Relations. Functions. Bijection and counting.
|
|
- Marsha Boone
- 5 years ago
- Views:
Transcription
1 Relations.. and counting.
2 s Given two sets A = {,, } B = {,,, 4} Their A B = {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, 4), (, 4), (, 4)} Question: What is the cartesian product of? ( is the set of all integers) p.
3 = (, ) (, 0) Origin (0, 0) All pairs of integers are in, for emaple (, ) (, ) p.
4 Is it a? p. 4
5 of p. 5
6 Is it a? p. 6
7 of = p. 7
8 Is it a? (, 0) (, 0) p. 8
9 Is it a? No (, 0) (, 0) R = (, ) ma(, ) = p. 9
10 Is it a? No (, 0) (, 0) A subset R of a Cartesian product is called a relation. p. 0
11 Relations Given an two sets A = {,,, } and B = {,,,...} Def. A subset R of the A B is called a relation from the set A to the set B. R = (, 99), (, 5), (, 0 5 ), (, ), (, 5) A B p.
12 Relations Eample of a relation: S = set of students C = set of classes R = {(s, c) student s takes class c} Alice Bob Eve Mark CS 50 CS 5 Math 54 Hist 5 Zack p.
13 Airline route map We connect two cities if the airline operates a flight from one to the other. Is it a relation? p.
14 Airline route map Routes Cities Cities p. 4
15 Airline route map p. 5
16 An subset of A B is a relation R R = (, ), (, ), (, ), (, ), (, ), (, ), (0, ), (, ), (, ) p. 6
17 An subset of A B is a relation (, ) R R = (, ) ( +) +( ) p. 7
18 A function is a relation too! R R = (, ) = f () p. 8
19 Relation {(, ) = } 0 0 p. 9
20 Relation {(, ) = rem } 0 0 p. 0
21 Def. A relation R A B is a function (a functional relation) if for ever a A, there is at most one b B so that (a, b) R. A = {,, }, B = {,,, 4} R = {(, ), (, ), (, )} R = {(, ), (, ), (, 4), (, ), (, 4), (, 4)} R = {(, ), (, ), (, 4)} Not a function p.
22 Functional relation R A B defines a unique wa to map each element from the set A to an element from the set B. There is a well-known and convenient notation for functions: f (a) = b where a A and b B It maps elements from A to B: f : A B A f B p.
23 Def. For the function f : A B, set A is called domain, and set B is called codomain. 4 f : A B z domain(f ) = A = {,, z} codomain(f ) = B = {,,, 4} image(f ) = f (A) = {, } Def. f (a) is the image of a point a A. Def. The image of a function f, denoted b f (A), is the set of all images of all points a A f (A) = { a A (f (a) = )}. The image of a function is also called range. p.
24 Onto Def. A function f : A B is called onto if and onl if for ever element b B there is an element a A with f (a) = b. In other words, the image f (A) is the whole codomain B. f : A B is onto g : A B is not onto z 4 z 4 w w v v p. 4
25 One-to-one Def. A function f : A B is said to be one-to-one if and onl if f () = f () implies that = for all, A. f : A B is one-to-one g : B C is not one-to-one z 4 4 /7 /6 w /5 0 0 p. 5
26 Def. The function f is a bijection (also called one-to-one correspondence) if and onl if it is both one-to-one and onto. f : A B is a bijection g : A B is not a bijection z 4 z 4 w w v 0 v 0 p. 6
27 f : A B is one-to-one, but not onto g : C D is onto, but not one-to-one z 4 z w w 0 v 0 So, both functions are not bijections. p. 7
28 . Observation Rule. Given two sets A and B, if there eists a bijection f : A B, then A = B. z w v 4 0 We can count the size of the set A, instead of the size of B! p. 8
29 Rule. Consider two similar problems: (a) How man bit strings contain eactl three s and two 0s? 00 (b) How man strings can be composed of three A s and five b s so that an A is alwas followed b a b? AbAbbAbb We show that this two problems are equivalent b constructing a bijection. p. 9
30 Rule. Let X be the set of bit strings X = {00,...} and Y be the set of A and b strings Y = {AbAbbAbb,...} We can construct a bijection f : X Y : gets replaced b Ab 0 gets replaced b b p. 0
31 Rule. f : 00 Ab Ab Ab b b 00 Ab Ab b Ab b 00 Ab Ab b b Ab 00 Ab b Ab Ab b 00 Ab b Ab b Ab 00 Ab b b Ab Ab 00 b Ab Ab Ab b 00 b Ab Ab b Ab 00 b Ab b Ab Ab 00 b b Ab Ab Ab Function f is one-to-one and onto, so it is a bijection. Therefore, the cardinalities of two sets are equal: X = Y = 5 = 0. p.
32 . Observation Observation For ever bijection f : A B, eists an inverse function f : B A f : A B z w v 4 0 The inverse function is a bijection too. p.
33 . Observation Observation For ever bijection f : A B, eists an inverse function f : B A f : A B f : B A z 4 4 z w w v 0 0 v The inverse function is a bijection too. p.
34 . Observation Given two bijections f : A B and g : B C. Consider their composition h() = g(f ()) f : A B g : B C z 4 4 4/5 /5 /5 w /5 h : A C is a bijection, and therefore A = C. p. 4
35 . Observation Given two bijections f : A B and g : B C. Consider their composition h() = g(f ()) f : A B g : B C h : A C z 4 4 4/5 z 4/5 /5 /5 /5 /5 w /5 w /5 h : A C is a bijection, and therefore A = C. p. 5
36 . Counting subsets Please, count the number of subsets of a set A = {a, b, c, d, e} b reducing the problem to counting bit strings. Let s find a bijection between the power set (A) = {, {a}, {b},...} and the set of bit stings of length 5: {0, } 5 = {00000, 0000, 0000, 000,...} p. 6
37 . Counting subsets Please, count the number of subsets of a set A = {a, b, c, d, e} b reducing the problem to counting bit strings. Let s find a bijection f between the power set (A) = {, {a}, {b},...} and the set of bit stings of length 5: {0, } 5 = {00000, 0000, 0000, 000,...} p. 7
38 . Counting subsets f : ({a, b, c, d, e}) {0, } 5 0s and s encode the membership of the five elements of {a, b, c, d, e} f : {a} 0000 {b} 0000 {a, b} 000 {c} 0000 {a, c} 000 {b, c} 000 {a, b, c} 00...skipping um.. subsets {a, b, c, d, e} The cardinalit {0, } 5 = 5 = Therefore, b the bijection rule, (A) = 5 p. 8
39 principle We remember the subtraction rule for the union of two sets A B = A + B A B Can it be generalized for a union of n sets Of course, it can! A... A n = A A n something? p. 9
40 principle We remember the subtraction rule for the union of two sets A B = A + B A B Can it be generalized for a union of n sets Of course, it can! A... A n = A A n something? p. 40
41 principle Union of three sets A A A = A + A + A A A A A A A + A A A {,, } {,, 4} {, 4, } = = 4 p. 4
42 principle Union of n sets A... A n = the sum of the sizes of the individual sets minus the sizes of all two-wa intersections plus the sizes of all three-wa intersections minus the sizes of all four-wa intersections plus the sizes of all five-wa intersections etc. p. 4
2.1 Sets. Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R.
2. Basic Structures 2.1 Sets Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R. Definition 2 Objects in a set are called elements or members of the set. A set is
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More information1.4 Cardinality. Tom Lewis. Fall Term Tom Lewis () 1.4 Cardinality Fall Term / 9
1.4 Cardinality Tom Lewis Fall Term 2006 Tom Lewis () 1.4 Cardinality Fall Term 2006 1 / 9 Outline 1 Functions 2 Cardinality 3 Cantor s theorem Tom Lewis () 1.4 Cardinality Fall Term 2006 2 / 9 Functions
More informationSets & Functions. x A. 7 Z, 2/3 Z, Z pow(r) Examples: {7, Albert R., π/2, T} x is a member of A: x A
Mathematics for Computer Science MIT 6.042J/18.062J Sets & Functions What is a Set? Informally: set is a collection of mathematical objects, with the collection treated as a single mathematical object.
More informationMATH 2050 Assignment 4 Fall Due: Thursday. u v v 2 v = P roj v ( u) = P roj u ( v) =
MATH 5 Assignment 4 Fall 8 Due: Thursday [5]. Let u = and v =. Find the projection of u onto v; and the projection of v onto u respectively. ANS: The projection of u onto v is P roj v ( u) = u v v. Note
More informationChapter Summary. Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability
Chapter 2 1 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability Sequences and Summations Types of Sequences Summation
More informationSets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions
Sets and Functions MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Notation x A means that element x is a member of set A. x / A means that x is not a member of A.
More informationMathematics Review for Business PhD Students
Mathematics Review for Business PhD Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationChapter Summary. Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (2.
Chapter 2 Chapter Summary Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (2.6) Section 2.1 Section Summary Definition of sets Describing
More informationSection 4.4 Functions. CS 130 Discrete Structures
Section 4.4 Functions CS 130 Discrete Structures Function Definitions Let S and T be sets. A function f from S to T, f: S T, is a subset of S x T where each member of S appears exactly once as the first
More informationDiscrete Mathematics. Kenneth A. Ribet. January 31, 2013
January 31, 2013 A non-constructive proof Two classical facts are that π and e 2.71828... are irrational. Further, it is known that neither satisfies a quadratic equation x 2 + ax + b = 0 where a and b
More informationMathematics Review for Business PhD Students Lecture Notes
Mathematics Review for Business PhD Students Lecture Notes Anthony M. Marino Department of Finance and Business Economics Marshall School of Business University of Southern California Los Angeles, CA 90089-0804
More information2. Polynomials. 19 points. 3/3/3/3/3/4 Clearly indicate your correctly formatted answer: this is what is to be graded. No need to justify!
1. Short Modular Arithmetic/RSA. 16 points: 3/3/3/3/4 For each question, please answer in the correct format. When an expression is asked for, it may simply be a number, or an expression involving variables
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Define and compute the cardinality of a set. Use functions to compare the sizes of sets. Classify sets
More informationPropositional Logic, Predicates, and Equivalence
Chapter 1 Propositional Logic, Predicates, and Equivalence A statement or a proposition is a sentence that is true (T) or false (F) but not both. The symbol denotes not, denotes and, and denotes or. If
More informationRELATIONS AND FUNCTIONS through
RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or
More informationPractice Exam 1 CIS/CSE 607, Spring 2009
Practice Exam 1 CIS/CSE 607, Spring 2009 Problem 1) Let R be a reflexive binary relation on a set A. Prove that R is transitive if, and only if, R = R R. Problem 2) Give an example of a transitive binary
More informationLESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.
LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v 4 5 6 () We might also write v as v Both notations refer to a vector (2) A vector can be man
More informationCS 455/555: Mathematical preliminaries
CS 455/555: Mathematical preliminaries Stefan D. Bruda Winter 2019 SETS AND RELATIONS Sets: Operations: intersection, union, difference, Cartesian product Big, powerset (2 A ) Partition (π 2 A, π, i j
More informationSection 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
More informationMath Fall 2014 Final Exam Solutions
Math 2001-003 Fall 2014 Final Exam Solutions Wednesday, December 17, 2014 Definition 1. The union of two sets X and Y is the set X Y consisting of all objects that are elements of X or of Y. The intersection
More information8. BOOLEAN ALGEBRAS x x
8. BOOLEAN ALGEBRAS 8.1. Definition of a Boolean Algebra There are man sstems of interest to computing scientists that have a common underling structure. It makes sense to describe such a mathematical
More informationMATH 243E Test #3 Solutions
MATH 4E Test # Solutions () Find a recurrence relation for the number of bit strings of length n that contain a pair of consecutive 0s. You do not need to solve this recurrence relation. (Hint: Consider
More informationChapter 2 - Basics Structures MATH 213. Chapter 2: Basic Structures. Dr. Eric Bancroft. Fall Dr. Eric Bancroft MATH 213 Fall / 60
MATH 213 Chapter 2: Basic Structures Dr. Eric Bancroft Fall 2013 Dr. Eric Bancroft MATH 213 Fall 2013 1 / 60 Chapter 2 - Basics Structures 2.1 - Sets 2.2 - Set Operations 2.3 - Functions 2.4 - Sequences
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.
MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection
More informationSection 7.2: One-to-One, Onto and Inverse Functions
Section 7.2: One-to-One, Onto and Inverse Functions In this section we shall developed the elementary notions of one-to-one, onto and inverse functions, similar to that developed in a basic algebra course.
More informationWhat is the limit of a function? Intuitively, we want the limit to say that as x gets closer to some value a,
Limits The notion of a limit is fundamental to the stud of calculus. It is one of the primar concepts that distinguishes calculus from mathematical subjects that ou saw prior to calculus, such as algebra
More information9/21/2018. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions
How can we prove that a function f is one-to-one? Whenever you want to prove something, first take a look at the relevant definition(s): x, y A (f(x) = f(y) x = y) f:r R f(x) = x 2 Disproof by counterexample:
More informationSets and Functions. (As we will see, in describing a set the order in which elements are listed is irrelevant).
Sets and Functions 1. The language of sets Informally, a set is any collection of objects. The objects may be mathematical objects such as numbers, functions and even sets, or letters or symbols of any
More informationCS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:
x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which
More information9/19/2018. Cartesian Product. Cartesian Product. Partitions
Cartesian Product The ordered n-tuple (a 1, a 2, a 3,, a n ) is an ordered collection of objects. Two ordered n-tuples (a 1, a 2, a 3,, a n ) and (b 1, b 2, b 3,, b n ) are equal if and only if they contain
More informationDiscrete Basic Structure: Sets
KS091201 MATEMATIKA DISKRIT (DISCRETE MATHEMATICS ) Discrete Basic Structure: Sets Discrete Math Team 2 -- KS091201 MD W-07 Outline What is a set? Set properties Specifying a set Often used sets The universal
More informationIntroduction to Automata
Introduction to Automata Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr 1 /
More informationBackground for Discrete Mathematics
Background for Discrete Mathematics Huck Bennett Northwestern University These notes give a terse summary of basic notation and definitions related to three topics in discrete mathematics: logic, sets,
More informationAlgebra/Pre-calc Review
Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge
More informationR = { } Fill-in-the-Table with the missing vocabulary terms: 1) 2) Fill-in-the-blanks: Function
Name: Date: / / QUIZ DAY! Fill-in-the-Table with the missing vocabular terms: ) ) Input Fill-in-the-blanks: 3) Output Function A special tpe of where there is one and onl one range () value for ever domain
More informationUnit 12 Study Notes 1 Systems of Equations
You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve
More informationReview of Functions. Functions. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Functions Current Semester 1 / 12
Review of Functions Functions Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Functions Current Semester 1 / 12 Introduction Students are expected to know the following concepts about functions:
More informationChapter 6: Systems of Equations and Inequalities
Chapter 6: Sstems of Equations and Inequalities 6-1: Solving Sstems b Graphing Objectives: Identif solutions of sstems of linear equation in two variables. Solve sstems of linear equation in two variables
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationMATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X.
MATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X. Notation 2 A set can be described using set-builder notation. That is, a set can be described
More informationSETS AND FUNCTIONS JOSHUA BALLEW
SETS AND FUNCTIONS JOSHUA BALLEW 1. Sets As a review, we begin by considering a naive look at set theory. For our purposes, we define a set as a collection of objects. Except for certain sets like N, Z,
More informationFUNCTIONS. Note: Example of a function may be represented diagrammatically. The above example can be written diagrammatically as follows.
FUNCTIONS Def : A relation f from a set A into a set is said to be a function or mapping from A into if for each A there eists a unique such that (, ) f. It is denoted b f : A. Note: Eample of a function
More informationAutomata and Languages
Automata and Languages Prof. Mohamed Hamada Software Engineering Lab. The University of Aizu Japan Mathematical Background Mathematical Background Sets Relations Functions Graphs Proof techniques Sets
More informationSets are one of the basic building blocks for the types of objects considered in discrete mathematics.
Section 2.1 Introduction Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. Important for counting. Programming languages have set operations. Set theory
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationVertex. March 23, Ch 9 Guided Notes.notebook
March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function
More informationMAT389 Fall 2016, Problem Set 2
MAT389 Fall 2016, Problem Set 2 Circles in the Riemann sphere Recall that the Riemann sphere is defined as the set Let P be the plane defined b Σ = { (a, b, c) R 3 a 2 + b 2 + c 2 = 1 } P = { (a, b, c)
More informationFlows and Connectivity
Chapter 4 Flows and Connectivit 4. Network Flows Capacit Networks A network N is a connected weighted loopless graph (G,w) with two specified vertices and, called the source and the sink, respectivel,
More informationChapter 2 - Basics Structures
Chapter 2 - Basics Structures 2.1 - Sets Definitions and Notation Definition 1 (Set). A set is an of. These are called the or of the set. We ll typically use uppercase letters to denote sets: S, A, B,...
More informationCS100: DISCRETE STRUCTURES
1 CS100: DISCRETE STRUCTURES Computer Science Department Lecture 2: Functions, Sequences, and Sums Ch2.3, Ch2.4 2.3 Function introduction : 2 v Function: task, subroutine, procedure, method, mapping, v
More informationToday s Topics. Methods of proof Relationships to logical equivalences. Important definitions Relationships to sets, relations Special functions
Today s Topics Set identities Methods of proof Relationships to logical equivalences Functions Important definitions Relationships to sets, relations Special functions Set identities help us manipulate
More informationConstant 2-labelling of a graph
Constant 2-labelling of a graph S. Gravier, and E. Vandomme June 18, 2012 Abstract We introduce the concept of constant 2-labelling of a graph and show how it can be used to obtain periodic sphere packing.
More informationSets. A set is a collection of objects without repeats. The size or cardinality of a set S is denoted S and is the number of elements in the set.
Sets A set is a collection of objects without repeats. The size or cardinality of a set S is denoted S and is the number of elements in the set. If A and B are sets, then the set of ordered pairs each
More information9/21/17. Functions. CS 220: Discrete Structures and their Applications. Functions. Chapter 5 in zybooks. definition. definition
Functions CS 220: Discrete Structures and their Applications Functions Chapter 5 in zybooks A function maps elements from one set X to elements of another set Y. A B Brian C Drew range: {A, B, D} Alan
More informationThree Profound Theorems about Mathematical Logic
Power & Limits of Logic Three Profound Theorems about Mathematical Logic Gödel's Completeness Theorem Thm 1, good news: only need to know* a few axioms & rules, to prove all validities. *Theoretically
More informationSets. We discuss an informal (naive) set theory as needed in Computer Science. It was introduced by G. Cantor in the second half of the nineteenth
Sets We discuss an informal (naive) set theory as needed in Computer Science. It was introduced by G. Cantor in the second half of the nineteenth century. Most students have seen sets before. This is intended
More informationCS 220: Discrete Structures and their Applications. Functions Chapter 5 in zybooks
CS 220: Discrete Structures and their Applications Functions Chapter 5 in zybooks Functions A function maps elements from one set X to elements of another set Y. A Brian Drew Alan Ben B C D F range: {A,
More information12.1 Systems of Linear equations: Substitution and Elimination
. Sstems of Linear equations: Substitution and Elimination Sstems of two linear equations in two variables A sstem of equations is a collection of two or more equations. A solution of a sstem in two variables
More informationRelations and Functions
artesian product Relations and Functions Let and be two sets. Then the set of all ordered pairs ab where a and b is called the artesian Product or ross Product or Product set of and and is denoted b X.
More informationCh 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations
Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4
More informationTrigonometry Outline
Trigonometr Outline Introduction Knowledge of the content of this outline is essential to perform well in calculus. The reader is urged to stud each of the three parts of the outline. Part I contains the
More informationCOMP9020 Lecture 3 Session 2, 2016 Sets, Functions, and Sequences. Revision: 1.3
1 COMP9020 Lecture 3 Session 2, 2016 Sets, Functions, and Sequences Revision: 1.3 2 Divisibility Let m, n Z. m n means m is a divisor of n, defined by n = km for some k Z (Also stated as: n is divisible
More information8.4 Inverse Functions
Section 8. Inverse Functions 803 8. Inverse Functions As we saw in the last section, in order to solve application problems involving eponential unctions, we will need to be able to solve eponential equations
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 27
CS 70 Discrete Mathematics for CS Spring 007 Luca Trevisan Lecture 7 Infinity and Countability Consider a function f that maps elements of a set A (called the domain of f ) to elements of set B (called
More informationWith Question/Answer Animations. Chapter 2
With Question/Answer Animations Chapter 2 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Sequences and Summations Types of
More informationPOL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005
POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1
More informationAutomata Theory and Formal Grammars: Lecture 1
Automata Theory and Formal Grammars: Lecture 1 Sets, Languages, Logic Automata Theory and Formal Grammars: Lecture 1 p.1/72 Sets, Languages, Logic Today Course Overview Administrivia Sets Theory (Review?)
More informationMATH STUDENT BOOK. 9th Grade Unit 8
MATH STUDENT BOOK 9th Grade Unit 8 Unit 8 Graphing Math 908 Graphing INTRODUCTION 3. USING TWO VARIABLES 5 EQUATIONS 5 THE REAL NUMBER PLANE TRANSLATIONS 5 SELF TEST. APPLYING GRAPHING TECHNIQUES 5 LINES
More informationCSCE 222 Discrete Structures for Computing
CSCE 222 Discrete Structures for Computing Sets and Functions Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Sets Sets are the most fundamental discrete structure on which all other discrete
More informationCourse 15 Numbers and Their Properties
Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.
More informationN E W S A N D L E T T E R S
N E W S A N D L E T T E R S 74th Annual William Lowell Putnam Mathematical Competition Editor s Note: Additional solutions will be printed in the Monthly later in the year. PROBLEMS A1. Recall that a regular
More informationSelected solutions to Discrete Mathematics and Functional Programming
Selected solutions to Discrete Mathematics and Functional Programming Thomas VanDrunen August 13, 2013 This document is to provide a resource for students studying Discrete Mathematics and Functional Programming
More information( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing
Section 5 : Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two or more linear equations that each represents the graph of a
More informationMath Review Packet #5 Algebra II (Part 2) Notes
SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of
More informationMGF 1106: Exam 1 Solutions
MGF 1106: Exam 1 Solutions 1. (15 points total) True or false? Explain your answer. a) A A B Solution: Drawn as a Venn diagram, the statement says: This is TRUE. The union of A with any set necessarily
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100
Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 5 1 Quadratics 5 The Discriminant 54 Completing the Square 55 4 Sketching Parabolas 57 5 Determining the Equation
More informationLinear Equations and Arithmetic Sequences
CONDENSED LESSON.1 Linear Equations and Arithmetic Sequences In this lesson, ou Write eplicit formulas for arithmetic sequences Write linear equations in intercept form You learned about recursive formulas
More information1. SET 10/9/2013. Discrete Mathematics Fajrian Nur Adnan, M.CS
1. SET 10/9/2013 Discrete Mathematics Fajrian Nur Adnan, M.CS 1 Discrete Mathematics 1. Set and Logic 2. Relation 3. Function 4. Induction 5. Boolean Algebra and Number Theory MID 6. Graf dan Tree/Pohon
More informationMATRIX TRANSFORMATIONS
CHAPTER 5. MATRIX TRANSFORMATIONS INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRIX TRANSFORMATIONS Matri Transformations Definition Let A and B be sets. A function f : A B
More information8.3 Zero, Negative, and Fractional Exponents
www.ck2.org Chapter 8. Eponents and Polynomials 8.3 Zero, Negative, and Fractional Eponents Learning Objectives Simplify epressions with zero eponents. Simplify epressions with negative eponents. Simplify
More informationMA123, Chapter 1: Equations, functions and graphs (pp. 1-15)
MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand
More informationWeek Some Warm-up Questions
1 Some Warm-up Questions Week 1-2 Abstraction: The process going from specific cases to general problem. Proof: A sequence of arguments to show certain conclusion to be true. If... then... : The part after
More informationFunctions Functions and Modeling A UTeach/TNT Course
Definition of a Function DEFINITION: Let A and B be sets. A function between A and B is a subset of A B with the property that if (a, b 1 )and(a, b 2 ) are both in the subset, then b 1 = b 2. The domain
More informationMathematical Structures Combinations and Permutations
Definitions: Suppose S is a (finite) set and n, k 0 are integers The set C(S, k) of k - combinations consists of all subsets of S that have exactly k elements The set P (S, k) of k - permutations consists
More informationStudy sheet for Final CS1100 by Matt in Wed night class
Study sheet for Final CS1100 by Matt in Wed night class Licensed under the GNU Free Documentation License (GFDL) http://www.gnu.org/copyleft/fdl.html Sequences have patterns; possible patterns are: 1.
More informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Suggested Problems for Sets and Functions The following problems are from Discrete Mathematics and Its Applications by Kenneth H. Rosen. 1. Define the
More informationEquivalence of Propositions
Equivalence of Propositions 1. Truth tables: two same columns 2. Sequence of logical equivalences: from one to the other using equivalence laws 1 Equivalence laws Table 6 & 7 in 1.2, some often used: Associative:
More information6 CARDINALITY OF SETS
6 CARDINALITY OF SETS MATH10111 - Foundations of Pure Mathematics We all have an idea of what it means to count a finite collection of objects, but we must be careful to define rigorously what it means
More informationNorthwest High School s Algebra 2/Honors Algebra 2
Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar
More informationMATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017
MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A
More informationAlgorithms: Lecture 2
1 Algorithms: Lecture 2 Basic Structures: Sets, Functions, Sequences, and Sums Jinwoo Kim jwkim@jjay.cuny.edu 2.1 Sets 2 1 2.1 Sets 3 2.1 Sets 4 2 2.1 Sets 5 2.1 Sets 6 3 2.1 Sets 7 2.2 Set Operations
More informationMath 205 April 28, 2009 Final Exam Review
1 Definitions 1. Some question will ask you to statea a definition and/or give an example of a defined term. The terms are: subset proper subset intersection union power set cartesian product relation
More informationGrade 7/8 Math Circles Winter March 20/21/22 Types of Numbers
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Winter 2018 - March 20/21/22 Types of Numbers Introduction Today, we take our number
More informationChapter 5: Integer Compositions and Partitions and Set Partitions
Chapter 5: Integer Compositions and Partitions and Set Partitions Prof. Tesler Math 184A Winter 2017 Prof. Tesler Ch. 5: Compositions and Partitions Math 184A / Winter 2017 1 / 32 5.1. Compositions A strict
More informationChapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
More information8.1 Exponents and Roots
Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents
More informationSTEP Support Programme. Hints and Partial Solutions for Assignment 17
STEP Support Programme Hints and Partial Solutions for Assignment 7 Warm-up You need to be quite careful with these proofs to ensure that you are not assuming something that should not be assumed. For
More informationTURING MAHINES
15-453 TURING MAHINES TURING MACHINE FINITE STATE q 10 CONTROL AI N P U T INFINITE TAPE read write move 0 0, R, R q accept, R q reject 0 0, R 0 0, R, L read write move 0 0, R, R q accept, R 0 0, R 0 0,
More informationCSE 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Today's learning goals Define and differentiate between important sets Use correct notation when describing sets: {...}, intervals
More information