Discrete Math Midterm 2014
|
|
- Alyson Carroll
- 5 years ago
- Views:
Transcription
1 Discrete Math Midterm 01 Name Student # Note: The numbers used in the question varied from exam to exam, so your exam might have a different numerical answer than the one given below. 1. (3 pts Sixteen people are to be seated at two circular tables, one seats 10 people and the other seats 6. How many different seating arrangements are possible? There are ( ways to divide the people amoung the two tables. The number of arrangements at the two tables are 9! and 5!. So ( ! 5!.. (8 pts (a Count the number of walks in the plane from (0, 0 to (, 9, using only steps that go one unit up or 1 unit to the right. (b How many from (a hit the point (, 5? (c How many from (a do not hit the point (, 8? (d The walks in (a are walks of 13 steps. How many walks of 15 steps are there from (0, 0 to (, 9? (You can now use steps going one unit down or left and you can visit points more than once. (a ( 13 : there are + 9 = 13 steps and any four of them can be chosen as the up steps. (b ( 7 ( 6 : we can go from (0, 0 to (, 5 in ( 7 ways, and from (, 5 to (, 9 in ( 13 7 ways. (c ( 1 3 : we must go from (0, 0 to (3, 9 and then go right. 15! (d 9!5! + 15! : we have either 5 steps right, 1 step left, and 9 steps up: 10!! 15! 9!5!, or steps right, 10 steps up, and 1 one step down: 15! 10!!. 3. (3 pts How many ways are there to colour the vertices of a path of length 7, (so having 8 vertices, so that three vertices are red, two are blue, and three are yellow?
2 ( 8 ( (8 pts (a How many integer solutions are there to the following x 1 + x + x 3 + x = 8, x i 0, i = 1,...,. Arrange three dividers and 8 dots: ( 8+3 ( 3 = (b How many ways are there to distribute 0 indentical dimes among children, if the youngest must get at least one dime, the second youngest must get at least two dimes, the second oldest must get at least 3 dimes, and the oldest must get at least dimes. This leaves 10 dimes to distrubute among the children with no restrictions, so can be done in ( 13 3 ways. (c For which positive integer n 19 will the equations x 1 + x + + x 19 = n, and x 1 + x + + x 6 = n, have the same number of positive integer solutions. (Note that this question is a lot harder if you allow solutions with zero. (A bit hard. The two equations have ( (n ( 18 = n 1 18 and ( (n 6+63 ( 63 = n 1 63 solutions respectively. These are equal when = n 1. That is, when n = 8. (d How many nonnegative integer solutions are there to the pair of equations x 1 + x + + x 7 = 37 and x 1 + x + x = 6?
3 There are ( 6+ solutions to the second equation, and once one is chosen the first equations reduces to x 3 + x 5 + x 6 + x 7 = 31, which has ( solutions. So ( 8 ( (3 pts Determine if (p q p is a tautology. The truth assignments p = 1 and q = 1 yield a false for the statements, so it isn t a tautology. (You could find this with a truth table. 6. (3 pts Define the connectives by (p q (p q, and by (p q (p q. Show that (p q ( p q. (p q ( (p qdefn ( p qdemorgan (p qdefn 7. (6 pts (a If A = {1,, 3,..., 10}, what is P(A? 10 (b With A as above, what is {B (B P(A ( B = }?
4 This is the number of -element subsets of a 10-element set, so is ( 10. (c Give an example of three sets X, Y, and Z, such that X Y and Y Z, but X Z, or show that three such sets cannot exist. Such sets cannot exist. We had a theorem in class that said that if X Y and Y Z then X Z. Three sets meeting the conditions above would be a counterexample to this theorem. (d Give an example of three sets X, Y, and Z, such that X Y and Y Z, but X Z, or show that three such sets cannot exist. For any X, X, Y = {X}, and Z = {{X}} are sets meeting these conditions, because X {X} = Y and Y = {X} {{X}} = Z, but X {{X}} = Z. 8. (6 pts Let A, B, C U. Recall that A C = A C. Prove or disprove the following two identities. (a (A C C = A C. (b [(A C = B C (A C = B C] A = B.
5 (i A C C = = (A C Cdefn = (A C (C Cdistribution = (A C Inverse = (A CIdentity = A Cdefn (ii Assume that (A C = B C and (A C = B C. Then A = A (U identity = A (C Cinverse = (A C (A Cdistribution = (A C (A Cdefn = (A C (A C Cpart (i = (B C (B C C assumption = B All previous steps in reverse 9. (5 pts Two integers are selected at random from {1,,..., 60} (without replacement. (a What is the probability that they are consecutive? (b What is the probability that their sum is odd? (a There are ( 60 possible choices of two numbers, and there are 59 pairs (i, i + 1 of consecutive numbers. So the probability is 59/ ( 60 = /60. (b To have an odd sum, we need an even and an odd number. There are 30 ways to choose these out of the ( 60 possible ways to choose two numbers. So the probability is 30 / ( 60 = 30/ (3 pts If the probability that a randomly chosen 6 element subset of [n] = {1,,..., n} contains the element n is 0., what is n.
6 The universe of choices contains ( n 6 outcomes and the event of chosing an n contains ( n 1 5 outcomes- five other elements will be chosen from the remaining n 1. So the probability of this event is ( n 1 5 (n 1! (n 6!6! 0. = ( n = = 6 (n 1 5!5! n! n. 6 Solving 1/5 = 6/n, we get n = (5 pts Show that for n 1, n i=1 i 3 = n (n + 1 We do this by induction on n. We take n = 1 for the base case. For this, observe that 1 i=1 i1 = 1 3 = 1 = 1 (. The identity holds for n = 1. Now assuming that it holds for n = α 1 we show that it holds for n = α. Indeed, α i 3 = i=1 α 1 i 3 + α 3 i=1 = (α 1 (α = (α 1 (α + α 3 ( = (α α + (α 1 ( = (α α α + 1 = (α (α 1 + α 3 by the induction hypoth. as needed.
MATH 61-02: PRACTICE PROBLEMS FOR FINAL EXAM
MATH 61-02: PRACTICE PROBLEMS FOR FINAL EXAM (FP1) The exclusive or operation, denoted by and sometimes known as XOR, is defined so that P Q is true iff P is true or Q is true, but not both. Prove (through
More informationProofs. Joe Patten August 10, 2018
Proofs Joe Patten August 10, 2018 1 Statements and Open Sentences 1.1 Statements A statement is a declarative sentence or assertion that is either true or false. They are often labelled with a capital
More informationPreparing for the CS 173 (A) Fall 2018 Midterm 1
Preparing for the CS 173 (A) Fall 2018 Midterm 1 1 Basic information Midterm 1 is scheduled from 7:15-8:30 PM. We recommend you arrive early so that you can start exactly at 7:15. Exams will be collected
More informationSome Review Problems for Exam 3: Solutions
Math 3355 Fall 018 Some Review Problems for Exam 3: Solutions I thought I d start by reviewing some counting formulas. Counting the Complement: Given a set U (the universe for the problem), if you want
More informationMATH 13 SAMPLE FINAL EXAM SOLUTIONS
MATH 13 SAMPLE FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers.
More informationMATH 13 FINAL EXAM SOLUTIONS
MATH 13 FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers. T F
More informationPractice Exam 1 CIS/CSE 607, Spring 2009
Practice Exam 1 CIS/CSE 607, Spring 2009 Problem 1) Let R be a reflexive binary relation on a set A. Prove that R is transitive if, and only if, R = R R. Problem 2) Give an example of a transitive binary
More informationOne side of each sheet is blank and may be used as scratch paper.
Math 301 Fall 2017 (Practice) Midterm Exam 1 10/3/2017 Time Limit: 1 hour and 15 minutes Name: One side of each sheet is blank and may be used as scratch paper. Show your work clearly. Grade Table (for
More informationThe grade table is on the following page.
Discrete Mathematics 1 TrevTutor.com Final Exam Time Limit: 180 Minutes Name: Class Section This exam contains 16 pages (including this cover page) and 17 questions. The total number of points is 142.
More information1.1 Inductive Reasoning filled in.notebook August 20, 2015
1.1 Inductive Reasoning 1 Vocabulary Natural or Counting Numbers Ellipsis Scientific Method Hypothesis or Conjecture Counterexample 2 Vocabulary Natural or Counting Numbers 1, 2, 3, 4, 5... positive whole
More informationSome Review Problems for Exam 3: Solutions
Math 3355 Spring 017 Some Review Problems for Exam 3: Solutions I thought I d start by reviewing some counting formulas. Counting the Complement: Given a set U (the universe for the problem), if you want
More informationExam in Discrete Mathematics
Exam in Discrete Mathematics First Year at the Faculty of Engineering and Science and the Technical Faculty of IT and Design June 4th, 018, 9.00-1.00 This exam consists of 11 numbered pages with 14 problems.
More informationMath 205 April 28, 2009 Final Exam Review
1 Definitions 1. Some question will ask you to statea a definition and/or give an example of a defined term. The terms are: subset proper subset intersection union power set cartesian product relation
More informationMathematics 220 Midterm Practice problems from old exams Page 1 of 8
Mathematics 220 Midterm Practice problems from old exams Page 1 of 8 1. (a) Write the converse, contrapositive and negation of the following statement: For every integer n, if n is divisible by 3 then
More informationHomework 7 Solutions, Math 55
Homework 7 Solutions, Math 55 5..36. (a) Since a is a positive integer, a = a 1 + b 0 is a positive integer of the form as + bt for some integers s and t, so a S. Thus S is nonempty. (b) Since S is nonempty,
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 informationSTRATEGIES OF PROBLEM SOLVING
STRATEGIES OF PROBLEM SOLVING Second Edition Maria Nogin Department of Mathematics College of Science and Mathematics California State University, Fresno 2014 2 Chapter 1 Introduction Solving mathematical
More informationMATH 363: Discrete Mathematics
MATH 363: Discrete Mathematics Learning Objectives by topic The levels of learning for this class are classified as follows. 1. Basic Knowledge: To recall and memorize - Assess by direct questions. The
More informationMath 3000 Section 003 Intro to Abstract Math Final Exam
Math 3000 Section 003 Intro to Abstract Math Final Exam Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Name: Problem 1a-j 2 3a-b 4a-b 5a-c 6a-c 7a-b 8a-j
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 informationExam Practice Problems
Math 231 Exam Practice Problems WARNING: This is not a sample test. Problems on the exams may or may not be similar to these problems. These problems are just intended to focus your study of the topics.
More informationHW MATH425/525 Lecture Notes 1
HW MATH425/525 Lecture Notes 1 Definition 4.1 If an experiment can be repeated under the same condition, its outcome cannot be predicted with certainty, and the collection of its every possible outcome
More informationCS1800 Discrete Structures Spring 2018 February CS1800 Discrete Structures Midterm Version A
CS1800 Discrete Structures Spring 2018 February 2018 CS1800 Discrete Structures Midterm Version A Instructions: 1. The exam is closed book and closed notes. You may not use a calculator or any other electronic
More informationMATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017
MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A
More informationMath 210 Exam 2 - Practice Problems. 1. For each of the following, determine whether the statement is True or False.
Math 20 Exam 2 - Practice Problems. For each of the following, determine whether the statement is True or False. (a) {a,b,c,d} TRE (b) {a,b,c,d} FLSE (c) {a,b, } TRE (d) {a,b, } TRE (e) {a,b} {a,b} FLSE
More informationMath Fall Final Exam. Friday, 14 December Show all work for full credit. The problems are worth 6 points each.
Name: Math 50 - Fall 2007 Final Exam Friday, 4 December 2007 Show all work for full credit. The problems are worth 6 points each.. Find the number of subsets of S = {, 2,... 0} that contain exactly 5 elements,
More informationMAT 300 RECITATIONS WEEK 7 SOLUTIONS. Exercise #1. Use induction to prove that for every natural number n 4, n! > 2 n. 4! = 24 > 16 = 2 4 = 2 n
MAT 300 RECITATIONS WEEK 7 SOLUTIONS LEADING TA: HAO LIU Exercise #1. Use induction to prove that for every natural number n 4, n! > 2 n. Proof. For any n N with n 4, let P (n) be the statement n! > 2
More informationCSC 125 :: Final Exam May 3 & 5, 2010
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 6-15. Match the following logical equivalences
More information{ 0! = 1 n! = n(n 1)!, n 1. n! =
Summations Question? What is the sum of the first 100 positive integers? Counting Question? In how many ways may the first three horses in a 10 horse race finish? Multiplication Principle: If an event
More informationHomework #2 Solutions Due: September 5, for all n N n 3 = n2 (n + 1) 2 4
Do the following exercises from the text: Chapter (Section 3):, 1, 17(a)-(b), 3 Prove that 1 3 + 3 + + n 3 n (n + 1) for all n N Proof The proof is by induction on n For n N, let S(n) be the statement
More informationMath 230 Final Exam, Spring 2009
IIT Dept. Applied Mathematics, May 13, 2009 1 PRINT Last name: Signature: First name: Student ID: Math 230 Final Exam, Spring 2009 Conditions. 2 hours. No book, notes, calculator, cell phones, etc. Part
More informationa + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationProperties of the Integers
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationSec$on Summary. Definition of sets Describing Sets
Section 2.1 Sec$on Summary Definition of sets Describing Sets Roster Method Set-Builder Notation Some Important Sets in Mathematics Empty Set and Universal Set Subsets and Set Equality Cardinality of Sets
More information(1) Which of the following are propositions? If it is a proposition, determine its truth value: A propositional function, but not a proposition.
Math 231 Exam Practice Problem Solutions WARNING: This is not a sample test. Problems on the exams may or may not be similar to these problems. These problems are just intended to focus your study of the
More information1.2 Inductive Reasoning
1.2 Inductive Reasoning Goal Use inductive reasoning to make conjectures. Key Words conjecture inductive reasoning counterexample Scientists and mathematicians look for patterns and try to draw conclusions
More informationPRINCIPLE OF MATHEMATICAL INDUCTION
Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION 4.1 Overview Mathematical induction is one of the techniques which can be used to prove variety of mathematical statements which are formulated in terms of
More informationCS1800 Discrete Structures Final Version A
CS1800 Discrete Structures Fall 2017 Profs. Aslam, Gold, & Pavlu December 11, 2017 CS1800 Discrete Structures Final Version A Instructions: 1. The exam is closed book and closed notes. You may not use
More informationMATH 2001 MIDTERM EXAM 1 SOLUTION
MATH 2001 MIDTERM EXAM 1 SOLUTION FALL 2015 - MOON Do not abbreviate your answer. Write everything in full sentences. Except calculators, any electronic devices including laptops and cell phones are not
More informationWith Question/Answer Animations. Chapter 2
With Question/Answer Animations Chapter 2 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Sequences and Summations Types of
More informationMath 230 Final Exam, Spring 2008
c IIT Dept. Applied Mathematics, May 15, 2008 1 PRINT Last name: Signature: First name: Student ID: Math 230 Final Exam, Spring 2008 Conditions. 2 hours. No book, notes, calculator, cell phones, etc. Part
More informationMATH 341, Section 001 FALL 2014 Introduction to the Language and Practice of Mathematics
MATH 341, Section 001 FALL 2014 Introduction to the Language and Practice of Mathematics Class Meetings: MW 9:30-10:45 am in EMS E424A, September 3 to December 10 [Thanksgiving break November 26 30; final
More informationMath 378 Spring 2011 Assignment 4 Solutions
Math 3 Spring 2011 Assignment 4 Solutions Brualdi 6.2. The properties are P 1 : is divisible by 4. P 2 : is divisible by 6. P 3 : is divisible by. P 4 : is divisible by 10. Preparing to use inclusion-exclusion,
More informationMath 430 Exam 2, Fall 2008
Do not distribute. IIT Dept. Applied Mathematics, February 16, 2009 1 PRINT Last name: Signature: First name: Student ID: Math 430 Exam 2, Fall 2008 These theorems may be cited at any time during the test
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 informationComputer Science Foundation Exam
Computer Science Foundation Exam May 6, 2016 Section II A DISCRETE STRUCTURES NO books, notes, or calculators may be used, and you must work entirely on your own. SOLUTION Question Max Pts Category Passing
More informationMATH 114 Fall 2004 Solutions to practice problems for Final Exam
MATH 11 Fall 00 Solutions to practice problems for Final Exam Reminder: the final exam is on Monday, December 13 from 11am - 1am. Office hours: Thursday, December 9 from 1-5pm; Friday, December 10 from
More informationMATH 215 Final. M4. For all a, b in Z, a b = b a.
MATH 215 Final We will assume the existence of a set Z, whose elements are called integers, along with a well-defined binary operation + on Z (called addition), a second well-defined binary operation on
More informationMidterm Preparation Problems
Midterm Preparation Problems The following are practice problems for the Math 1200 Midterm Exam. Some of these may appear on the exam version for your section. To use them well, solve the problems, then
More informationDiscrete Structures Homework 1
Discrete Structures Homework 1 Due: June 15. Section 1.1 16 Determine whether these biconditionals are true or false. a) 2 + 2 = 4 if and only if 1 + 1 = 2 b) 1 + 1 = 2 if and only if 2 + 3 = 4 c) 1 +
More informationSolving Problems by Inductive Reasoning
Solving Problems by Inductive Reasoning MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014 Inductive Reasoning (1 of 2) If we observe several similar problems
More informationCS1800 Discrete Structures Fall 2017 October, CS1800 Discrete Structures Midterm Version A
CS1800 Discrete Structures Fall 2017 October, 2017 CS1800 Discrete Structures Midterm Version A Instructions: 1. The exam is closed book and closed notes. You may not use a calculator or any other electronic
More informationRecall that the expression x > 3 is not a proposition. Why?
Predicates and Quantifiers Predicates and Quantifiers 1 Recall that the expression x > 3 is not a proposition. Why? Notation: We will use the propositional function notation to denote the expression "
More informationMath 13, Spring 2013, Lecture B: Midterm
Math 13, Spring 2013, Lecture B: Midterm Name Signature UCI ID # E-mail address Each numbered problem is worth 12 points, for a total of 84 points. Present your work, especially proofs, as clearly as possible.
More informationCS 70 Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Midterm 1 Solutions
CS 70 Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Midterm 1 Solutions PRINT Your Name: Answer: Oski Bear SIGN Your Name: PRINT Your Student ID: CIRCLE your exam room: Dwinelle
More informationShow Your Work! Point values are in square brackets. There are 35 points possible. Tables of tautologies and contradictions are on the last page.
Formal Methods Midterm 1, Spring, 2007 Name Show Your Work! Point values are in square brackets. There are 35 points possible. Tables of tautologies and contradictions are on the last page. 1. Use truth
More informationPreparing for the CSET. Sample Book. Mathematics
Preparing for the CSET Sample Book Mathematics by Jeff Matthew Dave Zylstra Preparing for the CSET Sample Book Mathematics We at CSETMath want to thank you for interest in Preparing for the CSET - Mathematics.
More informationConjunction: p q is true if both p, q are true, and false if at least one of p, q is false. The truth table for conjunction is as follows.
Chapter 1 Logic 1.1 Introduction and Definitions Definitions. A sentence (statement, proposition) is an utterance (that is, a string of characters) which is either true (T) or false (F). A predicate is
More informationShow Your Work! Point values are in square brackets. There are 35 points possible. Some facts about sets are on the last page.
Formal Methods Name: Key Midterm 2, Spring, 2007 Show Your Work! Point values are in square brackets. There are 35 points possible. Some facts about sets are on the last page.. Determine whether each of
More informationREVIEW FOR THIRD 3200 MIDTERM
REVIEW FOR THIRD 3200 MIDTERM PETE L. CLARK 1) Show that for all integers n 2 we have 1 3 +... + (n 1) 3 < 1 n < 1 3 +... + n 3. Solution: We go by induction on n. Base Case (n = 2): We have (2 1) 3 =
More informationASSIGNMENT 1 SOLUTIONS
MATH 271 ASSIGNMENT 1 SOLUTIONS 1. (a) Let S be the statement For all integers n, if n is even then 3n 11 is odd. Is S true? Give a proof or counterexample. (b) Write out the contrapositive of statement
More informationMath 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction
Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More information1 The Basic Counting Principles
1 The Basic Counting Principles The Multiplication Rule If an operation consists of k steps and the first step can be performed in n 1 ways, the second step can be performed in n ways [regardless of how
More informationWriting Assignment 2 Student Sample Questions
Writing Assignment 2 Student Sample Questions 1. Let P and Q be statements. Then the statement (P = Q) ( P Q) is a tautology. 2. The statement If the sun rises from the west, then I ll get out of the bed.
More informationProblem 1: Suppose A, B, C and D are finite sets such that A B = C D and C = D. Prove or disprove: A = B.
Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination III (Spring 2007) Problem 1: Suppose A, B, C and D are finite sets
More informationLet us think of the situation as having a 50 sided fair die; any one number is equally likely to appear.
Probability_Homework Answers. Let the sample space consist of the integers through. {, 2, 3,, }. Consider the following events from that Sample Space. Event A: {a number is a multiple of 5 5, 0, 5,, }
More informationChapter Summary. Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (2.
Chapter 2 Chapter Summary Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (2.6) Section 2.1 Section Summary Definition of sets Describing
More informationMATH 324 Summer 2011 Elementary Number Theory. Notes on Mathematical Induction. Recall the following axiom for the set of integers.
MATH 4 Summer 011 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More informationCMPSCI 611 Advanced Algorithms Midterm Exam Fall 2015
NAME: CMPSCI 611 Advanced Algorithms Midterm Exam Fall 015 A. McGregor 1 October 015 DIRECTIONS: Do not turn over the page until you are told to do so. This is a closed book exam. No communicating with
More informationCS1800 Discrete Structures Fall 2016 Profs. Gold & Schnyder April 25, CS1800 Discrete Structures Final
CS1800 Discrete Structures Fall 2016 Profs. Gold & Schnyder April 25, 2017 CS1800 Discrete Structures Final Instructions: 1. The exam is closed book and closed notes. You may not use a calculator or any
More informationFinals: Solutions. ECS 20 (Fall 2009) Patrice Koehl May 20, 2016
Finals: Solutions ECS 20 (Fall 2009) Patrice Koehl koehl@cs.ucdavis.edu May 20, 2016 Part I: Logic Exercise 1 On a distant island, every inhabitant is either a Knight or Knave. Knights only tell the truth.
More informationDiscrete Mathematics, Spring 2004 Homework 4 Sample Solutions
Discrete Mathematics, Spring 2004 Homework 4 Sample Solutions 4.2 #77. Let s n,k denote the number of ways to seat n persons at k round tables, with at least one person at each table. (The numbers s n,k
More information{ } all possible outcomes of the procedure. There are 8 ways this procedure can happen.
Probability with the 3-Kids Procedure Statistics Procedures and Events Definition A procedure is something that produces an outcome. When a procedure produces an outcome, it s called a trial or a run of
More informationMcGill University Faculty of Science. Solutions to Practice Final Examination Math 240 Discrete Structures 1. Time: 3 hours Marked out of 60
McGill University Faculty of Science Solutions to Practice Final Examination Math 40 Discrete Structures Time: hours Marked out of 60 Question. [6] Prove that the statement (p q) (q r) (p r) is a contradiction
More informationProbability. Part 1 - Basic Counting Principles. 1. References. (1) R. Durrett, The Essentials of Probability, Duxbury.
Probability Part 1 - Basic Counting Principles 1. References (1) R. Durrett, The Essentials of Probability, Duxbury. (2) L.L. Helms, Probability Theory with Contemporary Applications, Freeman. (3) J.J.
More informationCSI30. Chapter 2. Basic Structures: Sets, Functions, Sequences, Sums. 2.1 Sets and subsets 2.2 Sets of sets
Chapter 2. Basic Structures: Sets, Functions, Sequences, Sums 2.1 Sets and subsets 2.2 Sets of sets 1 Set is an unordered collection of objects. - used to group objects together, - often the objects with
More informationCITS2211: Test One. Student Number: 1. Use a truth table to prove or disprove the following statement.
CITS2211: Test One Name: Student Number: 1. Use a truth table to prove or disprove the following statement. ((P _ Q) ^ R)) is logically equivalent to ( P ) ^ ( Q ^ R) P Q R P _ Q (P _ Q) ^ R LHS P Q R
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 informationToday s class. Constrained optimization Linear programming. Prof. Jinbo Bi CSE, UConn. Numerical Methods, Fall 2011 Lecture 12
Today s class Constrained optimization Linear programming 1 Midterm Exam 1 Count: 26 Average: 73.2 Median: 72.5 Maximum: 100.0 Minimum: 45.0 Standard Deviation: 17.13 Numerical Methods Fall 2011 2 Optimization
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Express the set using the roster method. 1) {x x N and x is greater than 7} 1) A) {8,9,10,...}
More informationDepartment of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007)
Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007) Problem 1: Specify two different predicates P (x) and
More informationFINAL EXAM PRACTICE PROBLEMS CMSC 451 (Spring 2016)
FINAL EXAM PRACTICE PROBLEMS CMSC 451 (Spring 2016) The final exam will be on Thursday, May 12, from 8:00 10:00 am, at our regular class location (CSI 2117). It will be closed-book and closed-notes, except
More informationLogic. Facts (with proofs) CHAPTER 1. Definitions
CHAPTER 1 Logic Definitions D1. Statements (propositions), compound statements. D2. Truth values for compound statements p q, p q, p q, p q. Truth tables. D3. Converse and contrapositive. D4. Tautologies
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 informationDo not open this exam until you are told to begin. You will have 75 minutes for the exam.
Math 2603 Midterm 1 Spring 2018 Your Name Student ID # Section Do not open this exam until you are told to begin. You will have 75 minutes for the exam. Check that you have a complete exam. There are 5
More informationWhat you learned in Math 28. Rosa C. Orellana
What you learned in Math 28 Rosa C. Orellana Chapter 1 - Basic Counting Techniques Sum Principle If we have a partition of a finite set S, then the size of S is the sum of the sizes of the blocks of the
More information2. Transience and Recurrence
Virtual Laboratories > 15. Markov Chains > 1 2 3 4 5 6 7 8 9 10 11 12 2. Transience and Recurrence The study of Markov chains, particularly the limiting behavior, depends critically on the random times
More informationWrite the negation of each of the following propositions without using any form of the word not :
Write the negation of each of the following propositions without using any form of the word not : Today is Thursday Today is Monday or Tuesday or Wednesday or Friday or Saturday or Sunday 2 + 1 = 3 2+1
More informationTake the Anxiety Out of Word Problems
Take the Anxiety Out of Word Problems I find that students fear any problem that has words in it. This does not have to be the case. In this chapter, we will practice a strategy for approaching word problems
More informationIn Exercises 1 12, list the all of the elements of the given set. 2. The set of all positive integers whose square roots are less than or equal to 3
APPENDIX A EXERCISES In Exercises 1 12, list the all of the elements of the given set. 1. The set of all prime numbers less than 20 2. The set of all positive integers whose square roots are less than
More informationMath 3361-Modern Algebra Lecture 08 9/26/ Cardinality
Math 336-Modern Algebra Lecture 08 9/26/4. Cardinality I started talking about cardinality last time, and you did some stuff with it in the Homework, so let s continue. I said that two sets have the same
More informationMath 3000 Section 003 Intro to Abstract Math Midterm 1
Math 3000 Section 003 Intro to Abstract Math Midterm 1 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Name: Points: Read all problems carefully and write
More informationCSI 4105 MIDTERM SOLUTION
University of Ottawa CSI 4105 MIDTERM SOLUTION Instructor: Lucia Moura Feb 6, 2010 10:00 am Duration: 1:50 hs Closed book Last name: First name: Student number: There are 4 questions and 100 marks total.
More informationMATH 402 : 2017 page 1 of 6
ADMINISTRATION What? Math 40: Enumerative Combinatorics Who? Me: Professor Greg Smith You: students interested in combinatorics When and Where? lectures: slot 00 office hours: Tuesdays at :30 6:30 and
More informationCMSC Discrete Mathematics SOLUTIONS TO FIRST MIDTERM EXAM October 18, 2005 posted Nov 2, 2005
CMSC-37110 Discrete Mathematics SOLUTIONS TO FIRST MIDTERM EXAM October 18, 2005 posted Nov 2, 2005 Instructor: László Babai Ryerson 164 e-mail: laci@cs This exam contributes 20% to your course grade.
More informationMath 430 Exam 1, Fall 2006
c IIT Dept. Applied Mathematics, October 21, 2008 1 PRINT Last name: Signature: First name: Student ID: Math 430 Exam 1, Fall 2006 These theorems may be cited at any time during the test by stating By
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 informationAn Introduction to Proofs in Mathematics
An Introduction to Proofs in Mathematics The subject of mathematics is often regarded as a pinnacle in the achievement of human reasoning. The reason that mathematics is so highly regarded in the realm
More informationWhat can you prove by induction?
MEI CONFERENCE 013 What can you prove by induction? Martyn Parker M.J.Parker@keele.ac.uk Contents Contents iii 1 Splitting Coins.................................................. 1 Convex Polygons................................................
More informationPosted Thursday February 14. STUDY IN-DEPTH...the posted solutions to homeworks 1-3. Compare with your own solutions.
CIS 160 - Spring 2019 (Instructors Val Tannen, Clayton Greenberg Midterm 1 Review Posted Thursday February 14 1 Readings STUDY IN-DEPTH......the posted notes for lectures 1-8. STUDY IN-DEPTH......the posted
More information