What Is A Relation? Example. is a relation from A to B.

Size: px
Start display at page:

Download "What Is A Relation? Example. is a relation from A to B."

Transcription

1 3.3 Relations

2 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 a R b.

3 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 a R b. Example Let A = {1, 2, 3} and let B = {r, s}. Then R = {(1, r), (2, s), (3, r)} is a relation from A to B.

4 Examples Example Let A and B be sets of real numbers. We define the relation R equals from A to B as a R b a = b

5 Examples Example Let A and B be sets of real numbers. We define the relation R equals from A to B as a R b a = b Example Let A = {1, 2, 3, 4}. Define the relation R (less than) as follows: a R b a < b Then R = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}

6 Examples Example Let A be the set of positive integers. Define the following relation on A a R b a b Then 4 R 12 but 5 R 7.

7 Examples Example Let A be the set of positive integers. Define the following relation on A a R b a b Then 4 R 12 but 5 R 7. Example Let A be the set of all people in the world. We define the following relation on A: a R b if and only if there is a sequence a 0, a 1,... a n of people such that a = a 0, b = a n and a i 1 knows a i for i = 1, 2,... n.

8 Symmetry Anyone remember what the symmetric property does for us?

9 Symmetry Anyone remember what the symmetric property does for us? A relation R is said to be symmetric if for (x, y) R then (y, x) R.

10 Symmetry Anyone remember what the symmetric property does for us? A relation R is said to be symmetric if for (x, y) R then (y, x) R. A relation R is said to be asymmetric if for (x, y) R then (y, x) R.

11 Symmetry Anyone remember what the symmetric property does for us? A relation R is said to be symmetric if for (x, y) R then (y, x) R. A relation R is said to be asymmetric if for (x, y) R then (y, x) R. From our prior examples, can you think of any that were symmetric? Asymmetric?

12 Transitivity Does anyone remember what the transitive property does for us?

13 Transitivity Does anyone remember what the transitive property does for us? We say that a relation R is transitive if a R b and b R c implies a R c.

14 Transitivity Does anyone remember what the transitive property does for us? We say that a relation R is transitive if a R b and b R c implies a R c. In terms of the Cartesian products, we have that the relation R is transitive if for (x, y) R and (y, z) R that (x, z) R.

15 Transitivity Does anyone remember what the transitive property does for us? We say that a relation R is transitive if a R b and b R c implies a R c. In terms of the Cartesian products, we have that the relation R is transitive if for (x, y) R and (y, z) R that (x, z) R. Any of our examples transitive?

16 The Reflexive Property And how about the reflexive property - what does this property do for us?

17 The Reflexive Property And how about the reflexive property - what does this property do for us? A relation R is said to be reflexive if (x, x) R for all x in the domain.

18 The Reflexive Property And how about the reflexive property - what does this property do for us? A relation R is said to be reflexive if (x, x) R for all x in the domain. A relation R that is asymmetric, transitive and reflexive is called a partial order.

19 The Reflexive Property And how about the reflexive property - what does this property do for us? A relation R is said to be reflexive if (x, x) R for all x in the domain. A relation R that is asymmetric, transitive and reflexive is called a partial order. A relation R that is symmetric, transitive and reflexive is called an equivalence relation.

20 Inverses The inverse of a relation R, denoted R 1 is the set {(y, x) (x, y) R}

21 Inverses The inverse of a relation R, denoted R 1 is the set {(y, x) (x, y) R} Example For the relation we defined on the set A = {1, 2, 3, 4} where a R b a < b we have a R 1 b a b

22 Visual Representations - Digraphs A digraph, or directed graph, is a graph, or set of nodes connected by edges, where the edges have a direction associated with them.

23 Visual Representations - Digraphs A digraph, or directed graph, is a graph, or set of nodes connected by edges, where the edges have a direction associated with them. We can use digraphs to give visual representations of relations.

24 Visual Representations - Digraphs Example Give the relation represented by the digraph

25 Visual Representations - Digraphs Example Give the relation represented by the digraph R = {(1, 1), (1, 2), (1, 3), (2, 3)}

26 Visual Representations - Digraphs Example Give a visual representation for the relation R on A = {1, 2, 3, 4} where R = {(1, 2), (2, 3), (2, 4), (3, 3), (4, 2)}

27 Visual Representations - Digraphs Example Give a visual representation for the relation R on A = {1, 2, 3, 4} where R = {(1, 2), (2, 3), (2, 4), (3, 3), (4, 2)}

28 Visual Representations - Digraphs Example Give a visual representation for the relation R on A = {1, 2, 3, 4} where R = {(1, 1), (1, 2), (2, 2), (2, 3), (2, 4), (3, 4), (4, 1)}

29 Visual Representations - Digraphs Example Give a visual representation for the relation R on A = {1, 2, 3, 4} where R = {(1, 1), (1, 2), (2, 2), (2, 3), (2, 4), (3, 4), (4, 1)}

Binary Relations McGraw-Hill Education

Binary Relations McGraw-Hill Education Binary Relations A binary relation R from a set A to a set B is a subset of A X B Example: Let A = {0,1,2} and B = {a,b} {(0, a), (0, b), (1,a), (2, b)} is a relation from A to B. We can also represent

More information

1. Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order).

1. 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 information

Slides for Faculty Oxford University Press All rights reserved.

Slides 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 information

CHAPTER 8. Copyright Cengage Learning. All rights reserved.

CHAPTER 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 information

9.5 Equivalence Relations

9.5 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 information

CS 441 Discrete Mathematics for CS Lecture 24. Relations IV. CS 441 Discrete mathematics for CS. Equivalence relation

CS 441 Discrete Mathematics for CS Lecture 24. Relations IV. CS 441 Discrete mathematics for CS. Equivalence relation CS 441 Discrete Mathematics for CS Lecture 24 Relations IV Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Equivalence relation Definition: A relation R on a set A is called an equivalence relation

More information

Relational Database: The Relational Data Model; Operations on Database Relations

Relational Database: The Relational Data Model; Operations on Database Relations Relational Database: The Relational Data Model; Operations on Database Relations Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin Overview

More information

Properties. Comparing and Ordering Rational Numbers Using a Number Line

Properties. Comparing and Ordering Rational Numbers Using a Number Line Chapter 5 Summary Key Terms natural numbers (counting numbers) (5.1) whole numbers (5.1) integers (5.1) closed (5.1) rational numbers (5.1) irrational number (5.2) terminating decimal (5.2) repeating decimal

More information

CHAPTER 3 FUZZY RELATION and COMPOSITION

CHAPTER 3 FUZZY RELATION and COMPOSITION CHAPTER 3 FUZZY RELATION and COMPOSITION Crisp relation! Definition (Product set) Let A and B be two non-empty sets, the prod uct set or Cartesian product A B is defined as follows, A B = {(a, b) a A,

More information

1 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 information

Algorithm. Algorithm Analysis. Algorithm. Algorithm. Analyzing Sorting Algorithms (Insertion Sort) Analyzing Algorithms 8/31/2017

Algorithm. Algorithm Analysis. Algorithm. Algorithm. Analyzing Sorting Algorithms (Insertion Sort) Analyzing Algorithms 8/31/2017 8/3/07 Analysis Introduction to Analysis Model of Analysis Mathematical Preliminaries for Analysis Set Notation Asymptotic Analysis What is an algorithm? An algorithm is any well-defined computational

More information

Chapter 4. Relations & Graphs. 4.1 Relations. Exercises For each of the relations specified below:

Chapter 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 information

CHAPTER 3 FUZZY RELATION and COMPOSITION

CHAPTER 3 FUZZY RELATION and COMPOSITION CHAPTER 3 FUZZY RELATION and COMPOSITION The concept of fuzzy set as a generalization of crisp set has been introduced in the previous chapter. Relations between elements of crisp sets can be extended

More information

Introduction to Sets and Logic (MATH 1190)

Introduction 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 information

Digraphs and Relations

Digraphs and Relations Digraphs and Relations RAKESH PRUDHVI KASTHURI 07CS1035 Professor: Niloy Ganguly Department of Computer Science and Engineering IIT Kharagpur November 6, 2008 November 6, 2008 Section 1 1 Digraphs A graph

More information

TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3.

TOPOLOGY, 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 information

A set with only one member is called a SINGLETON. A set with no members is called the EMPTY SET or 2 N

A 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 information

1. (15 points) Solve the decanting problem for containers of sizes 199 and 179; that is find integers x and y satisfying.

1. (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 information

Logic and Discrete Mathematics. Section 2.5 Equivalence relations and partitions

Logic 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 information

Section 10.1: Graphs and Graph Models. Introduction to Graphs Definition of a Graph Types of Graphs Examples of Graphs

Section 10.1: Graphs and Graph Models. Introduction to Graphs Definition of a Graph Types of Graphs Examples of Graphs Graphs Chapter 10 Section 10.1: Graphs and Graph Models Introduction to Graphs Definition of a Graph Types of Graphs Examples of Graphs a b c e d Introduction to Graphs Graphs are Discrete Structures that

More information

Important!!! First homework is due on Monday, September 26 at 8:00 am.

Important!!! First homework is due on Monday, September 26 at 8:00 am. Important!!! First homework is due on Monday, September 26 at 8:00 am. You can solve and submit the homework on line using webwork: http://webwork.dartmouth.edu/webwork2/m3cod/. If you do not have a user

More information

10/11/2018. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings

10/11/2018. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings Sometimes, relations define an order on the elements in a set. Definition: A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set

More information

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.

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. 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 information

Sets 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. 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 information

Chapter 4 Graphs and Matrices. PAD637 Week 3 Presentation Prepared by Weijia Ran & Alessandro Del Ponte

Chapter 4 Graphs and Matrices. PAD637 Week 3 Presentation Prepared by Weijia Ran & Alessandro Del Ponte Chapter 4 Graphs and Matrices PAD637 Week 3 Presentation Prepared by Weijia Ran & Alessandro Del Ponte 1 Outline Graphs: Basic Graph Theory Concepts Directed Graphs Signed Graphs & Signed Directed Graphs

More information

T. Background material: Topology

T. 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 information

Power Set of a set and Relations

Power Set of a set and Relations Power Set of a set and Relations 1 Power Set (1) Definition: The power set of a set S, denoted P(S), is the set of all subsets of S. Examples Let A={a,b,c}, P(A)={,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}}

More information

Modular Arithmetic. Marizza Bailey. December 14, 2015

Modular Arithmetic. Marizza Bailey. December 14, 2015 Modular Arithmetic Marizza Bailey December 14, 2015 Introduction to Modular Arithmetic If someone asks you what day it is 145 days from now, what would you answer? Would you count 145 days, or find a quicker

More information

Functions. How is this definition written in symbolic logic notation?

Functions. 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 information

Graphing Trig Functions - Sine & Cosine

Graphing Trig Functions - Sine & Cosine Graphing Trig Functions - Sine & Cosine Up to this point, we have learned how the trigonometric ratios have been defined in right triangles using SOHCAHTOA as a memory aid. We then used that information

More information

36 Modular Arithmetic

36 Modular Arithmetic 36 Modular Arithmetic Tom Lewis Fall Term 2010 Tom Lewis () 36 Modular Arithmetic Fall Term 2010 1 / 10 Outline 1 The set Z n 2 Addition and multiplication 3 Modular additive inverse 4 Modular multiplicative

More information

Review of Sets. Review. Philippe B. Laval. Current Semester. Kennesaw State University. Philippe B. Laval (KSU) Sets Current Semester 1 / 16

Review of Sets. Review. Philippe B. Laval. Current Semester. Kennesaw State University. Philippe B. Laval (KSU) Sets Current Semester 1 / 16 Review of Sets Review Philippe B. Laval Kennesaw State University Current Semester Philippe B. Laval (KSU) Sets Current Semester 1 / 16 Outline 1 Introduction 2 Definitions, Notations and Examples 3 Special

More information

THREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.

THREE 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 information

The Further Mathematics Support Programme

The Further Mathematics Support Programme Degree Topics in Mathematics Groups A group is a mathematical structure that satisfies certain rules, which are known as axioms. Before we look at the axioms, we will consider some terminology. Elements

More information

MATH 271 Summer 2016 Assignment 4 solutions

MATH 271 Summer 2016 Assignment 4 solutions 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

More information

Math Analysis Chapter 1 Notes: Functions and Graphs

Math Analysis Chapter 1 Notes: Functions and Graphs Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian coordinate system) Practice: Label each on the

More information

SET DEFINITION 1 elements members

SET DEFINITION 1 elements members SETS SET DEFINITION 1 Unordered collection of objects, called elements or members of the set. Said to contain its elements. We write a A to denote that a is an element of the set A. The notation a A denotes

More information

Math Analysis Chapter 1 Notes: Functions and Graphs

Math Analysis Chapter 1 Notes: Functions and Graphs Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs; Section 1- Basics of Functions and Their Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian

More information

NAME UNIT 4 ALGEBRA II. NOTES PACKET ON RADICALS, RATIONALS d COMPLEX NUMBERS

NAME UNIT 4 ALGEBRA II. NOTES PACKET ON RADICALS, RATIONALS d COMPLEX NUMBERS NAME UNIT 4 ALGEBRA II NOTES PACKET ON RADICALS, RATIONALS d COMPLEX NUMBERS Properties for Algebra II Name: PROPERTIES OF EQUALITY EXAMPLE/MEANING Reflexive a - a Any quantity is equal to itself. Symmetric

More information

Theory of Computation

Theory of Computation Theory of Computation For Computer Science & Information Technology By www.thegateacademy.com Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, Context-Free Grammar s

More information

Complexity Theory. Compiled By : Hari Prasad Pokhrel Page 1 of 20. ioenotes.edu.np

Complexity 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 information

Sets MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Sets Fall / 31

Sets 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

Solutions to Some Examination Problems MATH 300 Monday 25 April 2016

Solutions 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 information

r=1 The Binomial Theorem. 4 MA095/98G Revision

r=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 information

The word zero has had a long and interesting history so far. The word comes

The word zero has had a long and interesting history so far. The word comes Worth 1000 Words Real Numbers and Their Properties Learning Goals In this lesson, you will: Classify numbers in the real number system. Understand the properties of real numbers. Key Terms real number

More information

CSC Discrete Math I, Spring Sets

CSC 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 information

1. Relations 2. Equivalence relations 3. Modular arithmetics. ! Think of relations as directed graphs! xry means there in an edge x!

1. Relations 2. Equivalence relations 3. Modular arithmetics. ! Think of relations as directed graphs! xry means there in an edge x! 2 Today s Topics: CSE 20: Discrete Mathematics for Computer Science Prof. Miles Jones 1. Relations 2. 3. Modular arithmetics 3 4 Relations are graphs! Think of relations as directed graphs! xry means there

More information

Graph Theory: Introduction

Graph Theory: Introduction Graph Theory: Introduction Pallab Dasgupta, Professor, Dept. of Computer Sc. and Engineering, IIT Kharagpur pallab@cse.iitkgp.ernet.in Resources Copies of slides available at: http://www.facweb.iitkgp.ernet.in/~pallab

More information

CS 1200 Discrete Math Math Preliminaries. A.R. Hurson 323 CS Building, Missouri S&T

CS 1200 Discrete Math Math Preliminaries. A.R. Hurson 323 CS Building, Missouri S&T CS 1200 Discrete Math A.R. Hurson 323 CS Building, Missouri S&T hurson@mst.edu 1 Course Objective: Mathematical way of thinking in order to solve problems 2 Variable: holder. A variable is simply a place

More information

Math 443/543 Graph Theory Notes

Math 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 information

ECS 20 Lecture 9 Fall Oct 2013 Phil Rogaway

ECS 20 Lecture 9 Fall Oct 2013 Phil Rogaway ECS 20 Lecture 9 Fall 2013 24 Oct 2013 Phil Rogaway Today: o Sets of strings (languages) o Regular expressions Distinguished Lecture after class : Some Hash-Based Data Structures and Algorithms Everyone

More information

CHAPTER 7. Copyright Cengage Learning. All rights reserved.

CHAPTER 7. Copyright Cengage Learning. All rights reserved. CHAPTER 7 FUNCTIONS Copyright Cengage Learning. All rights reserved. SECTION 7.1 Functions Defined on General Sets Copyright Cengage Learning. All rights reserved. Functions Defined on General Sets We

More information

Functions and Relations

Functions and Relations s and Relations Definitions and Examples E. Wenderholm Department of Computer Science SUNY Oswego c 2016 Elaine Wenderholm All rights Reserved E. Wenderholm s and Relations Outline 1 Java Methods Method

More information

Accelerated Precalculus 1.2 (Intercepts and Symmetry) Day 1 Notes. In 1.1, we discussed using t-charts to help graph functions. e.g.

Accelerated Precalculus 1.2 (Intercepts and Symmetry) Day 1 Notes. In 1.1, we discussed using t-charts to help graph functions. e.g. Accelerated Precalculus 1.2 (Intercepts and Symmetry) Day 1 Notes In 1.1, we discussed using t-charts to help graph functions. e.g., Graph: y = x 3 What are some other strategies that can make graphing

More information

Cardinality Lectures

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 information

Introduction to Graphs

Introduction to Graphs Graphs Introduction to Graphs Graph Terminology Directed Graphs Special Graphs Graph Coloring Representing Graphs Connected Graphs Connected Component Reading (Epp s textbook) 10.1-10.3 1 Introduction

More information

Math 443/543 Graph Theory Notes

Math 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 information

Graph Theory S 1 I 2 I 1 S 2 I 1 I 2

Graph 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 information

Graphs. Introduction To Graphs: Exercises. Definitions:

Graphs. Introduction To Graphs: Exercises. Definitions: Graphs Eng.Jehad Aldahdooh Introduction To Graphs: Definitions: A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated

More information

A 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.

A 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 information

Binary Relations Part One

Binary Relations Part One Binary Relations Part One Outline for Today Binary Relations Reasoning about connections between objects. Equivalence Relations Reasoning about clusters. A Fundamental Theorem How do we know we have the

More information

The Language of Sets and Functions

The Language of Sets and Functions MAT067 University of California, Davis Winter 2007 The Language of Sets and Functions Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (January 7, 2007) 1 The Language of Sets 1.1 Definition and Notation

More information

Resource Allocation. Pradipta De

Resource Allocation. Pradipta De Resource Allocation Pradipta De pradipta.de@sunykorea.ac.kr Outline Dining Philosophers Problem Drinking Philosophers Problem Dining Philosophers Problem f(5) 5 f(1) Each philosopher goes through, Think

More information

Ying Gao. May 18th, rd Annual PRIMES Conference. Depths of Posets Ordered by Refinement. Ying Gao. Mentored by Sergei Bernstein.

Ying Gao. May 18th, rd Annual PRIMES Conference. Depths of Posets Ordered by Refinement. Ying Gao. Mentored by Sergei Bernstein. of of 3rd Annual PRIMES Conference May 18th, 2013 of Partially-ordered sets, or posets, are sets in which any two elements may be related by a binary relation. of Partially-ordered sets, or posets, are

More information

Definition: two derivations are similar if one of them precedes the other.

Definition: two derivations are similar if one of them precedes the other. Parse Trees and Ambiguity (Chapter 3, ection 3.2) Cmc 365 Theory of Computation 1. Derivations and similarity Let G be a CFG. A string w L(G) may have many derivations, corresponding to how we choose the

More information

Semantics via Syntax. f (4) = if define f (x) =2 x + 55.

Semantics via Syntax. f (4) = if define f (x) =2 x + 55. 1 Semantics via Syntax The specification of a programming language starts with its syntax. As every programmer knows, the syntax of a language comes in the shape of a variant of a BNF (Backus-Naur Form)

More information

Directed Graph and Binary Trees

Directed Graph and Binary Trees and Dr. Nahid Sultana December 19, 2012 and Degrees Paths and Directed graphs are graphs in which the edges are one-way. This type of graphs are frequently more useful in various dynamic systems such as

More information

think of the molecular atoms or ligands L

think of the molecular atoms or ligands L 2 nd year Crystal and olecular Architecture course The Octahedral Point Group As an inorganic or materials chemist the octahedral point group is one of the most important, Figure o a vast number of T or

More information

Outline Purpose Disjoint Sets Maze Generation. Disjoint Sets. Seth Long. March 24, 2010

Outline Purpose Disjoint Sets Maze Generation. Disjoint Sets. Seth Long. March 24, 2010 March 24, 2010 Equivalence Classes Some Applications Forming Equivalence Classes Equivalence Classes Some Applications Forming Equivalence Classes Equivalence Classes The class of all things equivelant

More information

Graphs and Trees. An example. Graphs. Example 2

Graphs and Trees. An example. Graphs. Example 2 Graphs and Trees An example How would you describe this network? What kind of model would you write for it? What kind of information would you expect to obtain? Relationship between some of the apoptotic

More information

MT5821 Advanced Combinatorics

MT5821 Advanced Combinatorics MT5821 Advanced Combinatorics 4 Graph colouring and symmetry There are two colourings of a 4-cycle with two colours (red and blue): one pair of opposite vertices should be red, the other pair blue. There

More information

2.3: FUNCTIONS. abs( x)

2.3: FUNCTIONS. abs( x) 2.3: FUNCTIONS Definition: Let A and B be sets. A function f is a rule that assigns to each element x A exactly one element y B, written y = f (x). A is called the domain of f and f is said to be defined

More information

DS UNIT 4. Matoshri College of Engineering and Research Center Nasik Department of Computer Engineering Discrete Structutre UNIT - IV

DS 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 information

Advanced Set Representation Methods

Advanced Set Representation Methods Advanced Set Representation Methods AVL trees. 2-3(-4) Trees. Union-Find Set ADT DSA - lecture 4 - T.U.Cluj-Napoca - M. Joldos 1 Advanced Set Representation. AVL Trees Problem with BSTs: worst case operation

More information

A is any set of ordered pairs of real numbers. This is a set of ordered pairs of real numbers, so it is a.

A is any set of ordered pairs of real numbers. This is a set of ordered pairs of real numbers, so it is a. Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 1 Chapter 3A Rectangular Coordinate System A is any set of ordered pairs of real numbers. A relation can be finite: {(-3, 1), (-3, -1), (0,

More information

Notes 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 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 information

Missouri State University REU, 2013

Missouri State University REU, 2013 G. Hinkle 1 C. Robichaux 2 3 1 Department of Mathematics Rice University 2 Department of Mathematics Louisiana State University 3 Department of Mathematics Missouri State University Missouri State University

More information

Assignment 4 Solutions of graph problems

Assignment 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 information

Math 187 Sample Test II Questions

Math 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 information

Simple graph Complete graph K 7. Non- connected graph

Simple graph Complete graph K 7. Non- connected graph A graph G consists of a pair (V; E), where V is the set of vertices and E the set of edges. We write V (G) for the vertices of G and E(G) for the edges of G. If no two edges have the same endpoints we

More information

REDUNDANCY OF MULTISET TOPOLOGICAL SPACES

REDUNDANCY OF MULTISET TOPOLOGICAL SPACES Iranian Journal of Fuzzy Systems Vol. 14, No. 4, (2017) pp. 163-168 163 REDUNDANCY OF MULTISET TOPOLOGICAL SPACES A. GHAREEB Abstract. In this paper, we show the redundancies of multiset topological spaces.

More information

11 Sets II Operations

11 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 information

Practice Final. Read all the problems first before start working on any of them, so you can manage your time wisely

Practice 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 information

WHAT YOU SHOULD LEARN

WHAT YOU SHOULD LEARN GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of

More information

Graph and Digraph Glossary

Graph and Digraph Glossary 1 of 15 31.1.2004 14:45 Graph and Digraph Glossary A B C D E F G H I-J K L M N O P-Q R S T U V W-Z Acyclic Graph A graph is acyclic if it contains no cycles. Adjacency Matrix A 0-1 square matrix whose

More information

Chapter 3: Theory of Modular Arithmetic 1. Chapter 3: Theory of Modular Arithmetic

Chapter 3: Theory of Modular Arithmetic 1. Chapter 3: Theory of Modular Arithmetic Chapter 3: Theory of Modular Arithmetic 1 Chapter 3: Theory of Modular Arithmetic SECTION A Introduction to Congruences By the end of this section you will be able to deduce properties of large positive

More information

Introduction to Graph Theory

Introduction to Graph Theory Introduction to Graph Theory Tandy Warnow January 20, 2017 Graphs Tandy Warnow Graphs A graph G = (V, E) is an object that contains a vertex set V and an edge set E. We also write V (G) to denote the vertex

More information

Lecture 1: Examples, connectedness, paths and cycles

Lecture 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 information

A metric space is a set S with a given distance (or metric) function d(x, y) which satisfies the conditions

A metric space is a set S with a given distance (or metric) function d(x, y) which satisfies the conditions 1 Distance Reading [SB], Ch. 29.4, p. 811-816 A metric space is a set S with a given distance (or metric) function d(x, y) which satisfies the conditions (a) Positive definiteness d(x, y) 0, d(x, y) =

More information

Studying Graph Connectivity

Studying Graph Connectivity Studying Graph Connectivity Freeman Yufei Huang July 1, 2002 Submitted for CISC-871 Instructor: Dr. Robin Dawes Studying Graph Connectivity Freeman Yufei Huang Submitted July 1, 2002 for CISC-871 In some

More information

Directed Graphs (II) Hwansoo Han

Directed Graphs (II) Hwansoo Han Directed Graphs (II) Hwansoo Han Traversals of Directed Graphs To solve many problems dealing with digraphs, we need to visit vertexes and arcs in a systematic way Depth-first search (DFS) A generalization

More information

Graphs 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 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 information

Disjoint Sets and the Union/Find Problem

Disjoint Sets and the Union/Find Problem Disjoint Sets and the Union/Find Problem Equivalence Relations A binary relation R on a set S is a subset of the Cartesian product S S. If (a, b) R we write arb and say a relates to b. Relations can have

More information

Domination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes

Domination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes Domination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes Leonor Aquino-Ruivivar Mathematics Department, De La Salle University Leonorruivivar@dlsueduph

More information

The Real Number System

The Real Number System The Real Number System Pi is probably one of the most famous numbers in all of history. As a decimal, it goes on and on forever without repeating. Mathematicians have already calculated trillions of the

More information

Definition of Inverse Function

Definition of Inverse Function Definition of Inverse Function A function and its inverse function can be described as the "DO" and the "UNDO" functions. A function takes a starting value, performs some operation on this value, and creates

More information

(Refer Slide Time: 0:19)

(Refer Slide Time: 0:19) Theory of Computation. Professor somenath Biswas. Department of Computer Science & Engineering. Indian Institute of Technology, Kanpur. Lecture-15. Decision Problems for Regular Languages. (Refer Slide

More information

Precalculus Notes Unit 1 Day 1

Precalculus Notes Unit 1 Day 1 Precalculus Notes Unit Day Rules For Domain: When the domain is not specified, it consists of (all real numbers) for which the corresponding values in the range are also real numbers.. If is in the numerator

More information

Lecture 22 Tuesday, April 10

Lecture 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 information

Harvard School of Engineering and Applied Sciences CS 152: Programming Languages

Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 21 Tuesday, April 15, 2014 1 Static program analyses For the last few weeks, we have been considering type systems.

More information