs can fly. Santa Claus is real. Valid Invalid Valid Valid modus ponens {affirming the hypothesis} 35. If we pigs can fly, Santa Claus is real. Santa Claus is not real. Pigs cannot fly. Invalid 4
5 Valid Valid modus tollens {denying the conclusion} 36. If pigs can fly, elephants can sing. If elephants can sing, monkeys can compose music. If pigs can fly, monkeys can compose music. Invalid Valid hypothetical syllogism 37. Logic is either hard or it is nutty. Either logic is easy or it is impossible. Logic is either nutty or it is impossible. Valid Invalid Valid resolution (38 41) Let A = {1, 3, 5, 6} B = {1, 2, 4, 6, 7} 38. A B = {1, 2, 3, 4, 5, 6, 7} 39. A B = {1, 6} 40. A B = {3, 5} 41. A B = {2, 3, 4, 5, 7} 42. Let A = {a, b, c} Give the power set of A, P(A) = {, {a,b,c},{a}, {b}, {c}, {a,b}, {a,c}, {b,c}} (43 44) Let A = {a,b,c} B = {x,y}. Find 43. A B = {(a,x), (a,y), (b,x), (b,y), (c,x), (c,y)} Valid Invalid 5
6 44. B B = {(x,x), (x,y), (y,y), (y,x)} (45 48) Determine whether each sentence below is true or false. Circle either True or False. 45. {a, b, c} {a, b, c} True False 46. {a, b, c} {a, b, c} True False 47. {a, b} {a, b, c} True False 48. {0} True False (49 51) Determine whether each sentence below is always true or not (sometimes false). Circle either Always True or Not Always. 49. (A B) A Always True Not Always 50. (A B) A Always True Not Always 51. A B = B A Always True Not Always (52 53) Which region(s) in the Venn diagrams above represent: 52. A (B C) B C = 2,3,4,5,6,7 A (B C) = 2,4,5 6
7 53. cmp(a) cmp(b) cmp(c) Note: cmp(a) cmp(b) cmp(c) = cmp(a B C) A B C = 1,2,3,4,5,6,7 cmp(a B C) = 8 = cmp(a) cmp(b) cmp(c) (54 55) Give the following combinations in simplified form, if: f(x) = (x 2 5) g(x) = (3x + 1) 54. (f + g) (x) = (x 2 5) + (3x + 1) = x 2 + 3x (f o g) (x) = f(g(x)) = ((3x+1) 2 5) = 9x 2 + 6x = 9x 2 + 6x 6 (56 57) For the questions below, the notation n j=0 x represents the summation of x as j goes from 0 to n. (56 57) What are the values of these sums? k=0 (k+1) (0+1) + (1+1) + (2+1) + (3+1) + (4+1) + (5+1) = = j= = 5*3 = 15 (58 59) What is the value of each of these geometric progressions? j=0 2 j a = 1; r = 2; n = 5 a(r n+1 1)/(r 1) = 1(2 6 1)/(2 1) = 64 1 = j=0 ( 3) j a = 1; r = 3; n = 4 a(r n+1 1)/(r 1) = 1(( 3) 5 1)/( 3 1) = ( 243 1)/( 4) = 244/ 4 = 61 (60 61) Compute each of these double sums i=1 3 j=1 (i+j) 3 i=1 (i+1) + (i+2) + (i+3) = 3 i=1 3i+6 = (3*1 + 6) + (3*2 + 6) + (3*3 + 6) = = 36 7
8 61. 3 i=0 2 j=0 (3i+2j) 3 i=0 (3i + 2*0) + (3i + 2*1) +(3i + 2*2) = 3 i=0 (3i i i + 4) = 3 i=0 (9i + 6) = (9*0 + 6) + (9*1 + 6) + (9*2 + 6) + (9*3 + 6) = 9*6 + 4*6 = = Arrange these complexity classes in ascending order of complexity: O(n n ), O(n b ), O(n), O(1), O(n!), O(n log n), O(log n), O(b n ) Answer:: O(1), O(log n), O(n), O(n log n), O(n b ), O(b n ), O(n!), O(n n ) 63. P(9,3) = 9!/6! = /720 = C(10,4) = 10!/4!6! = /24*720 = The 8 th line of Pascal s triangle is given below. Give the next line Answer:: How many different bit strings of length 8 are there? mod 17 = mod 13 = Give the prime factorization of *7*11* Give the prime factorization of *5* Give the prime factorization of 7! 2 4 *3 2 *5*7 72. List 3 consecutive odd integers that are all prime. 8
9 3, 5, How many bit strings of length 8 contain exactly 3 1's? 8*7*6/3*2*1 = What is the minimum number of students, each of whom comes from one of the 50 states, that must be enrolled in a university to guarantee that there are at least 100 that come from the same state? What are the two basic components of a recursive function? base case recursive call 76. What are the two basic components of mathematical induction? base case inductive step 77. What are the steps in proof by mathematical induction? 1. prove the base case 2. assume true for k 3. prove that it follows for k+1 OR: 1. prove base case, P(0) 2. prove P(k) P(k+1) 78. Give the recursive definition of factorial: fact(n) = 1, if n = 0 n * fact(n), otherwise 79. Why is this not a circular definition? Because: 1. n-1 < n, and therefore: 2. We are guaranteed to eventually reach the base case in the definition Prove: The sum of two odd integers is even. Proof: Let: a = 2k + 1 b = 2l + 1 Then: a + b = 2k l + 1 = 2(k + l +1) EVEN 9
10 Prove: The product of two odd integers is odd. Proof: Let: a = 2k + 1 b = 2l + 1 Then: ab = (2k + 1)*(2 l + 1) = 4kl + 2k + 2l +1 = 2(2kl + k + l) + 1 ODD Prove: For every positive integer n, there are n consecutive composite integers. [Hint: Consider the n consecutive integers starting with (n+1)! + 2.] Proof: 2 (n+1)! (n+1)! (n+1)! (n+1)! n (n+1)! + n n+1 (n+1)! + n+1 Thus, there are n consecutive composites. Extra Credit EC-1. Give the values of C and k which can be used to show that f(x) = 3x + 7 is O(x). C = 4; k = 7 EC-2. Give the values of C and k which can be used to show that f(x) = x 4 + 9x 3 + 5x 2 + 4x + 7 is O(x 4 ). C = 5; k = 9 EC-3. How many bit strings of length 10 contain at least 4 1's? 2 10 at most 3 1's = 2 10 (C(10,0) + C(10,1) + C(10,2) + C(10,3)) = ( ) = 848 EC-4. How many bit strings of length 10 have an equal number of 1's and 0's? 10
11 To be equal #-1s = #-0s = 5 So question is, how many bit strings have exactly 5 1's? C(10,5) = 252 EC-5. What is the coefficient of x 5 y 8 in (x + y) 13? C(13,5) = 1287 EC-6. What is the coefficient of x 7 in (x + 1) 11? C(11,7) = 330 EC-7. What is the coefficient of y 3 x 5 in (3y + 2x) 8? (3y 3 )(2x 5 )*C(8,3) = *8*7*6/3*2*1 = 27*32*56 = EC-8. What is the coefficient of x 9 in (2 x) 19? *C(19,9) = *92378 = -94,595,072 EC-9. Prove or disprove: If a bc, with a,b & c positive integers, then either a b or a c. DISPROOF:: by counterexample: a = 15, b =9, c = 5, bc = but 15 9 & 15 5 OR: a = 6, b =2, c = 9, bc = but 6 2 & 6 9 EC-10. Prove by mathematical induction: n! > 2 n, for all n > c, where c is some positive integer. To Prove: n! > 2 n, for all n > 3. PROOF:: Prove base case: 4! = 24 > 16 = 2 4 Inductive step: Assume: k! > 2 k, for all k > 3. To prove: (k+1)!> 2 k+1. Proof: (k+1)! = (k+1)*k! > (k+1)* 2 k > 2*2 k = 2 k+1 EC-11. Prove by mathematical induction: The sum of the first n even nonzero integers is n(n+1). In other words, prove: n i=1 i = n(n+1) 11
12 PROOF:: Prove base case: To prove: 1 i=1 i = n(n+1) Proof: 1 i=1 i = 2*1 = 2 If n =1, then n(n+1) = 1*(1+1) = 2 Inductive step: Assume: k i=1 i = k(k+1) To prove: k+1 i=1 i = (k+1)[(k+1) + 1] Proof: k+1 i=1 i = k i=1 i + 2(k+1) = k(k+1) + 2(k+1) = (k+1)(k+2) = (k+1)[(k+1) + 1] EC-12. Prove by mathematical induction: n! > 25 n, for all n > c, where c is some positive integer. To Prove: n! > 25 n, for all n > 65. PROOF:: Prove base case: 65! = 8.2 x > 7.3 x = Inductive step: Assume: k! > 25 k, for all k > 65. To prove: (k+1)!> 25 k+1. Proof: (k+1)! = (k+1)*k! > (k+1)* 25 k > 25*25 k = 25 k+1 EC-13. Prove: For all b > 0, c b such that n! > b n, for all n > c b. PROOF:: It is very likely that c b = ceil(2*e) is the correct value. But for a Q&D solution, c b = b b^2+b will work. First, we look at two examples: b = 3 b b^2+b = 3 12 (b 2 +b)! = 1*2*3*4*5*6*7*8*9*10*11*12 3*4*5 > 3 3 6*7*8 > = > > > 3 2 All together, (b 2 +b)! > 3 2*3+4*2 = 3 14 >
13 b = 5 b b^2+b = 5 30 (b 2 +b)! = 1*2*3*4*5 *.. *29*30 5*6*7*8*9 > *11*12*13*14 > *16*17*18*19 > *21*22*23*24 > = > > > > > 5 2 All together, (b 2 +b)! > 5 4*5+6*2 = 5 32 > 5 30 In general, b b^2+b represents b multiplied by itself b 2 + b times (b 2 +b)! is a number greater than b multiplied by itself (b-1)*b + 2*(b+1) = b 2 b +2b +2 = b 2 + b +2 times Since b 2 + b +2 > b 2 + b, (b 2 + b)! > b b^2+b 13
CSC 125 :: Final Exam December 15, 2010
1-5. Complete the truth tables below: CSC 125 :: Final Exam December 15, 2010 p q p q p q p q p q p q T T F F T F T F 6-13. Fill in the missing portion of each of the rules of inference named below: 6.
More informationCSC 125 :: Final Exam December 14, 2011
1-5. Complete the truth tables below: CSC 125 :: Final Exam December 14, 2011 p q p q p q p q p q p q T T F F T F T F (6 9) Let p be: Log rolling is fun. q be: The sawmill is closed. Express these as English
More informationPacket #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics
CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus
More informationAt least one of us is a knave. What are A and B?
1. This is a puzzle about an island in which everyone is either a knight or a knave. Knights always tell the truth and knaves always lie. This problem is about two people A and B, each of whom is either
More informationSample Problems for all sections of CMSC250, Midterm 1 Fall 2014
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 1. Translate each of the following English sentences into formal statements using the logical operators (,,,,, and ). You may also use mathematical
More informationA. Propositional Logic
CmSc 175 Discrete Mathematics A. Propositional Logic 1. Statements (Propositions ): Statements are sentences that claim certain things. Can be either true or false, but not both. Propositional logic deals
More informationDISCRETE MATH: FINAL REVIEW
DISCRETE MATH: FINAL REVIEW DR. DANIEL FREEMAN 1) a. Does 3 = {3}? b. Is 3 {3}? c. Is 3 {3}? c. Is {3} {3}? c. Is {3} {3}? d. Does {3} = {3, 3, 3, 3}? e. Is {x Z x > 0} {x R x > 0}? 1. Chapter 1 review
More information1. Consider the conditional E = p q r. Use de Morgan s laws to write simplified versions of the following : The negation of E : 5 points
Introduction to Discrete Mathematics 3450:208 Test 1 1. Consider the conditional E = p q r. Use de Morgan s laws to write simplified versions of the following : The negation of E : The inverse of E : The
More informationDiscrete Structures CRN Test 3 Version 1 CMSC 2123 Autumn 2013
. Print your name on your scantron in the space labeled NAME. 2. Print CMSC 223 in the space labeled SUBJECT. 3. Print the date 2-2-203, in the space labeled DATE. 4. Print your CRN, 786, in the space
More informationDo not start until you are given the green signal
SOLUTIONS CSE 311 Winter 2011: Midterm Exam (closed book, closed notes except for 1-page summary) Total: 100 points, 5 questions. Time: 50 minutes Instructions: 1. Write your name and student ID on the
More informationDiscrete Mathematical Structures: Theory and Applications
Chapter 1: Foundations: Sets, Logic, and Algorithms Discrete Mathematical Structures: Theory and Applications Learning Objectives Learn about sets Explore various operations on sets Become familiar with
More informationWUCT121. Discrete Mathematics. Logic. Tutorial Exercises
WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the
More informationIntro to Logic and Proofs
Intro to Logic and Proofs Propositions A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. Examples: It is raining today. Washington
More information2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
More informationUNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) I Semester Core Course. FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) (2011 Admission Onwards) I Semester Core Course FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK 1) If A and B are two sets
More informationKS MATEMATIKA DISKRIT (DISCRETE MATHEMATICS ) RULES OF INFERENCE. Discrete Math Team
KS091201 MATEMATIKA DISKRIT (DISCRETE MATHEMATICS ) RULES OF INFERENCE Discrete Math Team 2 -- KS091201 MD W-04 Outline Valid Arguments Modus Ponens Modus Tollens Addition and Simplification More Rules
More informationLogic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics Solution Guide
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Discrete Mathematics Solution Guide Marcel B. Finan c All Rights Reserved 2015 Edition Contents
More informationLECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel
LECTURE NOTES on DISCRETE MATHEMATICS Eusebius Doedel 1 LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is
More informationA Quick Lesson on Negation
A Quick Lesson on Negation Several of the argument forms we have looked at (modus tollens and disjunctive syllogism, for valid forms; denying the antecedent for invalid) involve a type of statement which
More informationRules Build Arguments Rules Building Arguments
Section 1.6 1 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments Rules of Inference for Quantified Statements Building Arguments for Quantified
More informationCOMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
More informationLecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel
Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical
More informationCS 2336 Discrete Mathematics
CS 2336 Discrete Mathematics Lecture 3 Logic: Rules of Inference 1 Outline Mathematical Argument Rules of Inference 2 Argument In mathematics, an argument is a sequence of propositions (called premises)
More informationLECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel
LECTURE NOTES on DISCRETE MATHEMATICS Eusebius Doedel 1 LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is
More informationChapter 1 Elementary Logic
2017-2018 Chapter 1 Elementary Logic The study of logic is the study of the principles and methods used in distinguishing valid arguments from those that are not valid. The aim of this chapter is to help
More informationProofs. Example of an axiom in this system: Given two distinct points, there is exactly one line that contains them.
Proofs A mathematical system consists of axioms, definitions and undefined terms. An axiom is assumed true. Definitions are used to create new concepts in terms of existing ones. Undefined terms are only
More informationAnnouncements. Exam 1 Review
Announcements Quiz today Exam Monday! You are allowed one 8.5 x 11 in cheat sheet of handwritten notes for the exam (front and back of 8.5 x 11 in paper) Handwritten means you must write them by hand,
More informationMath 3336: Discrete Mathematics Practice Problems for Exam I
Math 3336: Discrete Mathematics Practice Problems for Exam I The upcoming exam on Tuesday, February 26, will cover the material in Chapter 1 and Chapter 2*. You will be provided with a sheet containing
More information2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic
CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares
More informationFirst order Logic ( Predicate Logic) and Methods of Proof
First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating
More informationBoolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012
March 5, 2012 Webwork Homework. The handout on Logic is Chapter 4 from Mary Attenborough s book Mathematics for Electrical Engineering and Computing. Proving Propositions We combine basic propositions
More informationReexam in Discrete Mathematics
Reexam in Discrete Mathematics First Year at the Faculty of Engineering and Science and the Technical Faculty of IT and Design August 15th, 2017, 9.00-13.00 This exam consists of 11 numbered pages with
More informationChapter 1, Logic and Proofs (3) 1.6. Rules of Inference
CSI 2350, Discrete Structures Chapter 1, Logic and Proofs (3) Young-Rae Cho Associate Professor Department of Computer Science Baylor University 1.6. Rules of Inference Basic Terminology Axiom: a statement
More informationSteinhardt School of Culture, Education, and Human Development Department of Teaching and Learning. Mathematical Proof and Proving (MPP)
Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning Terminology, Notations, Definitions, & Principles: Mathematical Proof and Proving (MPP) 1. A statement
More informationLogic and Proof. Aiichiro Nakano
Logic and Proof Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Department of Computer Science Department of Physics & Astronomy Department of Chemical Engineering & Materials Science
More informationCSI30. Chapter 1. The Foundations: Logic and Proofs Nested Quantifiers
Chapter 1. The Foundations: Logic and Proofs 1.9-1.10 Nested Quantifiers 1 Two quantifiers are nested if one is within the scope of the other. Recall one of the examples from the previous class: x ( P(x)
More informationReadings: Conjecture. Theorem. Rosen Section 1.5
Readings: Conjecture Theorem Lemma Lemma Step 1 Step 2 Step 3 : Step n-1 Step n a rule of inference an axiom a rule of inference Rosen Section 1.5 Provide justification of the steps used to show that a
More informationChapter 3. The Logic of Quantified Statements
Chapter 3. The Logic of Quantified Statements 3.1. Predicates and Quantified Statements I Predicate in grammar Predicate refers to the part of a sentence that gives information about the subject. Example:
More informationICS141: Discrete Mathematics for Computer Science I
ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross
More informationPredicate Logic & Quantification
Predicate Logic & Quantification Things you should do Homework 1 due today at 3pm Via gradescope. Directions posted on the website. Group homework 1 posted, due Tuesday. Groups of 1-3. We suggest 3. In
More informationTest 1 Solutions(COT3100) (1) Prove that the following Absorption Law is correct. I.e, prove this is a tautology:
Test 1 Solutions(COT3100) Sitharam (1) Prove that the following Absorption Law is correct. I.e, prove this is a tautology: ( q (p q) (r p)) r Solution. This is Modus Tollens applied twice, with transitivity
More informationTools for reasoning: Logic. Ch. 1: Introduction to Propositional Logic Truth values, truth tables Boolean logic: Implications:
Tools for reasoning: Logic Ch. 1: Introduction to Propositional Logic Truth values, truth tables Boolean logic: Implications: 1 Why study propositional logic? A formal mathematical language for precise
More informationMethods of Proof. 1.6 Rules of Inference. Argument and inference 12/8/2015. CSE2023 Discrete Computational Structures
Methods of Proof CSE0 Discrete Computational Structures Lecture 4 When is a mathematical argument correct? What methods can be used to construct mathematical arguments? Important in many computer science
More information1 The Foundation: Logic and Proofs
1 The Foundation: Logic and Proofs 1.1 Propositional Logic Propositions( 명제 ) a declarative sentence that is either true or false, but not both nor neither letters denoting propositions p, q, r, s, T:
More informationRules of Inference. Arguments and Validity
Arguments and Validity A formal argument in propositional logic is a sequence of propositions, starting with a premise or set of premises, and ending in a conclusion. We say that an argument is valid if
More informationSolutions to Exercises (Sections )
s to Exercises (Sections 1.11-1.12) Section 1.11 Exercise 1.11.1 (a) p q q r r p 1. q r Hypothesis 2. p q Hypothesis 3. p r Hypothetical syllogism, 1, 2 4. r Hypothesis 5. p Modus tollens, 3, 4. (b) p
More information5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction.
Statements Compounds and Truth Tables. Statements, Negations, Compounds, Conjunctions, Disjunctions, Truth Tables, Logical Equivalence, De Morgan s Law, Tautology, Contradictions, Proofs with Logical Equivalent
More informationThe Logic of Compound Statements cont.
The Logic of Compound Statements cont. CSE 215, Computer Science 1, Fall 2011 Stony Brook University http://www.cs.stonybrook.edu/~cse215 Refresh from last time: Logical Equivalences Commutativity of :
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs
More informationPacket #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should
More informationCS0441 Discrete Structures Recitation 3. Xiang Xiao
CS0441 Discrete Structures Recitation 3 Xiang Xiao Section 1.5 Q10 Let F(x, y) be the statement x can fool y, where the domain consists of all people in the world. Use quantifiers to express each of these
More information1 The Foundation: Logic and Proofs
1 The Foundation: Logic and Proofs 1.1 Propositional Logic Propositions( ) a declarative sentence that is either true or false, but not both nor neither letters denoting propostions p, q, r, s, T: true
More informationChapter 4, Logic using Propositional Calculus Handout
ECS 20 Chapter 4, Logic using Propositional Calculus Handout 0. Introduction to Discrete Mathematics. 0.1. Discrete = Individually separate and distinct as opposed to continuous and capable of infinitesimal
More informationsoftware design & management Gachon University Chulyun Kim
Gachon University Chulyun Kim 2 Outline Propositional Logic Propositional Equivalences Predicates and Quantifiers Nested Quantifiers Rules of Inference Introduction to Proofs 3 1.1 Propositional Logic
More information3. The Logic of Quantified Statements Summary. Aaron Tan August 2017
3. The Logic of Quantified Statements Summary Aaron Tan 28 31 August 2017 1 3. The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I Predicate; domain; truth set Universal quantifier,
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationReview: Potential stumbling blocks
Review: Potential stumbling blocks Whether the negation sign is on the inside or the outside of a quantified statement makes a big difference! Example: Let T(x) x is tall. Consider the following: x T(x)
More informationReview 1. Andreas Klappenecker
Review 1 Andreas Klappenecker Summary Propositional Logic, Chapter 1 Predicate Logic, Chapter 1 Proofs, Chapter 1 Sets, Chapter 2 Functions, Chapter 2 Sequences and Sums, Chapter 2 Asymptotic Notations,
More informationCSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 February 5, 2015 2 Announcements Homework 1 is due now. Homework 2 will be posted on the web site today. It is due Thursday, Feb. 12 at 10am in class.
More informationIII. Elementary Logic
III. Elementary Logic The Language of Mathematics While we use our natural language to transmit our mathematical ideas, the language has some undesirable features which are not acceptable in mathematics.
More informationWhat is the decimal (base 10) representation of the binary number ? Show your work and place your final answer in the box.
Question 1. [10 marks] Part (a) [2 marks] What is the decimal (base 10) representation of the binary number 110101? Show your work and place your final answer in the box. 2 0 + 2 2 + 2 4 + 2 5 = 1 + 4
More information1.1 Statements and Compound Statements
Chapter 1 Propositional Logic 1.1 Statements and Compound Statements A statement or proposition is an assertion which is either true or false, though you may not know which. That is, a statement is something
More informationPHI Propositional Logic Lecture 2. Truth Tables
PHI 103 - Propositional Logic Lecture 2 ruth ables ruth ables Part 1 - ruth unctions for Logical Operators ruth unction - the truth-value of any compound proposition determined solely by the truth-value
More informationMACM 101 Discrete Mathematics I. Exercises on Propositional Logic. Due: Tuesday, September 29th (at the beginning of the class)
MACM 101 Discrete Mathematics I Exercises on Propositional Logic. Due: Tuesday, September 29th (at the beginning of the class) SOLUTIONS 1. Construct a truth table for the following compound proposition:
More informationNatural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms)
More informationChapter 1: The Logic of Compound Statements. January 7, 2008
Chapter 1: The Logic of Compound Statements January 7, 2008 Outline 1 1.1 Logical Form and Logical Equivalence 2 1.2 Conditional Statements 3 1.3 Valid and Invalid Arguments Central notion of deductive
More informationCS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)
CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bio-conditional Converse Inverse Contrapositive Laws of
More informationMAT 243 Test 1 SOLUTIONS, FORM A
t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,
More informationDiscrete Mathematics Exam File Spring Exam #1
Discrete Mathematics Exam File Spring 2008 Exam #1 1.) Consider the sequence a n = 2n + 3. a.) Write out the first five terms of the sequence. b.) Determine a recursive formula for the sequence. 2.) Consider
More informationLecture 3. Logic Predicates and Quantified Statements Statements with Multiple Quantifiers. Introduction to Proofs. Reading (Epp s textbook)
Lecture 3 Logic Predicates and Quantified Statements Statements with Multiple Quantifiers Reading (Epp s textbook) 3.1-3.3 Introduction to Proofs Reading (Epp s textbook) 4.1-4.2 1 Propositional Functions
More informationSection A (not in the text) Which of the following are statements? Explain. 3. The President of the United States in 2089 will be a woman.
Math 299 Homework Assignment, Chapter 2 Section 2.1 2.A (not in the text) Which of the following are statements? Explain. 1. Let x be a positive integer. Then x is rational. 2. Mathematics is fun. 3. The
More information3/29/2017. Logic. Propositions and logical operations. Main concepts: propositions truth values propositional variables logical operations
Logic Propositions and logical operations Main concepts: propositions truth values propositional variables logical operations 1 Propositions and logical operations A proposition is the most basic element
More informationCHAPTER 1 - LOGIC OF COMPOUND STATEMENTS
CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS 1.1 - Logical Form and Logical Equivalence Definition. A statement or proposition is a sentence that is either true or false, but not both. ex. 1 + 2 = 3 IS a statement
More informationMathematical Induction
Mathematical Induction MAT30 Discrete Mathematics Fall 018 MAT30 (Discrete Math) Mathematical Induction Fall 018 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)
More informationSection 1.3. Let I be a set. When I is used in the following context,
Section 1.3. Let I be a set. When I is used in the following context, {B i } i I, we call I the index set. The set {B i } i I is the family of sets of the form B i where i I. One could also use set builder
More informationReview 3. Andreas Klappenecker
Review 3 Andreas Klappenecker Final Exam Friday, May 4, 2012, starting at 12:30pm, usual classroom Topics Topic Reading Algorithms and their Complexity Chapter 3 Logic and Proofs Chapter 1 Logic and Proofs
More informationMath 3320 Foundations of Mathematics
Math 3320 Foundations of Mathematics Chapter 1: Fundamentals Jesse Crawford Department of Mathematics Tarleton State University (Tarleton State University) Chapter 1 1 / 55 Outline 1 Section 1.1: Why Study
More informationNormal Forms Note: all ppts about normal forms are skipped.
Normal Forms Note: all ppts about normal forms are skipped. Well formed formula (wff) also called formula, is a string consists of propositional variables, connectives, and parenthesis used in the proper
More informationMathacle. PSet ---- Algebra, Logic. Level Number Name: Date: I. BASICS OF PROPOSITIONAL LOGIC
I. BASICS OF PROPOSITIONAL LOGIC George Boole (1815-1864) developed logic as an abstract mathematical system consisting of propositions, operations (conjunction, disjunction, and negation), and rules for
More informationDiscrete Mathematics
Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 9, 2009 Overview of ( 1.5-1.7, ~2 hours) Methods of mathematical argument (i.e., proof methods) can be formalized
More information(Refer Slide Time: 02:20)
Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 5 Logical Inference In the last class we saw about
More informationLecture 7 Feb 4, 14. Sections 1.7 and 1.8 Some problems from Sec 1.8
Lecture 7 Feb 4, 14 Sections 1.7 and 1.8 Some problems from Sec 1.8 Section Summary Proof by Cases Existence Proofs Constructive Nonconstructive Disproof by Counterexample Nonexistence Proofs Uniqueness
More informationProofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction
Introduction I Proofs Computer Science & Engineering 235 Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu A proof is a proof. What kind of a proof? It s a proof. A proof is a proof. And when
More informationFormal Logic: Quantifiers, Predicates, and Validity. CS 130 Discrete Structures
Formal Logic: Quantifiers, Predicates, and Validity CS 130 Discrete Structures Variables and Statements Variables: A variable is a symbol that stands for an individual in a collection or set. For example,
More informationLogic. Logic is a discipline that studies the principles and methods used in correct reasoning. It includes:
Logic Logic is a discipline that studies the principles and methods used in correct reasoning It includes: A formal language for expressing statements. An inference mechanism (a collection of rules) to
More informationCSE 1400 Applied Discrete Mathematics Proofs
CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationUNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS MATH 122: Logic and Foundations
UNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS 2013 MATH 122: Logic and Foundations Instructor and section (check one): K. Mynhardt [A01] CRN 12132 G. MacGillivray [A02] CRN 12133 NAME: V00#: Duration: 3
More informationProofs. Introduction II. Notes. Notes. Notes. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Fall 2007
Proofs Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.5, 1.6, and 1.7 of Rosen cse235@cse.unl.edu
More informationCSCE 222 Discrete Structures for Computing. Review for the Final. Hyunyoung Lee
CSCE 222 Discrete Structures for Computing Review for the Final! Hyunyoung Lee! 1 Final Exam Section 501 (regular class time 8:00am) Friday, May 8, starting at 1:00pm in our classroom!! Section 502 (regular
More informationPSU MATH RELAYS LOGIC & SET THEORY 2017
PSU MATH RELAYS LOGIC & SET THEORY 2017 MULTIPLE CHOICE. There are 40 questions. Select the letter of the most appropriate answer and SHADE in the corresponding region of the answer sheet. If the correct
More informationECOM Discrete Mathematics
ECOM 2311- Discrete Mathematics Chapter # 1 : The Foundations: Logic and Proofs Fall, 2013/2014 ECOM 2311- Discrete Mathematics - Ch.1 Dr. Musbah Shaat 1 / 85 Outline 1 Propositional Logic 2 Propositional
More informationMidterm Exam Solution
Midterm Exam Solution Name PID Honor Code Pledge: I certify that I am aware of the Honor Code in effect in this course and observed the Honor Code in the completion of this exam. Signature Notes: 1. This
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationFull file at Chapter 1
Chapter 1 Use the following to answer questions 1-5: In the questions below determine whether the proposition is TRUE or FALSE 1. 1 + 1 = 3 if and only if 2 + 2 = 3. 2. If it is raining, then it is raining.
More informationLogic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.
Math 0413 Appendix A.0 Logic Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another. This type of logic is called propositional.
More informationThinking of Nested Quantification
Section 1.5 Section Summary Nested Quantifiers Order of Quantifiers Translating from Nested Quantifiers into English Translating Mathematical Statements into Statements involving Nested Quantifiers. Translating
More informationPropositional Logic. Argument Forms. Ioan Despi. University of New England. July 19, 2013
Propositional Logic Argument Forms Ioan Despi despi@turing.une.edu.au University of New England July 19, 2013 Outline Ioan Despi Discrete Mathematics 2 of 1 Order of Precedence Ioan Despi Discrete Mathematics
More information