Review of Functions. Functions. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Functions Current Semester 1 / 12

Size: px
Start display at page:

Download "Review of Functions. Functions. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Functions Current Semester 1 / 12"

Transcription

1 Review of Functions Functions Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Functions Current Semester 1 / 12

2 Introduction Students are expected to know the following concepts about functions: s and notation. Operations on functions One-to-one and onto functions Inverse of a function You may review these concepts in the notes provided. We will only discuss in details the direct image and inverse image of a set under a function. Philippe B. Laval (KSU) Functions Current Semester 2 / 12

3 Review: s Let A and B denote two sets. 1 A function f from A to B is a rule which assigns a unique y B to each x A. We write y = f (x). f (x) denotes the value of the function at x. 2 x is called the independent variable (also called an input value), y is the dependent variable (also called an output value). Remember, if we say that y is a function of x, it implies that it depends on x. 3 The domain of f is A, the set of values of x. It is also denoted D (f ) or Dom f. When the domain of a function is not given, it is understood to be the largest set of real numbers for which the function is defined. 4 The range of f is the set R (f ) = {f (x) : x D (f )}. It is also denoted Range f. Philippe B. Laval (KSU) Functions Current Semester 3 / 12

4 Review: s Let A and B denote two sets. A function from A to B is a subset of A B that is a set of ordered pairs with the property that whenever (a, b) f and (a, c) f then b = c. If f is a function from A to B and y = f (x) (or if (x, y) f ), then we say that y is the image of x under f. If f is a function from A to B, we also say that f is a mapping from A into B, or that f maps A into B. We often write: f : A B Philippe B. Laval (KSU) Functions Current Semester 4 / 12

5 Review: Onto, and one-to-one functions If f is a mapping of A into B such that R (f ) = B, then we say that the mapping is onto. We also say that f is surjective, or that f is a surjection. Let f be a function from A into B. f is said to be injective, or an injection, or one-to-one if one of the three equivalent conditions below is satisfied. 1 f (a) = f (b) a = b 2 a b f (a) f (b) 3 (a, c) f and (b, c) f a = b Philippe B. Laval (KSU) Functions Current Semester 5 / 12

6 Review: Bijection A function which is both an injection (one-to-one) and a surjection (onto) is called a bijection. Philippe B. Laval (KSU) Functions Current Semester 6 / 12

7 Review: Inverse Function Let f be an injective function from A onto B. The function f 1 = {(b, a) B A : (a, b) f } is called the inverse of f. It is denoted f 1. f and f 1 are related in many different ways. We list a few of these relations below. 1 D (f ) = R ( f 1), R (f ) = D ( f 1). This can be seen from the definition. 2 y = f (x) x = f 1 (y). This can be seen from the definition. ( 3 f 1 f ) (x) = x x D (f ). See problems at the end of this section. ( 4 f f 1 ) (y) = y y R (f ). See problems at the end of this section. Philippe B. Laval (KSU) Functions Current Semester 7 / 12

8 Direct Image of a Set: s and Examples Let f : A B and let E be a set such that E A. The direct image of E, denoted f (E), is defined by: Example f (E) = {f (x) : x E} Remark It should be clear to the reader that f (E) B. Consider f : N N defined by f (n) = n 2 1. Let E = {2, 3, 4}. Find f (E). Philippe B. Laval (KSU) Functions Current Semester 8 / 12

9 Direct Image of a Set: Properties Theorem Let f : A B. Let E and F be subsets of A. The following is true: 1 If E F then f (E) f (F ) 2 f (E F ) f (E) f (F ) 3 f (E F ) = f (E) f (F ) 4 f (E \ F ) f (E) Philippe B. Laval (KSU) Functions Current Semester 9 / 12

10 Inverse Image of a Set: s and Examples Let f : A B and let G be a set such that G B. The inverse image of G, denoted f 1 (G) is defined by: Example f 1 (G) = {x A : f (x) G} Remark The above definition does not require that f be injective or have an inverse. f 1 (G) is simply the notation for the inverse image of G. The reader should never think we are talking about the inverse of f. Remark It should be clear to the reader that f 1 (G) A. Consider f : N N defined by f (n) = n 2 1. Let G = {3, 4, 5, 6, 7, 8}, find f 1 (G). Philippe B. Laval (KSU) Functions Current Semester 10 / 12

11 Inverse Image of a Set: Properties Theorem Let f : A B. Let G and H be subsets of B. The following is true: 1 If G H then f 1 (G) f 1 (H) 2 f 1 (G H) = f 1 (G) f 1 (H) 3 f 1 (G H) = f 1 (G) f 1 (H) 4 f 1 (G \ H) = f 1 (G) \ f 1 (H) Philippe B. Laval (KSU) Functions Current Semester 11 / 12

12 Exercises See the problems at the end of my notes on functions. Philippe B. Laval (KSU) Functions Current Semester 12 / 12

1.2 Functions What is a Function? 1.2. FUNCTIONS 11

1.2 Functions What is a Function? 1.2. FUNCTIONS 11 1.2. FUNCTIONS 11 1.2 Functions 1.2.1 What is a Function? In this section, we only consider functions of one variable. Loosely speaking, a function is a special relation which exists between two variables.

More information

Consequences of the Completeness Property

Consequences of the Completeness Property Consequences of the Completeness Property Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of the Completeness Property Today 1 / 10 Introduction In this section, we use the fact that R

More information

Differentiation - Quick Review From Calculus

Differentiation - Quick Review From Calculus Differentiation - Quick Review From Calculus Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 1 / 13 Introduction In this section,

More information

The Laplace Transform

The Laplace Transform The Laplace Transform Inverse of the Laplace Transform Philippe B. Laval KSU Today Philippe B. Laval (KSU) Inverse of the Laplace Transform Today 1 / 12 Outline Introduction Inverse of the Laplace Transform

More information

Representation of Functions as Power Series

Representation of Functions as Power Series Representation of Functions as Power Series Philippe B. Laval KSU Today Philippe B. Laval (KSU) Functions as Power Series Today / Introduction In this section and the next, we develop several techniques

More information

Sequences: Limit Theorems

Sequences: Limit Theorems Sequences: Limit Theorems Limit Theorems Philippe B. Laval KSU Today Philippe B. Laval (KSU) Limit Theorems Today 1 / 20 Introduction These limit theorems fall in two categories. 1 The first category deals

More information

Functions of Several Variables

Functions of Several Variables Functions of Several Variables Extreme Values Philippe B Laval KSU April 9, 2012 Philippe B Laval (KSU) Functions of Several Variables April 9, 2012 1 / 13 Introduction In Calculus I (differential calculus

More information

Testing Series with Mixed Terms

Testing Series with Mixed Terms Testing Series with Mixed Terms Philippe B. Laval KSU Today Philippe B. Laval (KSU) Series with Mixed Terms Today 1 / 17 Outline 1 Introduction 2 Absolute v.s. Conditional Convergence 3 Alternating Series

More information

Introduction to Vector Functions

Introduction to Vector Functions Introduction to Vector Functions Limits and Continuity Philippe B. Laval KSU Today Philippe B. Laval (KSU) Vector Functions Today 1 / 14 Introduction Until now, the functions we studied took a real number

More information

Functions of Several Variables

Functions of Several Variables Functions of Several Variables Extreme Values Philippe B. Laval KSU Today Philippe B. Laval (KSU) Extreme Values Today 1 / 18 Introduction In Calculus I (differential calculus for functions of one variable),

More information

Sets 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. 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 information

Lagrange s Theorem. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10

Lagrange s Theorem. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10 Lagrange s Theorem Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10 Introduction In this chapter, we develop new tools which will allow us to extend

More information

The Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring /

The Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring / The Cross Product Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) The Cross Product Spring 2012 1 / 15 Introduction The cross product is the second multiplication operation between vectors we will

More information

Functions Functions and Modeling A UTeach/TNT Course

Functions 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 information

Introduction to Partial Differential Equations

Introduction to Partial Differential Equations Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 1 / 19 Introduction The derivation of the heat

More information

Differentiation - Important Theorems

Differentiation - Important Theorems Differentiation - Important Theorems Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Differentiation - Important Theorems Spring 2012 1 / 10 Introduction We study several important theorems related

More information

RELATIONS AND FUNCTIONS

RELATIONS AND FUNCTIONS For more important questions visit : www.4ono.com CHAPTER 1 RELATIONS AND FUNCTIONS IMPORTANT POINTS TO REMEMBER Relation R from a set A to a set B is subset of A B. A B = {(a, b) : a A, b B}. If n(a)

More information

1.4 Cardinality. Tom Lewis. Fall Term Tom Lewis () 1.4 Cardinality Fall Term / 9

1.4 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 information

Introduction to Vector Functions

Introduction to Vector Functions Introduction to Vector Functions Differentiation and Integration Philippe B. Laval KSU Today Philippe B. Laval (KSU) Vector Functions Today 1 / 14 Introduction In this section, we study the differentiation

More information

Integration Using Tables and Summary of Techniques

Integration Using Tables and Summary of Techniques Integration Using Tables and Summary of Techniques Philippe B. Laval KSU Today Philippe B. Laval (KSU) Summary Today 1 / 13 Introduction We wrap up integration techniques by discussing the following topics:

More information

The Laplace Transform

The Laplace Transform The Laplace Transform Laplace Transform Philippe B. Laval KSU Today Philippe B. Laval (KSU) Definition of the Laplace Transform Today 1 / 16 Outline General idea behind the Laplace transform and other

More information

Relationship Between Integration and Differentiation

Relationship Between Integration and Differentiation Relationship Between Integration and Differentiation Fundamental Theorem of Calculus Philippe B. Laval KSU Today Philippe B. Laval (KSU) FTC Today 1 / 16 Introduction In the previous sections we defined

More information

Arc Length. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Arc Length Today 1 / 12

Arc Length. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Arc Length Today 1 / 12 Philippe B. Laval KSU Today Philippe B. Laval (KSU) Arc Length Today 1 / 12 Introduction In this section, we discuss the notion of curve in greater detail and introduce the very important notion of arc

More information

Differentiation and Integration of Fourier Series

Differentiation and Integration of Fourier Series Differentiation and Integration of Fourier Series Philippe B. Laval KSU Today Philippe B. Laval (KSU) Fourier Series Today 1 / 12 Introduction When doing manipulations with infinite sums, we must remember

More information

Introduction to Vector Functions

Introduction to Vector Functions Introduction to Vector Functions Limits and Continuity Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Introduction to Vector Functions Spring 2012 1 / 14 Introduction In this section, we study

More information

Functions of Several Variables: Limits and Continuity

Functions of Several Variables: Limits and Continuity Functions of Several Variables: Limits and Continuity Philippe B. Laval KSU Today Philippe B. Laval (KSU) Limits and Continuity Today 1 / 24 Introduction We extend the notion of its studied in Calculus

More information

SECTION 1.8 : x = f LEARNING OBJECTIVES

SECTION 1.8 : x = f LEARNING OBJECTIVES SECTION 1.8 : x = f (Section 1.8: x = f ( y) ( y)) 1.8.1 LEARNING OBJECTIVES Know how to graph equations of the form x = f ( y). Compare these graphs with graphs of equations of the form y = f ( x). Recognize

More information

First Order Differential Equations

First Order Differential Equations First Order Differential Equations Linear Equations Philippe B. Laval KSU Philippe B. Laval (KSU) 1st Order Linear Equations 1 / 11 Introduction We are still looking at 1st order equations. In today s

More information

Relations, Functions & Binary Operations

Relations, Functions & Binary Operations Relations, Functions & Binary Operations Important Terms, Definitions & Formulae 0 TYPES OF INTERVLS a) Open interval: If a and b be two real numbers such that a b then, the set of all the real numbers

More information

Functions of Several Variables

Functions of Several Variables Functions of Several Variables Partial Derivatives Philippe B Laval KSU March 21, 2012 Philippe B Laval (KSU) Functions of Several Variables March 21, 2012 1 / 19 Introduction In this section we extend

More information

2.1 Sets. Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R.

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 information

Section 4.4 Functions. CS 130 Discrete Structures

Section 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 information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Spring

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Spring c Dr Oksana Shatalov, Spring 2016 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element in the set

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2016 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from the set A to the set B is a correspondence that assigns to

More information

Introduction to Partial Differential Equations

Introduction to Partial Differential Equations Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Key Concepts Current Semester 1 / 25 Introduction The purpose of this section is to define

More information

ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS

ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements

More information

Consequences of Orthogonality

Consequences of Orthogonality Consequences of Orthogonality Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of Orthogonality Today 1 / 23 Introduction The three kind of examples we did above involved Dirichlet, Neumann

More information

Multiple Integrals. Introduction and Double Integrals Over Rectangular Regions. Philippe B. Laval KSU. Today

Multiple Integrals. Introduction and Double Integrals Over Rectangular Regions. Philippe B. Laval KSU. Today Multiple Integrals Introduction and Double Integrals Over Rectangular Regions Philippe B. Laval KSU Today Philippe B. Laval (KSU) Double Integrals Today 1 / 21 Introduction In this section we define multiple

More information

Topics in Logic, Set Theory and Computability

Topics in Logic, Set Theory and Computability Topics in Logic, Set Theory and Computability Homework Set #3 Due Friday 4/6 at 3pm (by email or in person at 08-3234) Exercises from Handouts 7-C-2 7-E-6 7-E-7(a) 8-A-4 8-A-9(a) 8-B-2 8-C-2(a,b,c) 8-D-4(a)

More information

SETS AND FUNCTIONS JOSHUA BALLEW

SETS 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 information

Multiple Integrals. Introduction and Double Integrals Over Rectangular Regions. Philippe B. Laval. Spring 2012 KSU

Multiple Integrals. Introduction and Double Integrals Over Rectangular Regions. Philippe B. Laval. Spring 2012 KSU Multiple Integrals Introduction and Double Integrals Over Rectangular Regions Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Multiple Integrals Spring 2012 1 / 21 Introduction In this section

More information

5 FUNCTIONS. 5.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

5 FUNCTIONS. 5.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2017 1 5 FUNCTIONS 5.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from the set A to the set B is a correspondence that assigns to

More information

9 FUNCTIONS. 9.1 The Definition of Function. c Dr Oksana Shatalov, Fall

9 FUNCTIONS. 9.1 The Definition of Function. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2018 1 9 FUNCTIONS 9.1 The Definition of Function DEFINITION 1. Let X and Y be nonempty sets. A function f from the set X to the set Y is a correspondence that assigns to each

More information

Cardinality and ordinal numbers

Cardinality and ordinal numbers Cardinality and ordinal numbers The cardinality A of a finite set A is simply the number of elements in it. When it comes to infinite sets, we no longer can speak of the number of elements in such a set.

More information

Chapter 2 Invertible Mappings

Chapter 2 Invertible Mappings Chapter 2 Invertible Mappings 2. Injective, Surjective and Bijective Mappings Given the map f : A B, and I A, theset f (I ) ={f (x) : x I } is called the image of I under f.ifi = A, then f (A) is called

More information

Relations, Functions, and Sequences

Relations, Functions, and Sequences MCS-236: Graph Theory Handout #A3 San Skulrattanakulchai Gustavus Adolphus College Sep 13, 2010 Relations, Functions, and Sequences Relations An ordered pair can be constructed from any two mathematical

More information

CS100: DISCRETE STRUCTURES

CS100: 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 information

Functions. Given a function f: A B:

Functions. Given a function f: A B: Functions Given a function f: A B: We say f maps A to B or f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f.

More information

and The important theorem which connects these various spaces with each other is the following: (with the notation above)

and The important theorem which connects these various spaces with each other is the following: (with the notation above) When F : U V is a linear transformation there are two special subspaces associated to F which are very important. One is a subspace of U and the other is a subspace of V. They are: kerf U (the kernel of

More information

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that:

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: Equivalence MA Relations 274 and Partitions Scott Taylor 1. EQUIVALENCE RELATIONS Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: (1) is reflexive. That is, (2) is

More information

MATH 224 FOUNDATIONS DR. M. P. M. M. M C LOUGHLIN HAND-OUT XIV FUNCTIONS REVISED 2016

MATH 224 FOUNDATIONS DR. M. P. M. M. M C LOUGHLIN HAND-OUT XIV FUNCTIONS REVISED 2016 ath 224 Dr. c Loughlin Hand-out 14 Functions, page 1 of 9 ATH 224 FOUNDATIONS DR.. P... C LOUGHLIN HAND-OUT XIV FUNCTIONS REVISED 2016 Recall: Let U be a well defined universe. Let V be the well defined

More information

The Principal Component Analysis

The Principal Component Analysis The Principal Component Analysis Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) PCA Fall 2017 1 / 27 Introduction Every 80 minutes, the two Landsat satellites go around the world, recording images

More information

Chapter One. The Real Number System

Chapter One. The Real Number System Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and

More information

Cartesian Products and Relations

Cartesian Products and Relations Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) : (a A) and (b B)}. The following points are worth special

More information

Homework for MATH 4603 (Advanced Calculus I) Fall Homework 13: Due on Tuesday 15 December. Homework 12: Due on Tuesday 8 December

Homework for MATH 4603 (Advanced Calculus I) Fall Homework 13: Due on Tuesday 15 December. Homework 12: Due on Tuesday 8 December Homework for MATH 4603 (Advanced Calculus I) Fall 2015 Homework 13: Due on Tuesday 15 December 49. Let D R, f : D R and S D. Let a S (acc S). Assume that f is differentiable at a. Let g := f S. Show that

More information

Solving Linear Systems

Solving Linear Systems Solving Linear Systems Iterative Solutions Methods Philippe B. Laval KSU Fall 207 Philippe B. Laval (KSU) Linear Systems Fall 207 / 2 Introduction We continue looking how to solve linear systems of the

More information

Notes. Functions. Introduction. Notes. Notes. Definition Function. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y.

Notes. Functions. Introduction. Notes. Notes. Definition Function. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Functions Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 Computer Science & Engineering 235 Introduction to Discrete Mathematics Section 2.3 of Rosen cse235@cse.unl.edu Introduction

More information

Section Summary. Definition of a Function.

Section Summary. Definition of a Function. Section 2.3 Section Summary Definition of a Function. Domain, Cdomain Image, Preimage Injection, Surjection, Bijection Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial

More information

Section 7.2: One-to-One, Onto and Inverse Functions

Section 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 information

Pullbacks, Isometries & Conformal Maps

Pullbacks, Isometries & Conformal Maps Pullbacks, Isometries & Conformal Maps Outline 1. Pullbacks Let V and W be vector spaces, and let T : V W be an injective linear transformation. Given an inner product, on W, the pullback of, is the inner

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2014 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element in the set

More information

T ((x 1, x 2,..., x n )) = + x x 3. , x 1. x 3. Each of the four coordinates in the range is a linear combination of the three variables x 1

T ((x 1, x 2,..., x n )) = + x x 3. , x 1. x 3. Each of the four coordinates in the range is a linear combination of the three variables x 1 MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

4) Have you met any functions during our previous lectures in this course?

4) Have you met any functions during our previous lectures in this course? Definition: Let X and Y be sets. A function f from the set X to the set Y is a rule which associates to each element x X a unique element y Y. Notation: f : X Y f defined on X with values in Y. x y y =

More information

Connectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).

Connectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ). Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.

More information

Functions as Relations

Functions as Relations Functions as Relations Definition Recall that if A and B are sets, then a relation from A to B is a subset of A B. A function from A to B is a relation f from A to B with the following properties (i) The

More information

1 Equivalence Relations

1 Equivalence Relations Mitchell Faulk June 12, 2014 Equivalence Relations and Quotient Spaces 1 Equivalence Relations Recall that a relation on a set X is a subset S of the cartesian product X X. Problem Set 2 has several examples

More information

1. Decide for each of the following expressions: Is it a function? If so, f is a function. (i) Domain: R. Codomain: R. Range: R. (iii) Yes surjective.

1. Decide for each of the following expressions: Is it a function? If so, f is a function. (i) Domain: R. Codomain: R. Range: R. (iii) Yes surjective. Homework 2 2/14/2018 SOLUTIONS Exercise 6. 1. Decide for each of the following expressions: Is it a function? If so, (i) what is its domain, codomain, and image? (iii) is it surjective? (ii) is it injective?

More information

0 Logical Background. 0.1 Sets

0 Logical Background. 0.1 Sets 0 Logical Background 0.1 Sets In this course we will use the term set to simply mean a collection of things which have a common property such as the totality of positive integers or the collection of points

More information

Handout 2 (Correction of Handout 1 plus continued discussion/hw) Comments and Homework in Chapter 1

Handout 2 (Correction of Handout 1 plus continued discussion/hw) Comments and Homework in Chapter 1 22M:132 Fall 07 J. Simon Handout 2 (Correction of Handout 1 plus continued discussion/hw) Comments and Homework in Chapter 1 Chapter 1 contains material on sets, functions, relations, and cardinality that

More information

Homework #05, due 2/17/10 = , , , , , Additional problems recommended for study: , , 10.2.

Homework #05, due 2/17/10 = , , , , , Additional problems recommended for study: , , 10.2. Homework #05, due 2/17/10 = 10.3.1, 10.3.3, 10.3.4, 10.3.5, 10.3.7, 10.3.15 Additional problems recommended for study: 10.2.1, 10.2.2, 10.2.3, 10.2.5, 10.2.6, 10.2.10, 10.2.11, 10.3.2, 10.3.9, 10.3.12,

More information

1 Background and Review Material

1 Background and Review Material 1 Background and Review Material 1.1 Vector Spaces Let V be a nonempty set and consider the following two operations on elements of V : (x,y) x + y maps V V V (addition) (1.1) (α, x) αx maps F V V (scalar

More information

Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)

Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010) http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product

More information

Integration. Darboux Sums. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Darboux Sums Today 1 / 13

Integration. Darboux Sums. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Darboux Sums Today 1 / 13 Integration Darboux Sums Philippe B. Laval KSU Today Philippe B. Laval (KSU) Darboux Sums Today 1 / 13 Introduction The modern approach to integration is due to Cauchy. He was the first to construct a

More information

Sets, Functions and Relations

Sets, Functions and Relations Chapter 2 Sets, Functions and Relations A set is any collection of distinct objects. Here is some notation for some special sets of numbers: Z denotes the set of integers (whole numbers), that is, Z =

More information

9/21/2018. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions

9/21/2018. 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 information

2 Functions. 2.1 What is a function?

2 Functions. 2.1 What is a function? Functions After working through this section, you should be able to: (a) determine the image of a given function; (b) determine whether a given function is one-one and/or onto; (c) find the inverse of

More information

Structural-Morphism System Property: atis Homomorphismness

Structural-Morphism System Property: atis Homomorphismness ATIS Glossary 1 Structural-Morphism System Property: atis Homomorphismness (Structural-morphism system properties are those properties that are part of the theory and define the mapping-relatedness of

More information

A Logician s Toolbox

A Logician s Toolbox A Logician s Toolbox 461: An Introduction to Mathematical Logic Spring 2009 We recast/introduce notions which arise everywhere in mathematics. All proofs are left as exercises. 0 Notations from set theory

More information

Func%ons. function f to the element a of A. Functions are sometimes called mappings or transformations.

Func%ons. function f to the element a of A. Functions are sometimes called mappings or transformations. Section 2.3 Func%ons Definition: Let A and B be nonempty sets. A function f from A to B, denoted f: A B is an assignment of each element of A to exactly one element of B. We write f(a) = b if b is the

More information

Mathematics Review for Business PhD Students

Mathematics 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 information

can only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4

can only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4 .. Conditions for Injectivity and Surjectivity In this section, we discuss what we can say about linear maps T : R n R m given only m and n. We motivate this problem by looking at maps f : {,..., n} {,...,

More information

Lesson 12: Solving Equations

Lesson 12: Solving Equations Exploratory Exercises 1. Alonzo was correct when he said the following equations had the same solution set. Discuss with your partner why Alonzo was correct. (xx 1)(xx + 3) = 17 + xx (xx 1)(xx + 3) = xx

More information

A Little Beyond: Linear Algebra

A Little Beyond: Linear Algebra A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of non-zero polynomials in [x], no two

More information

106 CHAPTER 3. TOPOLOGY OF THE REAL LINE. 2. The set of limit points of a set S is denoted L (S)

106 CHAPTER 3. TOPOLOGY OF THE REAL LINE. 2. The set of limit points of a set S is denoted L (S) 106 CHAPTER 3. TOPOLOGY OF THE REAL LINE 3.3 Limit Points 3.3.1 Main Definitions Intuitively speaking, a limit point of a set S in a space X is a point of X which can be approximated by points of S other

More information

(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k)

(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k) MATH 111 Optional Exam 3 lutions 1. (0 pts) We define a relation on Z as follows: a b if a + b is divisible by 3. (a) (1 pts) Prove that is an equivalence relation. (b) (8 pts) Determine all equivalence

More information

MATH 403 MIDTERM ANSWERS WINTER 2007

MATH 403 MIDTERM ANSWERS WINTER 2007 MAH 403 MIDERM ANSWERS WINER 2007 COMMON ERRORS (1) A subset S of a ring R is a subring provided that x±y and xy belong to S whenever x and y do. A lot of people only said that x + y and xy must belong

More information

UMASS AMHERST MATH 300 SP 05, F. HAJIR HOMEWORK 8: (EQUIVALENCE) RELATIONS AND PARTITIONS

UMASS AMHERST MATH 300 SP 05, F. HAJIR HOMEWORK 8: (EQUIVALENCE) RELATIONS AND PARTITIONS UMASS AMHERST MATH 300 SP 05, F. HAJIR HOMEWORK 8: (EQUIVALENCE) RELATIONS AND PARTITIONS 1. Relations Recall the concept of a function f from a source set X to a target set Y. It is a rule for mapping

More information

447 HOMEWORK SET 1 IAN FRANCIS

447 HOMEWORK SET 1 IAN FRANCIS 7 HOMEWORK SET 1 IAN FRANCIS For each n N, let A n {(n 1)k : k N}. 1 (a) Determine the truth value of the statement: for all n N, A n N. Justify. This statement is false. Simply note that for 1 N, A 1

More information

Sec$on Summary. Definition of a Function.

Sec$on Summary. Definition of a Function. Section 2.3 Sec$on Summary Definition of a Function. Domain, Codomain Image, Preimage Injection, Surjection, Bijection Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial

More information

Discrete Structures - CM0246 Cardinality

Discrete Structures - CM0246 Cardinality Discrete Structures - CM0246 Cardinality Andrés Sicard-Ramírez Universidad EAFIT Semester 2014-2 Cardinality Definition (Cardinality (finite sets)) Let A be a set. The number of (distinct) elements in

More information

The Singular Value Decomposition

The Singular Value Decomposition The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will

More information

4.1 Real-valued functions of a real variable

4.1 Real-valued functions of a real variable Chapter 4 Functions When introducing relations from a set A to a set B we drew an analogy with co-ordinates in the x-y plane. Instead of coming from R, the first component of an ordered pair comes from

More information

Set theory. Math 304 Spring 2007

Set theory. Math 304 Spring 2007 Math 304 Spring 2007 Set theory Contents 1. Sets 2 1.1. Objects and set formation 2 1.2. Unions and intersections 3 1.3. Differences 4 1.4. Power sets 4 1.5. Ordered pairs and binary,amscdcartesian products

More information

MATH 3300 Test 1. Name: Student Id:

MATH 3300 Test 1. Name: Student Id: Name: Student Id: There are nine problems (check that you have 9 pages). Solutions are expected to be short. In the case of proofs, one or two short paragraphs should be the average length. Write your

More information

Solving Linear Systems

Solving Linear Systems Solving Linear Systems Iterative Solutions Methods Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Linear Systems Fall 2015 1 / 12 Introduction We continue looking how to solve linear systems of

More information

ECS 120 Lesson 18 Decidable Problems, the Halting Problem

ECS 120 Lesson 18 Decidable Problems, the Halting Problem ECS 120 Lesson 18 Decidable Problems, the Halting Problem Oliver Kreylos Friday, May 11th, 2001 In the last lecture, we had a look at a problem that we claimed was not solvable by an algorithm the problem

More information

NOTES WEEK 13 DAY 2 SCOT ADAMS

NOTES WEEK 13 DAY 2 SCOT ADAMS NOTES WEEK 13 DAY 2 SCOT ADAMS Recall: Let px, dq be a metric space. Then, for all S Ď X, we have p S is sequentially compact q ñ p S is closed and bounded q. DEFINITION 0.1. Let px, dq be a metric space.

More information

CW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes.

CW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes. CW complexes Soren Hansen This note is meant to give a short introduction to CW complexes. 1. Notation and conventions In the following a space is a topological space and a map f : X Y between topological

More information

Independent Component Analysis

Independent Component Analysis Independent Component Analysis Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) ICA Fall 2017 1 / 18 Introduction Independent Component Analysis (ICA) falls under the broader topic of Blind Source

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Functions

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Functions ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 2017/2018 DR. ANTHONY BROWN 4. Functions 4.1. What is a Function: Domain, Codomain and Rule. In the course so far, we

More information