MATH 271 Summer 2016 Assignment 4 solutions
|
|
- Ada Janis Nash
- 6 years ago
- Views:
Transcription
1 MATH 7 ummer 06 Assignment 4 solutions Problem Let A, B, and C be some sets and suppose that f : A B and g : B C are functions Prove or disprove each of the following statements (a) If f is one-to-one then g f is one-to-one olution This statement is false Consider the functions f and g defined by the following diagrams: A f B g C Then f is clearly one-to-one, but g f is not one-to-one (b) If both f and g are one-to-one then g f is one-to-one olution This statement is true Proof Assume that f is one-to-one and g is one-to-one (We will show that g f is one-to-one) Let a, a A and assume that (g f)(a ) = (g f)(a ) (We will show that a = a ) Then g ( f(a ) ) = g ( f(a ) ) which means that f(a ) = f(a ) since g is one-to-one a = a But f is also one-to-one, which means that (c) If g f is one-to-one then f is one-to-one olution This statement is true Proof Assume that g f is one-to-one (We will show that f is one-to-one) Let a, a A ans assume that f(a ) = f(a ) (We will show that a = a ) ince f(a ) = f(a ), applying g to both sides gives us g ( f(a ) ) = g ( f(a ) ) or (g f)(a ) = (g f)(a ) But g f is one-to-one, so a = a (d) If g f is one-to-one then g is one-to-one olution This statement is false Consider the functions defined by the following diagrams: A f B g C
2 Then g f is clearly one-to-one, but g is not one-to-one (e) If g f is one-to-one and f is onto then g is one-to-one olution This statement is true Proof Assume that g f is one-to-one and that f is onto (We will show that g is one-to-one) Let b, b B and assume that g(b ) = g(b ) (We will show that b = b ) ince f is onto, there exists an a A so that f(a ) = b imilarly, there exists an a A so that f(a ) = b ince g(b ) = g(b ), this gives us g ( f(a ) ) = g ( f(a ) ), which means (g f)(a ) = (g f)(a ) But g f is one-to-one, which means that a = a Hence f(a ) = f(a ) and thus b = f(a ) = f(a ) = b, so b = b Problem Consider the set = {,,, 4, 5, 6, 7, 8, 9, 0} and let F denote the set of all functions from to Define a relation R on F by: for all f, g F, f R g if and only if x so that f(x) = g(x) Let α F be the function defined by α(x) = for each x Let h F be the function defined by h(x) = x+ for each x (a) Is R reflexive? ymmetric, Transitive? Prove your answers olution The relation R is reflexive and symmetric, but not transitive Proof (that R is reflexive) Let f F be arbitrary Then f() = f(), so there is at least one x so that f(x) = f(x) Hence f R f, so R is reflexive Proof (that R is symmetric) Let f, g F be arbitrary and assume that f R g Then there is at least one x so that f(x) = g(x) Hence g(x) = f(x) for that x, which means that g R f, so R is symmetric Proof (that R is not transitive) Let f, g F be the functions defined by f(x) = {, if x =,, if x and g(x) = {, if x =,, if x Function diagrams for these two functions are the following: f g Then f R α since f() = = α() and α R g since g() = = α() But f /R g since f(x) g(x) for all x Thus R is not transitive
3 (b) Prove or disprove the statement: f F so that g F, f R g olution This statement is false Its negation is: f F, g { so that f /R g Proof (of the negation) Let f F be arbitrary Pick g to be the function defined by {, if f(x) =, g(x) =, if f(x) It is clear that g(x) f(x) for all x Indeed, let x be an element of Then f(x) = or f(x) If f(x) =, then g(x) = If f(x), then g(x) = f(x) In either case, g(x) f(x) for all x This proves the statement (c) How many functions f F are there so that f R α? Explain olution There are The reasoning is as follows Consider the set A = {f F f R α} We want to count A It is easier to count the complement A c = F A, which is the set of functions that are not related to α That is, A c = {f F f /R α} Note that A = F A c The recipe for counting the functions that are not related to α is as follows: Choose a value for f() It cannot be, otherwise f() = α() There are 9 other choices Choose a value for f() It cannot be, otherwise f() = α() There are 9 other choices 0 Choose a value for f(0) It cannot be, otherwise f(0) = α(0) There are 9 other choices Hence A c = 9 0 There are 0 0 total functions from to, so A = F A c = (d) How many functions f F are there so that f R h? Explain olution There are The reasoning is as follows Consider the set H = {f F f R h} We want to count H It is easier to count the complement H c = F H, which is the set of functions that are not related to h That is, H c = {f F f /R h} Note that H = F H c The recipe for counting the functions that are not related to h is as follows: Choose a value for f() It cannot be (since h() = ), otherwise f() = h() There are 9 other choices Choose a value for f() It cannot be (since h() = ), otherwise f() = h() There are 9 other choices Choose a value for f() It cannot be (since h() = ), otherwise f() = h() There are 9 other choices 0 Choose a value for f(0) It cannot be 6 (since h(0) = 6), otherwise f(0) = h(0) There are 9 other choices Hence H c = 9 0 There are 0 0 total functions from to, so H = F H c =
4 (e) How many functions f F are there so that f /R α or f /R h? Explain 4 olution The answer is The reasoning is as follows The functions that we want to count are the functions f F such that f /R α or f /R h Note that f /R α f / A f A c imilarly, note that f /R h f / H f H c This means that we want to count the functions f so that f A c or f H c That is, we want to count the functions f in A c H c Consider the following picture: F A c H c We want to count the union of A c and H c That is, we want to count the shaeded region of the following diagram: F A c H c The number of elements in this set is A c H c = A c + H c A c H c, where A c H c is the set that is the shaded region of the following diagram: F A c H c Note that f A c H c means that f /R α and f /R h We can count the nubmer of functions that are not related to both α and h by the following recipe: Choose a value for f() It cannot be, otherwise f() = α() =, and it cannot be (since h() = ), otherwise f() = h() There are 8 other choices Choose a value for f() It cannot be, otherwise f() = α() =, and it cannot be (since h() = ), otherwise f() = h() There are 8 other choices
5 5 Choose a value for f() It cannot be, otherwise f() = α() =, and it cannot be (since h() = ), otherwise f() = h() There are 8 other choices 0 Choose a value for f(0) It cannot be, otherwise f(0) = α(0) =, and it cannot be 6 (since h(0) = 6), otherwise f(0) = h(0) There are 8 other choices Hence A c H c = 8 0 Now A c H c = A c + H c A c H c = Problem Consider the set Z + of all positive integers Let be the relation on Z + Z + defined by for all (a, b) and (c, d) in Z + Z + (a, b) (c, d) if and only if a + b = c + d (a) Prove that is an equivalence relation on Z + Z + olution Proof We prove that is reflexive, symmetric, and transitive (Reflexive) Let (a, b) Z + Z + be an arbitrary pair of positive integers Then a + b = a + b, so (a, b) (a, b) Hence is reflexive (ymmetric) Let (a, b) Z + Z + and (c, d) Z + Z + be arbitrary pairs of positive integers Assume that (a, b) (c, d) Then a + b = c + d which means that c + d = a + b Hence (c, d) (a, b) and thus is symmetric (Transitive) Let (a, b), (c, d), (e, f) Z + Z + be arbitrary pairs of positive integers Assume that (a, b) (b, c) and (b, d) (e, f) Then a + b = c + d and c + d = e + f and thus a + b = d + f since = is transitive Hence (a, b) (e, f) and thus is transitive Thus is an equivalence relation because it is reflexive, symmetric, and transitive (b) List all elements of [(, )] and all elements of [(4, 4)] olution Note that + = 9 Then (a, b) (, ) if and only if a + b = 9 The elements of [(, )] are [(, )] = {(, 4), (, ), (5, ), (7, )} Note that = Then (a, b) (4, 4) if and only if a + b = The elements of [(4, 4)] are [(4, 4)] = {(, 5), (4, 4), (6, ), (8, ), (0, )} (c) Is there an equivalence class of that has exactly 7 elements? Explain olution Yes Consider the equivalence class of (, 7) The elements are [(, 7)] = {(, 7), (, 70), (5, 69),, (54, )} = {(54 k, k) k =,,, 7}, which has 7 elements (d) How many equivalence classes of are there that contain at most 7 elements?
6 6 olution There are 7 = 54 equivalence classes that contain at most 7 elements Note that for each n Z +, there are two exactly equivalence classes that contain exactly n elements Indeed, we see that [(, )] = {(, )} and [(, )] = {(, )} are the only equivalence classes that contain exactly element For each n, the equivalence classes and [(, n)] = {(, n), (, n ), (5, n ),, (n, )} [(, n)] = {(, n), (4, n ), (6, n ),, (n, )} are the only classes that contain exactly n elements For each n =,,, 7 there are exactly equivalence classes containig n elements o there are a total of 7 classes that have at most 7 elements
Cardinality Lectures
Cardinality Lectures Enrique Treviño March 8, 014 1 Definition of cardinality The cardinality of a set is a measure of the size of a set. When a set A is finite, its cardinality is the number of elements
More informationStandard Boolean Forms
Standard Boolean Forms In this section, we develop the idea of standard forms of Boolean expressions. In part, these forms are based on some standard Boolean simplification rules. Standard forms are either
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 informationSolutions to Some Examination Problems MATH 300 Monday 25 April 2016
Name: s to Some Examination Problems MATH 300 Monday 25 April 2016 Do each of the following. (a) Let [0, 1] denote the unit interval that is the set of all real numbers r that satisfy the contraints that
More informationCSE 215: Foundations of Computer Science Recitation Exercises Set #9 Stony Brook University. Name: ID#: Section #: Score: / 4
CSE 215: Foundations of Computer Science Recitation Exercises Set #9 Stony Brook University Name: ID#: Section #: Score: / 4 Unit 14: Set Theory: Definitions and Properties 1. Let C = {n Z n = 6r 5 for
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 informationMATH 22 MORE ABOUT FUNCTIONS. Lecture M: 10/14/2003. Form follows function. Louis Henri Sullivan
MATH 22 Lecture M: 10/14/2003 MORE ABOUT FUNCTIONS Form follows function. Louis Henri Sullivan This frightful word, function, was born under other skies than those I have loved. Le Corbusier D ora innanzi
More informationCHAPTER 8. Copyright Cengage Learning. All rights reserved.
CHAPTER 8 RELATIONS Copyright Cengage Learning. All rights reserved. SECTION 8.3 Equivalence Relations Copyright Cengage Learning. All rights reserved. The Relation Induced by a Partition 3 The Relation
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 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 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 informationAlgebra of Sets. Aditya Ghosh. April 6, 2018 It is recommended that while reading it, sit with a pen and a paper.
Algebra of Sets Aditya Ghosh April 6, 2018 It is recommended that while reading it, sit with a pen and a paper. 1 The Basics This article is only about the algebra of sets, and does not deal with the foundations
More informationCantor-Schröder-Bernstein Theorem, Part 1
Cantor-Schröder-ernstein Theorem, Part 1 Jean. Larson and Christopher C. Porter MHF 3202 December 4, 2015 Proving Equinumerousity Up to this point, the main method we have had for proving that two sets
More informationA graph is finite if its vertex set and edge set are finite. We call a graph with just one vertex trivial and all other graphs nontrivial.
2301-670 Graph theory 1.1 What is a graph? 1 st semester 2550 1 1.1. What is a graph? 1.1.2. Definition. A graph G is a triple (V(G), E(G), ψ G ) consisting of V(G) of vertices, a set E(G), disjoint from
More informationLecture 17: Continuous Functions
Lecture 17: Continuous Functions 1 Continuous Functions Let (X, T X ) and (Y, T Y ) be topological spaces. Definition 1.1 (Continuous Function). A function f : X Y is said to be continuous if the inverse
More information1. (15 points) Solve the decanting problem for containers of sizes 199 and 179; that is find integers x and y satisfying.
May 9, 2003 Show all work Name There are 260 points available on this test 1 (15 points) Solve the decanting problem for containers of sizes 199 and 179; that is find integers x and y satisfying where
More informationChapter 3. Set Theory. 3.1 What is a Set?
Chapter 3 Set Theory 3.1 What is a Set? A set is a well-defined collection of objects called elements or members of the set. Here, well-defined means accurately and unambiguously stated or described. Any
More informationIt is important that you show your work. There are 134 points available on this test.
Math 1165 Discrete Math Test April 4, 001 Your name It is important that you show your work There are 134 points available on this test 1 (10 points) Show how to tile the punctured chess boards below with
More information2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Fall
c Dr Oksana Shatalov, Fall 2014 1 2. Sets 2.1&2.2: Sets and Subsets. Combining Sets. Set Terminology and Notation DEFINITIONS: Set is well-defined collection of objects. Elements are objects or members
More informationCSC Discrete Math I, Spring Sets
CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its
More informationFinal Exam, F11PE Solutions, Topology, Autumn 2011
Final Exam, F11PE Solutions, Topology, Autumn 2011 Question 1 (i) Given a metric space (X, d), define what it means for a set to be open in the associated metric topology. Solution: A set U X is open if,
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 informationGenerell Topologi. Richard Williamson. May 6, 2013
Generell Topologi Richard Williamson May 6, 2013 1 8 Thursday 7th February 8.1 Using connectedness to distinguish between topological spaces I Proposition 8.1. Let (, O ) and (Y, O Y ) be topological spaces.
More informationLocally convex topological vector spaces
Chapter 4 Locally convex topological vector spaces 4.1 Definition by neighbourhoods Let us start this section by briefly recalling some basic properties of convex subsets of a vector space over K (where
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 information2.2 Set Operations. Introduction DEFINITION 1. EXAMPLE 1 The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, EXAMPLE 2
2.2 Set Operations 127 2.2 Set Operations Introduction Two, or more, sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of
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 informationWhat Is A Relation? Example. is a relation from A to B.
3.3 Relations What Is A Relation? Let A and B be nonempty sets. A relation R from A to B is a subset of the Cartesian product A B. If R A B and if (a, b) R, we say that a is related to b by R and we write
More information1. Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order).
Exercises Exercises 1. Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order). a) {(1, 1), (1, 2), (1, 3)} b) {(1, 2), (2, 1), (2, 2), (3,
More informationJohns Hopkins Math Tournament Proof Round: Point Set Topology
Johns Hopkins Math Tournament 2019 Proof Round: Point Set Topology February 9, 2019 Problem Points Score 1 3 2 6 3 6 4 6 5 10 6 6 7 8 8 6 9 8 10 8 11 9 12 10 13 14 Total 100 Instructions The exam is worth
More information1KOd17RMoURxjn2 CSE 20 DISCRETE MATH Fall
CSE 20 https://goo.gl/forms/1o 1KOd17RMoURxjn2 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Explain the steps in a proof by mathematical and/or structural
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 informationSAT-CNF Is N P-complete
SAT-CNF Is N P-complete Rod Howell Kansas State University November 9, 2000 The purpose of this paper is to give a detailed presentation of an N P- completeness proof using the definition of N P given
More informationAntisymmetric Relations. Definition A relation R on A is said to be antisymmetric
Antisymmetric Relations Definition A relation R on A is said to be antisymmetric if ( a, b A)(a R b b R a a = b). The picture for this is: Except For Example The relation on R: if a b and b a then a =
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 informationMATH 54 - LECTURE 10
MATH 54 - LECTURE 10 DAN CRYTSER The Universal Mapping Property First we note that each of the projection mappings π i : j X j X i is continuous when i X i is given the product topology (also if the product
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 informationSolutions to Exercises 9
Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 9 (1) There are 5 cities. The cost of building a road directly between i and j is the entry a i,j in the matrix below. An indefinite entry indicates
More informationComplexity Theory. Compiled By : Hari Prasad Pokhrel Page 1 of 20. ioenotes.edu.np
Chapter 1: Introduction Introduction Purpose of the Theory of Computation: Develop formal mathematical models of computation that reflect real-world computers. Nowadays, the Theory of Computation can be
More informationSets 1. The things in a set are called the elements of it. If x is an element of the set S, we say
Sets 1 Where does mathematics start? What are the ideas which come first, in a logical sense, and form the foundation for everything else? Can we get a very small number of basic ideas? Can we reduce it
More information1.1 - Introduction to Sets
1.1 - Introduction to Sets Math 166-502 Blake Boudreaux Department of Mathematics Texas A&M University January 18, 2018 Blake Boudreaux (Texas A&M University) 1.1 - Introduction to Sets January 18, 2018
More informationQED Q: Why is it called the triangle inequality? A: Analogue with euclidean distance in the plane: picture Defn: Minimum Distance of a code C:
Lecture 3: Lecture notes posted online each week. Recall Defn Hamming distance: for words x = x 1... x n, y = y 1... y n of the same length over the same alphabet, d(x, y) = {1 i n : x i y i } i.e., d(x,
More information1 of 7 7/15/2009 3:40 PM Virtual Laboratories > 1. Foundations > 1 2 3 4 5 6 7 8 9 1. Sets Poincaré's quote, on the title page of this chapter could not be more wrong (what was he thinking?). Set theory
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 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 informationNotes on metric spaces and topology. Math 309: Topics in geometry. Dale Rolfsen. University of British Columbia
Notes on metric spaces and topology Math 309: Topics in geometry Dale Rolfsen University of British Columbia Let X be a set; we ll generally refer to its elements as points. A distance function, or metric
More informationChapter 10 Part 1: Reduction
//06 Polynomial-Time Reduction Suppose we could solve Y in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Chapter 0 Part : Reduction Reduction. Problem X
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 information2. Metric and Topological Spaces
2 Metric and Topological Spaces Topology begins where sets are implemented with some cohesive properties enabling one to define continuity Solomon Lefschetz In order to forge a language of continuity,
More information(d) If the moon shares nothing and the sun does not share our works, then the earth is alive with creeping men.
Math 15 - Spring 17 Chapters 1 and 2 Test Solutions 1. Consider the declaratives statements, P : The moon shares nothing. Q: It is the sun that shares our works. R: The earth is alive with creeping men.
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 information1 Connected components in undirected graphs
Lecture 10 Connected components of undirected and directed graphs Scribe: Luke Johnston (2016) and Mary Wootters (2017) Date: October 25, 2017 Much of the following notes were taken from Tim Roughgarden
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 informationLogic and Discrete Mathematics. Section 2.5 Equivalence relations and partitions
Logic and Discrete Mathematics Section 2.5 Equivalence relations and partitions Slides version: January 2015 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence
More informationSets. {1, 2, 3, Calvin}.
ets 2-24-2007 Roughly speaking, a set is a collection of objects. he objects are called the members or the elements of the set. et theory is the basis for mathematics, and there are a number of axiom systems
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 information751 Problem Set I JWR. Due Sep 28, 2004
751 Problem Set I JWR Due Sep 28, 2004 Exercise 1. For any space X define an equivalence relation by x y iff here is a path γ : I X with γ(0) = x and γ(1) = y. The equivalence classes are called the path
More informationSets. Mukulika Ghosh. Fall Based on slides by Dr. Hyunyoung Lee
Sets Mukulika Ghosh Fall 2018 Based on slides by Dr. Hyunyoung Lee Sets Sets A set is an unordered collection of objects, called elements, without duplication. We write a A to denote that a is an element
More information= [ U 1 \ U 2 = B \ [ B \ B.
5. Mon, Sept. 8 At the end of class on Friday, we introduced the notion of a topology, and I asked you to think about how many possible topologies there are on a 3-element set. The answer is... 29. The
More informationLecture 15: The subspace topology, Closed sets
Lecture 15: The subspace topology, Closed sets 1 The Subspace Topology Definition 1.1. Let (X, T) be a topological space with topology T. subset of X, the collection If Y is a T Y = {Y U U T} is a topology
More informationBoolean Algebra & Digital Logic
Boolean Algebra & Digital Logic Boolean algebra was developed by the Englishman George Boole, who published the basic principles in the 1854 treatise An Investigation of the Laws of Thought on Which to
More informationTOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3.
TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 301. Definition. Let m be a positive integer, and let X be a set. An m-tuple of elements of X is a function x : {1,..., m} X. We sometimes use x i instead
More informationWe can determine this with derivatives: the graph rises where its slope is positive.
Math 1 Derivatives and Graphs Stewart. Increasing and decreasing functions. We will see how to determine the important features of a graph y = f(x) from the derivatives f (x) and f (x), summarizing our
More informationWe show that the composite function h, h(x) = g(f(x)) is a reduction h: A m C.
219 Lemma J For all languages A, B, C the following hold i. A m A, (reflexive) ii. if A m B and B m C, then A m C, (transitive) iii. if A m B and B is Turing-recognizable, then so is A, and iv. if A m
More informationCONNECTED SPACES AND HOW TO USE THEM
CONNECTED SPACES AND HOW TO USE THEM 1. How to prove X is connected Checking that a space X is NOT connected is typically easy: you just have to find two disjoint, non-empty subsets A and B in X, such
More information[Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics
400 lecture note #4 [Ch 6] Set Theory 1. Basic Concepts and Definitions 1) Basics Element: ; A is a set consisting of elements x which is in a/another set S such that P(x) is true. Empty set: notated {
More informationSets MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Sets Fall / 31
Sets MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Sets Fall 2014 1 / 31 Outline 1 Sets Introduction Cartesian Products Subsets Power Sets Union, Intersection, Difference
More information/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18
601.433/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18 22.1 Introduction We spent the last two lectures proving that for certain problems, we can
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 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 informationLecture : Topological Space
Example of Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Example of 1 2 3 Example of 4 5 6 Example of I Topological spaces and continuous
More information1. (10 points) Draw the state diagram of the DFA that recognizes the language over Σ = {0, 1}
CSE 5 Homework 2 Due: Monday October 6, 27 Instructions Upload a single file to Gradescope for each group. should be on each page of the submission. All group members names and PIDs Your assignments in
More information13 th Annual Johns Hopkins Math Tournament Saturday, February 18, 2012 Explorations Unlimited Round Automata Theory
13 th Annual Johns Hopkins Math Tournament Saturday, February 18, 2012 Explorations Unlimited Round Automata Theory 1. Introduction Automata theory is an investigation of the intersection of mathematics,
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 informationMore NP-Complete Problems [HMU06,Chp.10b] Node Cover Independent Set Knapsack Real Games
More NP-Complete Problems [HMU06,Chp.10b] Node Cover Independent Set Knapsack Real Games 1 Next Steps We can now reduce 3SAT to a large number of problems, either directly or indirectly. Each reduction
More informationMATH 139 W12 Review 1 Checklist 1. Exam Checklist. 1. Introduction to Predicates and Quantified Statements (chapters ).
MATH 139 W12 Review 1 Checklist 1 Exam Checklist 1. Introduction to Predicates and Quantified Statements (chapters 3.1-3.4). universal and existential statements truth set negations of universal and existential
More informationCS446: Machine Learning Fall Problem Set 4. Handed Out: October 17, 2013 Due: October 31 th, w T x i w
CS446: Machine Learning Fall 2013 Problem Set 4 Handed Out: October 17, 2013 Due: October 31 th, 2013 Feel free to talk to other members of the class in doing the homework. I am more concerned that you
More informationMath 187 Sample Test II Questions
Math 187 Sample Test II Questions Dr. Holmes October 2, 2008 These are sample questions of kinds which might appear on Test II. There is no guarantee that all questions on the test will look like these!
More informationA set with only one member is called a SINGLETON. A set with no members is called the EMPTY SET or 2 N
Mathematical Preliminaries Read pages 529-540 1. Set Theory 1.1 What is a set? A set is a collection of entities of any kind. It can be finite or infinite. A = {a, b, c} N = {1, 2, 3, } An entity is an
More informationMath236 Discrete Maths with Applications
Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 19 Degree Sequences Let G be a graph with vertex set V (G) = {v 1, v 2, v
More informationHomework Set #2 Math 440 Topology Topology by J. Munkres
Homework Set #2 Math 440 Topology Topology by J. Munkres Clayton J. Lungstrum October 26, 2012 Exercise 1. Prove that a topological space X is Hausdorff if and only if the diagonal = {(x, x) : x X} is
More informationTaibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103. Chapter 2. Sets
Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 2 Sets Slides are adopted from Discrete Mathematics and It's Applications Kenneth H.
More informationTopology 550A Homework 3, Week 3 (Corrections: February 22, 2012)
Topology 550A Homework 3, Week 3 (Corrections: February 22, 2012) Michael Tagare De Guzman January 31, 2012 4A. The Sorgenfrey Line The following material concerns the Sorgenfrey line, E, introduced in
More information11 Sets II Operations
11 Sets II Operations Tom Lewis Fall Term 2010 Tom Lewis () 11 Sets II Operations Fall Term 2010 1 / 12 Outline 1 Union and intersection 2 Set operations 3 The size of a union 4 Difference and symmetric
More informationModule 7. Independent sets, coverings. and matchings. Contents
Module 7 Independent sets, coverings Contents and matchings 7.1 Introduction.......................... 152 7.2 Independent sets and coverings: basic equations..... 152 7.3 Matchings in bipartite graphs................
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 informationDefinition 1 (Hand-shake model). A hand shake model is an incidence geometry for which every line has exactly two points.
Math 3181 Dr. Franz Rothe Name: All3181\3181_spr13t1.tex 1 Solution of Test I Definition 1 (Hand-shake model). A hand shake model is an incidence geometry for which every line has exactly two points. Definition
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 informationFinding a -regular Supergraph of Minimum Order
Finding a -regular Supergraph of Minimum Order Hans L. Bodlaender a, Richard B. Tan a,b and Jan van Leeuwen a a Department of Computer Science Utrecht University Padualaan 14, 3584 CH Utrecht The Netherlands
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 informationNP and computational intractability. Kleinberg and Tardos, chapter 8
NP and computational intractability Kleinberg and Tardos, chapter 8 1 Major Transition So far we have studied certain algorithmic patterns Greedy, Divide and conquer, Dynamic programming to develop efficient
More informationFINITELY GENERATED CLASSES OF SETS OF NATURAL NUMBERS
FINITELY GENERATED CLASSES OF SETS OF NATURAL NUMBERS JULIA ROBINSON We say that a set S of natural numbers is generated by a class S of functions if S is the range of a function obtained by composition
More informationLECTURE 1 Basic definitions, the intersection poset and the characteristic polynomial
R. STANLEY, HYPERPLANE ARRANGEMENTS LECTURE Basic definitions, the intersection poset and the characteristic polynomial.. Basic definitions The following notation is used throughout for certain sets of
More informationCommando: Solution. Solution 3 O(n): Consider two decisions i<j, we choose i instead of j if and only if : A S j S i
Commando: Solution Commando: Solution Solution 1 O(n 3 ): Using dynamic programming, let f(n) indicate the maximum battle effectiveness after adjustment. We have transfer equations below: n f(n) = max
More information1-3 Continuity, End Behavior, and Limits
Determine whether each function is continuous at the given x-value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. 1. f (x)
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 informationFRUCHT S THEOREM FOR THE DIGRAPH FACTORIAL
Discussiones Mathematicae Graph Theory 33 (2013) 329 336 doi:10.7151/dmgt.1661 FRUCHT S THEOREM FOR THE DIGRAPH FACTORIAL Richard H. Hammack Department of Mathematics and Applied Mathematics Virginia Commonwealth
More informationFunctions. Jason Filippou UMCP. Jason Filippou UMCP) Functions / 19
Functions Jason Filippou CMSC250 @ UMCP 06-22-2016 Jason Filippou (CMSC250 @ UMCP) Functions 06-22-2016 1 / 19 Outline 1 Basic definitions and examples 2 Properties of functions 3 The pigeonhole principle
More informationChapter 1. Preliminaries
Chapter 1 Preliminaries 1.1 Topological spaces 1.1.1 The notion of topological space The topology on a set X is usually defined by specifying its open subsets of X. However, in dealing with topological
More informationLecture 2 September 3
EE 381V: Large Scale Optimization Fall 2012 Lecture 2 September 3 Lecturer: Caramanis & Sanghavi Scribe: Hongbo Si, Qiaoyang Ye 2.1 Overview of the last Lecture The focus of the last lecture was to give
More information