MATH 139 W12 Review 1 Checklist 1. Exam Checklist. 1. Introduction to Predicates and Quantified Statements (chapters ).
|
|
- Dinah Cook
- 5 years ago
- Views:
Transcription
1 MATH 139 W12 Review 1 Checklist 1 Exam Checklist 1. Introduction to Predicates and Quantified Statements (chapters ). universal and existential statements truth set negations of universal and existential statements. contrapositive, converse and inverse of a universal statement. statements with multiple quantifiers forming negations of statements with multiple quantifiers. translating statements with multiple quantifiers into English and the reverse. Tarski world examples arguments with quantified statements universal instantiation. universal modus ponens. universal modus tollens. valid argument. using Venn diagrams to show validity or invalidity of an argument. converse error, inverse error. 2. Number Theory and Proofs (chapters ) prime number, composite number, rational number, irrational number. divisibility of numbers, when does one number divide another. proving an existential statement proof by example. proving a universal statement proof by exhaustion. disproof by counterexample. method of direct proof(eg. proof that the sum of two rational numbers is rational) properties of integers every integer n > 1 is divisible by a prime. every integer n > 1 can be factored into primes, n = p e 1 1 p e 2 2 p e k k. the quotient-remainder theorem: n = qd + r, 0 r < d. div and mod. what is an even number? odd number? the product of any two odd integers is odd the floor x and ceiling x functions. proofs by contradiction and contrapositive.
2 MATH 139 W12 Review 1 Checklist 2 proof that the sum of a rational and an irrational number is irrational. proof of the irrationality of 2 (and 3, 5, etc). proof that there are infinitely many primes. 3. Mathematical Induction (chapters ) Base step and inductive step in mathematical induction. proofs involving sums, eg n = n(n+1) 2. sum of a geometric series: 1 + r + r r n = rn+1 1 r 1. proofs involving divisibility, eg n 1 for all integers n 0. strong mathematical induction what is the difference in the assumption made in the inductive step? proofs involving sequences, eg. given b 1, b 2, b 3,... where b 1 = 4, b 2 = 12, and b k = b k 2 + b k 1 prove that 4 bn for all n = 1, 2, 3, Correctness of Algorithms (chapter 5.5) pre-conditions and post-conditions for an algorithm. loop invariants Guard, pre- and post-conditions for a loop. The Loop Invariant Theorem Basis property Inductive property Eventual falsity of the guard Correctness of the post-condition. Using the loop invariant theorem to show that certain simple algorithms are correct. 5. Functions (chapters ). domain, codomain, range, inverse image f 1 (y). arrow diagrams some basic functions: identity function i X, logarithmic function log a (x), Hamming distance function. One-to-one functions (injective), onto functions (surjective), bijective functions. the inverse function f 1 (x), how to find the inverse of a bijective function (eg. f(x) = 3x + 2). 6. Composition of Functions (chapter 7.3). Definition of g f. Various properties, such as f 1 f(x) = x and f f 1 (y) = y.
3 MATH 139 W12 Review 1 Checklist 3 If f : X Y and g : Y Z are both one-to-one functions, then g f is also one-to-one. If f : X Y and g : Y Z are both onto functions, then g f is also onto. If f and g are bijections, then g f is also a bijection. 7. Cardinality of Sets (chapter 7.4) How does one define sets of equal cardinality in terms of bijections? If the cardinality of a finite set is n, what does this mean in terms of bijections? Countable sets. Examples of countable infinite sets one should know: the set of integers, the set of rational numbers, the set of prime numbers. Uncountable sets eg. the set of real numbers. 8. Relations (chapter 8.1). What is a (binary) relation R for a pair of sets A and B. Arrow diagrams for a relation between two sets. Why is a function f : A B also a relation between A and B? When is a relation a function? The inverse R 1 of a relation R. The directed graph for a relation on a set. 9. Reflexivity, Symmetry, Transitivity, and Equivalence relations (chapters ) Certain properties of relations: reflexive. symmetric. transitive. How to test for reflexivity, symmetry, and transitivity by looking at the directed graph of a relation on a set. The transitive closure R t of a relation R. Equivalence relations relations which are reflexive, symmetric, and transitive. equivalence classes. certain basic equivalence relations, for example, the equivalence relation R defined on the set of integers where xry if x y mod d (where d is some fixed integer). How to find equivalence classes by looking at the directed graph of an equivalence relation on a set. 10. Graphs (chapter 10.1). vertex, edge, endpoints of an edge.
4 MATH 139 W12 Review 1 Checklist 4 loops, parallel edges, adjacent vertices, adjacent edges, isolated vertices. simple graph, complete graph, bipartite graph. subgraph of a graph. degree of a vertex, deg(v). total degree of a graph. Handshake Theorem total degree = twice the number of edges. Some basic corollaries eg. (i) total degree of a graph is always even (ii) number of odd vertices is even. 11. Trails, Paths and Circuits (chapter 10.2). walk, closed walk, path, simple path, circuit, simple circuit. connected graph. components of a graph, bridge of a graph. Euler circuit. Eulers Theorem a graph has an Euler circuit if and only if it is connected and all its vertices have even degree. Fleury s Algorithm for finding an Euler circuit. Hamilton circuit. 12. Matrix Representation Graphs (chapter 10.3) Adjacency matrix of a graph. Reconstructing a graph with help of its adjacency matrix. Counting walks using A 2, A 3, etc. 13. Trees (chapter ). What is a tree? Trivial tree. leaf (or terminal vertex) of a tree. internal (or branch) vertex of a tree. Theorem Every tree has at least 2 vertices of degree one. Theorem If a tree has n vertices, then it has n 1 edges. Rooted tree, root vertex, level of a vertex, height of a rooted tree. binary tree, full binary tree. Theorem If T is a full binary tree with k internal vertices, then T has k + 1 leaves. Theorem If T is a full binary tree with height h, then T has at most 2 h leaves. 14. Spanning Trees (chapter 10.7).
5 MATH 139 W12 Review 1 Checklist 5 Spanning tree of a graph. Weighted graph Minimum weight spanning tree of a weighted graph. Kruskal s Algorithm for finding a min. weight spanning tree. 15. Languages, Regular Expressions, and Automata (chapters ). Languages alphabet Σ, string over Σ. null string ϵ. language over Σ. operations on strings concatenation of two strings. concatenation of two languages. union of two languages. the Kleene closure L of a language L. regular expression over an alphabet Σ. How does a regular expression represent a language? Given a language, find a regular expression representing it. Finite state automata input alphabet states of an automaton, initial state S 0, accepting states. next-state function, next-state table. transition diagram when does an automaton accept an input string? language accepted by an automaton. Given an automaton, find the language accepted by it. Given a language, find an automaton which accepts it. Constructing an automaton which accepts the language represented by a regular expression. Kleene s theorem. 16. Real-Valued Functions (chapters ) real-valued function, graph of a real-valued function. power functions: x, x 2, x 3, x 4,.... graphs of basic power functions. multiple M f of a function f. increasing function, decreasing function. Big O, Ω notation.
6 MATH 139 W12 Review 1 Checklist 6 f(x) is order at most g(x): ie. f(x) is O(g(x)). g(x) is order at least f(x): ie. g(x) is Ω(f(x)). Given functions f(x) and g(x), show that f(x) is O(g(x)), or show that f(x) is Ω(g(x)).
DISCRETE MATHEMATICS
DISCRETE MATHEMATICS WITH APPLICATIONS THIRD EDITION SUSANNA S. EPP DePaul University THOIVISON * BROOKS/COLE Australia Canada Mexico Singapore Spain United Kingdom United States CONTENTS Chapter 1 The
More informationSummary of Course Coverage
CS-227, Discrete Structures I Spring 2006 Semester Summary of Course Coverage 1) Propositional Calculus a) Negation (logical NOT) b) Conjunction (logical AND) c) Disjunction (logical inclusive-or) d) Inequalities
More informationr=1 The Binomial Theorem. 4 MA095/98G Revision
Revision Read through the whole course once Make summary sheets of important definitions and results, you can use the following pages as a start and fill in more yourself Do all assignments again Do the
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Final exam The final exam is Saturday March 18 8am-11am. Lecture A will take the exam in GH 242 Lecture B will take the exam
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Final exam The final exam is Saturday December 16 11:30am-2:30pm. Lecture A will take the exam in Lecture B will take the exam
More informationMATHEMATICAL STRUCTURES FOR COMPUTER SCIENCE
MATHEMATICAL STRUCTURES FOR COMPUTER SCIENCE A Modern Approach to Discrete Mathematics SIXTH EDITION Judith L. Gersting University of Hawaii at Hilo W. H. Freeman and Company New York Preface Note to the
More informationDiscrete Mathematics SECOND EDITION OXFORD UNIVERSITY PRESS. Norman L. Biggs. Professor of Mathematics London School of Economics University of London
Discrete Mathematics SECOND EDITION Norman L. Biggs Professor of Mathematics London School of Economics University of London OXFORD UNIVERSITY PRESS Contents PART I FOUNDATIONS Statements and proofs. 1
More informationMath 776 Graph Theory Lecture Note 1 Basic concepts
Math 776 Graph Theory Lecture Note 1 Basic concepts Lectured by Lincoln Lu Transcribed by Lincoln Lu Graph theory was founded by the great Swiss mathematician Leonhard Euler (1707-178) after he solved
More informationNotation Index. Probability notation. (there exists) (such that) Fn-4 B n (Bell numbers) CL-27 s t (equivalence relation) GT-5.
Notation Index (there exists) (for all) Fn-4 Fn-4 (such that) Fn-4 B n (Bell numbers) CL-27 s t (equivalence relation) GT-5 ( n ) k (binomial coefficient) CL-15 ( n m 1,m 2,...) (multinomial coefficient)
More informationDiscrete Mathematics Lecture 4. Harper Langston New York University
Discrete Mathematics Lecture 4 Harper Langston New York University Sequences Sequence is a set of (usually infinite number of) ordered elements: a 1, a 2,, a n, Each individual element a k is called a
More informationAbout the Author. Dependency Chart. Chapter 1: Logic and Sets 1. Chapter 2: Relations and Functions, Boolean Algebra, and Circuit Design
Preface About the Author Dependency Chart xiii xix xxi Chapter 1: Logic and Sets 1 1.1: Logical Operators: Statements and Truth Values, Negations, Conjunctions, and Disjunctions, Truth Tables, Conditional
More informationSlides for Faculty Oxford University Press All rights reserved.
Oxford University Press 2013 Slides for Faculty Assistance Preliminaries Author: Vivek Kulkarni vivek_kulkarni@yahoo.com Outline Following topics are covered in the slides: Basic concepts, namely, symbols,
More informationIntroduction to Sets and Logic (MATH 1190)
Introduction to Sets and Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University Dec 4, 2014 Outline 1 2 3 4 Definition A relation R from a set A to a set
More information2. Functions, sets, countability and uncountability. Let A, B be sets (often, in this module, subsets of R).
2. Functions, sets, countability and uncountability I. Functions Let A, B be sets (often, in this module, subsets of R). A function f : A B is some rule that assigns to each element of A a unique element
More informationAssignment 4 Solutions of graph problems
Assignment 4 Solutions of graph problems 1. Let us assume that G is not a cycle. Consider the maximal path in the graph. Let the end points of the path be denoted as v 1, v k respectively. If either of
More informationMATH 363 Final Wednesday, April 28. Final exam. You may use lemmas and theorems that were proven in class and on assignments unless stated otherwise.
Final exam This is a closed book exam. No calculators are allowed. Unless stated otherwise, justify all your steps. You may use lemmas and theorems that were proven in class and on assignments unless stated
More informationGraph Theory S 1 I 2 I 1 S 2 I 1 I 2
Graph Theory S I I S S I I S Graphs Definition A graph G is a pair consisting of a vertex set V (G), and an edge set E(G) ( ) V (G). x and y are the endpoints of edge e = {x, y}. They are called adjacent
More informationAbout the Tutorial. Audience. Prerequisites. Copyright & Disclaimer. Discrete Mathematics
About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics
More informationMath 778S Spectral Graph Theory Handout #2: Basic graph theory
Math 778S Spectral Graph Theory Handout #: Basic graph theory Graph theory was founded by the great Swiss mathematician Leonhard Euler (1707-178) after he solved the Königsberg Bridge problem: Is it possible
More informationNotation Index 9 (there exists) Fn-4 8 (for all) Fn-4 3 (such that) Fn-4 B n (Bell numbers) CL-25 s ο t (equivalence relation) GT-4 n k (binomial coef
Notation 9 (there exists) Fn-4 8 (for all) Fn-4 3 (such that) Fn-4 B n (Bell numbers) CL-25 s ο t (equivalence relation) GT-4 n k (binomial coefficient) CL-14 (multinomial coefficient) CL-18 n m 1 ;m 2
More informationLet A(x) be x is an element of A, and B(x) be x is an element of B.
Homework 6. CSE 240, Fall, 2014 Due, Tuesday October 28. Can turn in at the beginning of class, or earlier in the mailbox labelled Pless in Bryan Hall, room 509c. Practice Problems: 1. Given two arbitrary
More informationExample: NFA to DFA Conversion
CPSC 121 Lecture 36 April 8, 2009 Menu April 8, 2009 Topics: Example: NFA to DFA Conversion Final Reading List Summary The End! Reminders: On-line Quiz 12 deadline 5:00pm TODAY Teaching evaluation survey
More informationGraph Theory CS/Math231 Discrete Mathematics Spring2015
1 Graphs Definition 1 A directed graph (or digraph) G is a pair (V, E), where V is a finite set and E is a binary relation on V. The set V is called the vertex set of G, and its elements are called vertices
More informationFunctions 2/1/2017. Exercises. Exercises. Exercises. and the following mathematical appetizer is about. Functions. Functions
Exercises Question 1: Given a set A = {x, y, z} and a set B = {1, 2, 3, 4}, what is the value of 2 A 2 B? Answer: 2 A 2 B = 2 A 2 B = 2 A 2 B = 8 16 = 128 Exercises Question 2: Is it true for all sets
More informationAdjacent: Two distinct vertices u, v are adjacent if there is an edge with ends u, v. In this case we let uv denote such an edge.
1 Graph Basics What is a graph? Graph: a graph G consists of a set of vertices, denoted V (G), a set of edges, denoted E(G), and a relation called incidence so that each edge is incident with either one
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/ SECTION 5.5 Application: Correctness of Algorithms Copyright Cengage Learning. All
More informationMath 170- Graph Theory Notes
1 Math 170- Graph Theory Notes Michael Levet December 3, 2018 Notation: Let n be a positive integer. Denote [n] to be the set {1, 2,..., n}. So for example, [3] = {1, 2, 3}. To quote Bud Brown, Graph theory
More informationPractice Final. Read all the problems first before start working on any of them, so you can manage your time wisely
PRINT your name here: Practice Final Print your name immediately on the cover page, as well as each page of the exam, in the space provided. Each time you are caught working on a page without your name
More informationDiscrete mathematics , Fall Instructor: prof. János Pach
Discrete mathematics 2016-2017, Fall Instructor: prof. János Pach - covered material - Lecture 1. Counting problems To read: [Lov]: 1.2. Sets, 1.3. Number of subsets, 1.5. Sequences, 1.6. Permutations,
More informationIntroduction III. Graphs. Motivations I. Introduction IV
Introduction I Graphs Computer Science & Engineering 235: Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu Graph theory was introduced in the 18th century by Leonhard Euler via the Königsberg
More informationCS 3512, Spring Instructor: Doug Dunham. Textbook: James L. Hein, Discrete Structures, Logic, and Computability, 3rd Ed. Jones and Barlett, 2010
CS 3512, Spring 2011 Instructor: Doug Dunham Textbook: James L. Hein, Discrete Structures, Logic, and Computability, 3rd Ed. Jones and Barlett, 2010 Prerequisites: Calc I, CS2511 Rough course outline:
More informationLOGIC AND DISCRETE MATHEMATICS
LOGIC AND DISCRETE MATHEMATICS A Computer Science Perspective WINFRIED KARL GRASSMANN Department of Computer Science University of Saskatchewan JEAN-PAUL TREMBLAY Department of Computer Science University
More informationSTUDENT NUMBER: MATH Final Exam. Lakehead University. April 13, Dr. Adam Van Tuyl
Page 1 of 13 NAME: STUDENT NUMBER: MATH 1281 - Final Exam Lakehead University April 13, 2011 Dr. Adam Van Tuyl Instructions: Answer all questions in the space provided. If you need more room, answer on
More informationCS3102 Theory of Computation Solutions to Problem Set 1, Spring 2012 Department of Computer Science, University of Virginia
CS3102 Theory of Computation Solutions to Problem Set 1, Spring 2012 Department of Computer Science, University of Virginia Gabriel Robins Please start solving these problems immediately, and work in study
More informationCHAPTER 2. Graphs. 1. Introduction to Graphs and Graph Isomorphism
CHAPTER 2 Graphs 1. Introduction to Graphs and Graph Isomorphism 1.1. The Graph Menagerie. Definition 1.1.1. A simple graph G = (V, E) consists of a set V of vertices and a set E of edges, represented
More informationA Survey of Mathematics with Applications 8 th Edition, 2009
A Correlation of A Survey of Mathematics with Applications 8 th Edition, 2009 South Carolina Discrete Mathematics Sample Course Outline including Alternate Topics and Related Objectives INTRODUCTION This
More informationDO NOT RE-DISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 104, Graph Theory Homework 2 Solutions February 7, 2013 Introduction to Graph Theory, West Section 1.2: 26, 38, 42 Section 1.3: 14, 18 Section 2.1: 26, 29, 30 DO NOT RE-DISTRIBUTE
More informationVarying Applications (examples)
Graph Theory Varying Applications (examples) Computer networks Distinguish between two chemical compounds with the same molecular formula but different structures Solve shortest path problems between cities
More information13 th Annual Johns Hopkins Math Tournament Saturday, February 19, 2011 Automata Theory EUR solutions
13 th Annual Johns Hopkins Math Tournament Saturday, February 19, 011 Automata Theory EUR solutions Problem 1 (5 points). Prove that any surjective map between finite sets of the same cardinality is a
More informationGraphs and Isomorphisms
Graphs and Isomorphisms Discrete Structures (CS 173) Backyards of Old Houses in Antwerp in the Snow Van Gogh Madhusudan Parthasarathy, University of Illinois Proof techniques: Direct Contrapositive Disproving
More informationIntroduction to Automata Theory. BİL405 - Automata Theory and Formal Languages 1
Introduction to Automata Theory BİL405 - Automata Theory and Formal Languages 1 Automata, Computability and Complexity Automata, Computability and Complexity are linked by the question: What are the fundamental
More informationDS UNIT 4. Matoshri College of Engineering and Research Center Nasik Department of Computer Engineering Discrete Structutre UNIT - IV
Sr.No. Question Option A Option B Option C Option D 1 2 3 4 5 6 Class : S.E.Comp Which one of the following is the example of non linear data structure Let A be an adjacency matrix of a graph G. The ij
More informationLecture 1. 1 Notation
Lecture 1 (The material on mathematical logic is covered in the textbook starting with Chapter 5; however, for the first few lectures, I will be providing some required background topics and will not be
More informationγ(ɛ) (a, b) (a, d) (d, a) (a, b) (c, d) (d, d) (e, e) (e, a) (e, e) (a) Draw a picture of G.
MAD 3105 Spring 2006 Solutions for Review for Test 2 1. Define a graph G with V (G) = {a, b, c, d, e}, E(G) = {r, s, t, u, v, w, x, y, z} and γ, the function defining the edges, is given by the table ɛ
More informationDiscrete mathematics II. - Graphs
Emil Vatai April 25, 2018 Basic definitions Definition of an undirected graph Definition (Undirected graph) An undirected graph or (just) a graph is a triplet G = (ϕ, E, V ), where V is the set of vertices,
More informationInfinity and Uncountability. Countable Countably infinite. Enumeration
Infinity and Uncountability. Countable Countably infinite. Enumeration How big is the set of reals or the set of integers? Infinite! Is one bigger or smaller? Same size? Same number? Make a function f
More informationAbout the Tutorial. Audience. Prerequisites. Copyright & Disclaimer. Discrete Mathematics
About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics
More informationThe self-minor conjecture for infinite trees
The self-minor conjecture for infinite trees Julian Pott Abstract We prove Seymour s self-minor conjecture for infinite trees. 1. Introduction P. D. Seymour conjectured that every infinite graph is a proper
More information8.2 Paths and Cycles
8.2 Paths and Cycles Degree a b c d e f Definition The degree of a vertex is the number of edges incident to it. A loop contributes 2 to the degree of the vertex. (G) is the maximum degree of G. δ(g) is
More informationMaterial from Recitation 1
Material from Recitation 1 Darcey Riley Frank Ferraro January 18, 2011 1 Introduction In CSC 280 we will be formalizing computation, i.e. we will be creating precise mathematical models for describing
More informationCS 341 Homework 1 Basic Techniques
CS 341 Homework 1 Basic Techniques 1. What are these sets? Write them using braces, commas, numerals, (for infinite sets), and only. (a) ({1, 3, 5} {3, 1}) {3, 5, 7} (b) {{3}, {3, 5}, {{5, 7}, {7, 9}}}
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Copyright Cengage Learning. All rights reserved. SECTION 5.5 Application: Correctness of Algorithms Copyright Cengage Learning. All rights reserved.
More informationElements of Graph Theory
Elements of Graph Theory Quick review of Chapters 9.1 9.5, 9.7 (studied in Mt1348/2008) = all basic concepts must be known New topics we will mostly skip shortest paths (Chapter 9.6), as that was covered
More informationFunctions. Def. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A.
Functions functions 1 Def. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A. a A! b B b is assigned to a a A! b B f ( a) = b Notation: If
More information9.5 Equivalence Relations
9.5 Equivalence Relations You know from your early study of fractions that each fraction has many equivalent forms. For example, 2, 2 4, 3 6, 2, 3 6, 5 30,... are all different ways to represent the same
More informationChapter 4. Relations & Graphs. 4.1 Relations. Exercises For each of the relations specified below:
Chapter 4 Relations & Graphs 4.1 Relations Definition: Let A and B be sets. A relation from A to B is a subset of A B. When we have a relation from A to A we often call it a relation on A. When we have
More informationTHREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.
THREE LECTURES ON BASIC TOPOLOGY PHILIP FOTH 1. Basic notions. Let X be a set. To make a topological space out of X, one must specify a collection T of subsets of X, which are said to be open subsets of
More informationCombinatorics Summary Sheet for Exam 1 Material 2019
Combinatorics Summary Sheet for Exam 1 Material 2019 1 Graphs Graph An ordered three-tuple (V, E, F ) where V is a set representing the vertices, E is a set representing the edges, and F is a function
More informationFunctions. How is this definition written in symbolic logic notation?
functions 1 Functions Def. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by
More informationCS3102 Theory of Computation Solutions to Selected Problems from Set 1 Department of Computer Science, University of Virginia
CS3102 Theory of Computation Solutions to Selected Problems from Set 1 Department of Computer Science, University of Virginia Gabriel Robins Please start solving these problems immediately, don t procrastinate,
More informationDiscrete Mathematics Exam File Fall Exam #1
Discrete Mathematics Exam File Fall 2015 Exam #1 1.) Which of the following quantified predicate statements are true? Justify your answers. a.) n Z, k Z, n + k = 0 b.) n Z, k Z, n + k = 0 2.) Prove that
More informationT. Background material: Topology
MATH41071/MATH61071 Algebraic topology Autumn Semester 2017 2018 T. Background material: Topology For convenience this is an overview of basic topological ideas which will be used in the course. This material
More informationcorrelated to the Michigan High School Mathematics Content Expectations
correlated to the Michigan High School Mathematics Content Expectations McDougal Littell Algebra 1 Geometry Algebra 2 2007 correlated to the STRAND 1: QUANTITATIVE LITERACY AND LOGIC (L) STANDARD L1: REASONING
More informationMath 443/543 Graph Theory Notes
Math 443/543 Graph Theory Notes David Glickenstein September 3, 2008 1 Introduction We will begin by considering several problems which may be solved using graphs, directed graphs (digraphs), and networks.
More informationCPSC 121 Some Sample Questions for the Final Exam Tuesday, April 15, 2014, 8:30AM
CPSC 121 Some Sample Questions for the Final Exam Tuesday, April 15, 2014, 8:30AM Name: Student ID: Signature: Section (circle one): George Steve Your signature acknowledges your understanding of and agreement
More informationCharacterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4)
S-72.2420/T-79.5203 Basic Concepts 1 S-72.2420/T-79.5203 Basic Concepts 3 Characterizing Graphs (1) Characterizing Graphs (3) Characterizing a class G by a condition P means proving the equivalence G G
More informationTopic 10 Part 2 [474 marks]
Topic Part 2 [474 marks] The complete graph H has the following cost adjacency matrix Consider the travelling salesman problem for H a By first finding a minimum spanning tree on the subgraph of H formed
More information(l) Represent each of the sets A, B and C using bit strings. Then, use bit string representation and bitwise logical operations to find
Fall 2004 Ahmed Elgammal CS 205: Sample Final Exam December 6th, 2004 1. [10 points] Let A = {1, 3, 5, 7, 9}, B = {4, 5, 6, 7, 8}, C = {2, 4, 6, 8, 10}, D = {1, 2, 3} and let the universal set be U = {1,
More information1. a graph G = (V (G), E(G)) consists of a set V (G) of vertices, and a set E(G) of edges (edges are pairs of elements of V (G))
10 Graphs 10.1 Graphs and Graph Models 1. a graph G = (V (G), E(G)) consists of a set V (G) of vertices, and a set E(G) of edges (edges are pairs of elements of V (G)) 2. an edge is present, say e = {u,
More informationPoint-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS
Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS Definition 1.1. Let X be a set and T a subset of the power set P(X) of X. Then T is a topology on X if and only if all of the following
More informationMath 485, Graph Theory: Homework #3
Math 485, Graph Theory: Homework #3 Stephen G Simpson Due Monday, October 26, 2009 The assignment consists of Exercises 2129, 2135, 2137, 2218, 238, 2310, 2313, 2314, 2315 in the West textbook, plus the
More informationA B. bijection. injection. Section 2.4: Countability. a b c d e g
Section 2.4: Countability We can compare the cardinality of two sets. A = B means there is a bijection between A and B. A B means there is an injection from A to B. A < B means A B and A B Example: Let
More informationChapter 3: Paths and Cycles
Chapter 3: Paths and Cycles 5 Connectivity 1. Definitions: Walk: finite sequence of edges in which any two consecutive edges are adjacent or identical. (Initial vertex, Final vertex, length) Trail: walk
More informationLecture 1: Examples, connectedness, paths and cycles
Lecture 1: Examples, connectedness, paths and cycles Anders Johansson 2011-10-22 lör Outline The course plan Examples and applications of graphs Relations The definition of graphs as relations Connectedness,
More information4. (a) Draw the Petersen graph. (b) Use Kuratowski s teorem to prove that the Petersen graph is non-planar.
UPPSALA UNIVERSITET Matematiska institutionen Anders Johansson Graph Theory Frist, KandMa, IT 010 10 1 Problem sheet 4 Exam questions Solve a subset of, say, four questions to the problem session on friday.
More informationComparing sizes of sets
Comparing sizes of sets Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A
More informationCHAPTER 10 GRAPHS AND TREES. Alessandro Artale UniBZ - artale/z
CHAPTER 10 GRAPHS AND TREES Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/z SECTION 10.1 Graphs: Definitions and Basic Properties Copyright Cengage Learning. All rights reserved. Graphs: Definitions
More informationquanüfied Statements; valid well-formed formulas; comparison of propositional and predicate wffs. Exercises
Contents Preface xix Note to the Student xxv 1 Formal Logic 1 Chapter Objectives 1 1.1 Statements, Symbolic-Representation, and Tautologies 1 Statements and logical connectives; truth tables; well-formed
More informationCS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK
CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1 UNIT I INTRODUCTION CS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK 1. Define Graph. 2. Define Simple graph. 3. Write few problems
More informationFundamentals of Discrete Mathematical Structures
Fundamentals of Discrete Mathematical Structures THIRD EDITION K.R. Chowdhary Campus Director JIET School of Engineering and Technology for Girls Jodhpur Delhi-110092 2015 FUNDAMENTALS OF DISCRETE MATHEMATICAL
More informationCardinality of Sets MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Cardinality of Sets Fall / 15
Cardinality of Sets MAT Transition to Higher Mathematics Fall 0 MAT (Transition to Higher Math) Cardinality of Sets Fall 0 / Outline Sets with Equal Cardinality Countable and Uncountable Sets MAT (Transition
More informationChapter 11: Graphs and Trees. March 23, 2008
Chapter 11: Graphs and Trees March 23, 2008 Outline 1 11.1 Graphs: An Introduction 2 11.2 Paths and Circuits 3 11.3 Matrix Representations of Graphs 4 11.5 Trees Graphs: Basic Definitions Informally, a
More informationLecture 5: Graphs. Rajat Mittal. IIT Kanpur
Lecture : Graphs Rajat Mittal IIT Kanpur Combinatorial graphs provide a natural way to model connections between different objects. They are very useful in depicting communication networks, social networks
More informationTHEORY OF COMPUTATION
THEORY OF COMPUTATION UNIT-1 INTRODUCTION Overview This chapter begins with an overview of those areas in the theory of computation that are basic foundation of learning TOC. This unit covers the introduction
More informationCardinality of Sets. Washington University Math Circle 10/30/2016
Cardinality of Sets Washington University Math Circle 0/0/06 The cardinality of a finite set A is just the number of elements of A, denoted by A. For example, A = {a, b, c, d}, B = {n Z : n } = {,,, 0,,,
More informationIndicate the option which most accurately completes the sentence.
Discrete Structures, CSCI 246, Fall 2015 Final, Dec. 10 Indicate the option which most accurately completes the sentence. 1. Say that Discrete(x) means that x is a discrete structures exam and Well(x)
More informationTo illustrate what is intended the following are three write ups by students. Diagonalization
General guidelines: You may work with other people, as long as you write up your solution in your own words and understand everything you turn in. Make sure to justify your answers they should be clear
More informationFoundations of Computer Science Spring Mathematical Preliminaries
Foundations of Computer Science Spring 2017 Equivalence Relation, Recursive Definition, and Mathematical Induction Mathematical Preliminaries Mohammad Ashiqur Rahman Department of Computer Science College
More informationGraphs and trees come up everywhere. We can view the internet as a graph (in many ways) Web search views web pages as a graph
Graphs and Trees Graphs and trees come up everywhere. We can view the internet as a graph (in many ways) who is connected to whom Web search views web pages as a graph Who points to whom Niche graphs (Ecology):
More informationELEMENTARY NUMBER THEORY AND METHODS OF PROOF
CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright Cengage Learning. All rights reserved. SECTION 4.3 Direct Proof and Counterexample III: Divisibility Copyright Cengage Learning. All rights
More informationA Random Walk through CS70
A Random Walk through CS70 CS70 Summer 2016 - Lecture 8B David Dinh 09 August 2016 UC Berkeley 1 Today (and tomorrow, and Wednesday) Review: what have we done in class? Future classes: where do you go
More informationMath 443/543 Graph Theory Notes
Math 443/543 Graph Theory Notes David Glickenstein September 8, 2014 1 Introduction We will begin by considering several problems which may be solved using graphs, directed graphs (digraphs), and networks.
More informationELEMENTARY NUMBER THEORY AND METHODS OF PROOF
CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright Cengage Learning. All rights reserved. SECTION 4.3 Direct Proof and Counterexample III: Divisibility Copyright Cengage Learning. All rights
More informationDefinition For vertices u, v V (G), the distance from u to v, denoted d(u, v), in G is the length of a shortest u, v-path. 1
Graph fundamentals Bipartite graph characterization Lemma. If a graph contains an odd closed walk, then it contains an odd cycle. Proof strategy: Consider a shortest closed odd walk W. If W is not a cycle,
More informationArtificial Intelligence
Artificial Intelligence Graph theory G. Guérard Department of Nouvelles Energies Ecole Supérieur d Ingénieurs Léonard de Vinci Lecture 1 GG A.I. 1/37 Outline 1 Graph theory Undirected and directed graphs
More informationGraphs (MTAT , 6 EAP) Lectures: Mon 14-16, hall 404 Exercises: Wed 14-16, hall 402
Graphs (MTAT.05.080, 6 EAP) Lectures: Mon 14-16, hall 404 Exercises: Wed 14-16, hall 402 homepage: http://courses.cs.ut.ee/2012/graafid (contains slides) For grade: Homework + three tests (during or after
More informationCSL105: Discrete Mathematical Structures. Ragesh Jaiswal, CSE, IIT Delhi
is another way of showing that an argument is correct. Definitions: Literal: A variable or a negation of a variable is called a literal. Sum and Product: A disjunction of literals is called a sum and a
More information14 More Graphs: Euler Tours and Hamilton Cycles
14 More Graphs: Euler Tours and Hamilton Cycles 14.1 Degrees The degree of a vertex is the number of edges coming out of it. The following is sometimes called the First Theorem of Graph Theory : Lemma
More informationLecture 22 Tuesday, April 10
CIS 160 - Spring 2018 (instructor Val Tannen) Lecture 22 Tuesday, April 10 GRAPH THEORY Directed Graphs Directed graphs (a.k.a. digraphs) are an important mathematical modeling tool in Computer Science,
More informationv V Question: How many edges are there in a graph with 10 vertices each of degree 6?
ECS20 Handout Graphs and Trees March 4, 2015 (updated 3/9) Notion of a graph 1. A graph G = (V,E) consists of V, a nonempty set of vertices (or nodes) and E, a set of pairs of elements of V called edges.
More information