# Supplementary Material for MTH 299 Online Edition

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Supplementary Material for MTH 299 Online Edition Abstract This document contains supplementary material, such as definitions, explanations, examples, etc., to complement that of the text, How to Think Like a Mathematician. Contents 1 Useful Sets and Spaces 2 2 Logic Supplement to Chapter 6 - Statements, Negation of Compound Statements Supplement to Chapter 7 - Implications Various facts about real numbers, integers, and rationals 5 4 Functions Some definitions about functions Some frequently used functions Real Analysis A few properties of the real numbers Properties of subsets of the real numbers Sequences of real numbers Linear Algebra 10

2 1 Useful Sets and Spaces Theorem 1.1. A = B is equivalent to (A B and B A) Definition 1.2. The power set of a set, A, is the set which contains all possible subsets of A as its elements. The power set is denoted as P(A) = {E : E A}. Definition 1.3. P n is the collection of all (real) polynomials of degree less than or equal to n. That is P n = {p : p(x) = a 0 + a 1 x + + a n x n, where a 0,..., a n R}. Definition 1.4. Let I be any fixed set, and assume that for all r I, E r, is a set. The operations of union and intersection over a general index set are defined as E r = {x : x E r for at least one r I} and r I E r = {x : x E r for all r I}. r I Note, another way to write this is ( x r I E r ) ( r 0 I such that x E r0 ) and ( x r I E r ) ( r I, x E r ). As a notational convenience, when I is a simple set that can be listed easily, we sometimes write or 21 E i E i = i=1 i N E i = E j. j=5 j {5,6,...,21} Proposition 1.5. Suppose A, B are subsets of a universal set U. Their union and intersection satisfy the so-called De Morgan s Laws for sets: i) (A B) c = A c B c, and ii) (A B) c = A c B c. Remark 1.6. The same results hold for Boolean algebra (propositional logic). State this result and prove it.

3 Definition 1.7. Let E be a set. A partition of E is a collection, T P(E), such that A 1, A 2 T such that A 1 A 2, A 1 A 2 = and E = A T A. (Recall that P(E) is the power set of E, defined earlier in this section.) Remark 1.8. It is very useful to simply write out what it means to be a partition in plain words. A partition of E is a collection of subsets of E such that two things happen: (1) the union over all of the sets in the collection covers all of E and (2) if you take any two distinct subsets in the collection, they have an empty intersection. So, what you are doing is breaking up E into disjoint pieces and not leaving any piece of E out. This is incredibly useful for many areas of mathematics. You can also think of it as exactly what happens with the hard drive on your computer. The hard drive is one disc, but many people choose to break it up into separate partitions that don t interact with each other. Remark 1.9. It is very important to note a possible simplification in your proofs involving partitions. Suppose you want to show that T is a partition. If you first confirm that it is true that all sets in T are also elements of P(E), then you have confirmed that A E for all A in T. You may then tell your reader that you only need to check that E A T A, because the reverse set containment is already true by the previous observation. 2 Logic 2.1 Supplement to Chapter 6 - Statements, Negation of Compound Statements Definition 2.1. A compound statement is a statement composed of one or more given statements, and at least one connective:,,,,. For example, (P Q) P is a compound statement. Definition 2.2. Two compound statements are logically equivalent if they have the same truth table. Definition 2.3. (see page 54 of textbook) A conditional statement is a statement that requires an input to become a statement. For example, the expression, P (x) : x is even, is to be understood as something that becomes a statement after plugging in a value of x. (See pg. 81 of our text for an extremely brief description.) Formally, we need to define the collection of legal values that can be plugged in, which we call the domain of the conditional statement. For this problem, we will use the domain Z. For example, P (2) is true, and P (5) is false.

4 2.2 Supplement to Chapter 7 - Implications Definition 2.4. Statements of the form If statement A is true, then statement B is true. are called implications. Mathematically this is denoted by A B. In English, this statement can be expressed in the following equivalent ways. (i) If A then B (ii) A implies B (iii) A only if B (iv) B if A (v) B whenever A (vi) A is sufficient for B (vii) B is necessary for A When is the statement A B true? For example, is the following statement true: If pigs could fly, then I am on Mars.? Note that A B says nothing about whether the statements A, B themselves are true or false. The following cases are possible for the implication A B to be true. A - true and B - true A - false and B - false A - false and B - true In other words, if the assumption is false, the conclusion could be anything! Theorem 2.5. The negation of A B is equivalent to A and (not B). (A B) (A ( B))

5 3 Various facts about real numbers, integers, and rationals Axiom 3.1. Always assume all of the associativity and commutativity and distributivity as you have done since you were six years old. We will assume the following properties of the integers: 1. closed under addition 2. closed under multiplication e.g. if x, y Z, then so is (5y 10 xy + x 2 11) Z. Axiom 3.2. We will assume the following properties of the rationals: 1. closed under addition 2. closed under multiplication 3. addition is invertible 4. multiplication is invertible (with the exception of the zero element). Definition 3.3. The set of even integers is the set even integers = {n Z : n = 2k for some k Z}. An integer is said to be odd if it is not even. You can check that the set of odd integers is odd integers = {n Z : n = 2k + 1 for some k Z}.

6 4 Functions 4.1 Some definitions about functions Definition 4.1. A function from a set X to a set Y is an assignment such for all x X, there is assigned exactly one element of Y. This assignment can re-use elements of Y for the corresponding values of x. In concise terms, we write f : X Y, and for all x X, f(x) Y. We call f the function, x the input variable, f(x) the output, X is the domain, and Y is the co-domain. Remark 4.2. It is very important to note, you cannot assume you have precisely defined a function until you have specified three items: the domain, the co-domain, and the assignment rule / formula. This is a very frequently confused issue, and it can make things very difficult when investigating properties involving injective, surjective, and bijective. Definition 4.3. If f : X Y is a function, we define the graph of f to be the set graph(f) := {(x, y) X Y y = f(x)}. Definition 4.4. If f : X Y is a function, we define the range of f to be the set range(f) = {y Y : y = f(x) for some x X}. Definition 4.5. Assume that f is a function, f : X Y. We say the image of A under f is f(a), and it is defined as f(a) = {y Y : y = f(x) for some x A}. Definition 4.6. Assume that f and g are both functions, f : X R and g : X R. We define the addition f + g as the new function, (f + g)(x) = f(x) + g(x). Definition 4.7. Assume that f : X Y is a function. We say that it is injective (or one to one) if for all x 1, x 2 X, f(x 1 ) = f(x 2 ) implies x 1 = x 2. Definition 4.8. A function f : X Y is surjective if for all y Y, there exists x X such that f(x) = y. Definition 4.9. A function, f : X Y, for X and Y both subsets of R, is said to be increasing if and only if it holds that x 1, x 2 X and x 1 x 2 = f(x 1 ) f(x 2 ). We say f is strictly increasing if the above implication holds with x 1 < x 2 implies f(x 1 ) < f(x 2 ). Lemma The following functions are strictly increasing: (i) f : [0, ) [0, ), f(x) = x 2. (Note the restricted domain!) (ii) f : [0, ) [0, ), f(x) = x 1/2. (iii) f : [0, ) [0, ), f(x) = x p, where p > 0 is a real number. (iv) f : R (0, ), f(x) = e x. (v) f : (0, ) (0, ), f(x) = ln(x).

7 4.2 Some frequently used functions Definition A polynomial of degree n or less is a function that can be expressed as p(x) = n a i x i where a i R for i {0, 1,..., n}. i=0 Definition Take p(x) = i {0, 1,..., n} then: (a) (p + q)(x) := (b) (cp)(x) := n (a i + b i )x i i=0 n (c a i )x i i=0 n a i x i and q(x) = i=0 n b i x i where a i, b i, c R for Definition P n is the collection of all (real) polynomials of degree less than or equal to n. That is P n = {p : p(x) = a 0 + a 1 x + + a n x n, where a 0,..., a n R}. Definition The absolute value of a real number, x, is defined as { x if x 0 x = x if x < 0. You can think of as a function (here we use the dot,, as a placeholder), : R R, given by the assignment rule listed in the previous sentence. Definition The maximum of two real numbers x and y can be defined as: { x if x y max{x, y} = y if x < y. This can be expanded to 3 or more entries max{x, y, z} intuitively. Definition The minimum of two real numbers x and y can be defined as: { x if x y min{x, y} = y if x > y. This can be expanded to three or more entries min{x, y, z} intuitively. Definition The ceiling function x is a function from R Z defined by i=0 x := min{z Z n x}

8 5 Real Analysis 5.1 A few properties of the real numbers Axiom 5.1. The real numbers are endowed with a number of properties that we will take for granted. Below, we name a few of them: 1. The Archimedean property says that if x R, then there exists an N N such that x < N. That is, if you are challenged with a real number, then you can find a larger integer. 2. If x, y R, then one, and only one of the following cases can happen: Case I: x < y, Case II: x = y, Case III: x > y. 3. Additive and multiplicative properties: (a) a + (b + c) = (a + b) + c for all a, b, c R. (b) a + b = b + a for all a, b R. (c) a + 0 = a for all a R. (d) For each a R, there exists a unique a R, with a + ( a) = 0. (e) a(bc) = (ab)c for all a, b, c R. (f) ab = ba for any a, b R. (g) a 1 = a for any a R. (h) For each a 0, there exists a unique a 1 R, with aa 1 = 1. (i) a(b + c) = ab + ac for all a, b, c R. 4. Ordered properties: (a) If a, b R, then a b or b a. (b) If a b and b c, then a c. (c) For any x, y, c R, if x y then x + c y + c. (d) If a b, and 0 c, then ac bc. Remark 5.2. Assume the following property of the power function. If b a 0 and t > 0 then 1. a t b t, and 2. if also a > 0 and b > 0, then a t b t. For example, you know well from previous classes that if 0 < a b, then a 1/3 b 1/3, a 1 b 1, a 79 b 79, and so on.

9 5.2 Properties of subsets of the real numbers Definition 5.3. A set, E R, is said to be bounded if there exists an M R such that x M for all x E. Definition 5.4. We say the set A R, is open, if for all x A, there exists an r > 0, such that the interval (x r, x + r) A. Definition 5.5. We say a set C R is closed, if its complement, C c R is open. (Recall, the complement is defined via the shorthand notation, C c = R \ C.) 5.3 Sequences of real numbers A sequence of real numbers is an ordered list {a n } n N. Formally, a sequence is a function a : N R, but think of them as simply a list. Definition 5.6. We say a sequence {a n } n N converges to the real number L R if for every ɛ > 0, there exists an N N, such that for all n N, we have a n L < ɛ. We will refer to this definition as the ɛ-n definition of convergence. Definition 5.7. We say a sequence converges if there exists an L R, such that the sequence converges to L. Definition 5.8. We say a sequence, {a n } n N, is bounded if there exists a real number, say M, such that for all n N, a n M Note this is the same thing as saying the set E = n N{a n } is a bounded subset of R.

11 Proposition 6.5. If V is a vector space, and W V is a non-empty subset, then W is a subspace of V if it satisfies the following two properties: 1. Closure under addition: if u, v W, then u + v W, and 2. Closure under scalar multiplication: if c is a scalar, and u W, then cu W. Definition 6.6. If V and W are vector spaces over the reals, R, and L : V W, then L is a linear map (also called a linear function) if and only if L satisfies i) Additivity: L(u + v) = L(u) + L(v), for all u, v V, and ii) Scalar multiplication: L(au) = al(u), for all scalars a R, and u V. ( ) a b Definition 6.7. Matrix-vector multiplication in R 2 is defined for a matrix, A =, and an c d element, x R 2, with x = (x 1, x 2 ). The formula which defines matrix multiplication is ( ) ( ) a b x1 Ax := := (ax c d 1 + bx 2, cx 1 + dx 2 ). Here, we require that a, b, c, d R. Proposition 6.8. If we define the space of all 2 2 matrices {( ) } a b V = : a, b, c, d R, c d with the addition operation as ( ) ( ) ( ) a b e f a + e b + f + =, c d g h c + g d + h and the scalar multiplication operation, for λ R, as ( ) ( ) a b λa λb λ =, c d λc λd then V is a vector space over R. x 2

### Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

### August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei. 1.

August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei 1. Vector spaces 1.1. Notations. x S denotes the fact that the element x

### General Notation. Exercises and Problems

Exercises and Problems The text contains both Exercises and Problems. The exercises are incorporated into the development of the theory in each section. Additional Problems appear at the end of most sections.

### Mathematics 220 Midterm Practice problems from old exams Page 1 of 8

Mathematics 220 Midterm Practice problems from old exams Page 1 of 8 1. (a) Write the converse, contrapositive and negation of the following statement: For every integer n, if n is divisible by 3 then

### 1 Take-home exam and final exam study guide

Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number

### 2. Two binary operations (addition, denoted + and multiplication, denoted

Chapter 2 The Structure of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference between

### Proofs. Chapter 2 P P Q Q

Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,

### Math 115A: Linear Algebra

Math 115A: Linear Algebra Michael Andrews UCLA Mathematics Department February 9, 218 Contents 1 January 8: a little about sets 4 2 January 9 (discussion) 5 2.1 Some definitions: union, intersection, set

### Introduction to Topology

Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

### MATH 13 SAMPLE FINAL EXAM SOLUTIONS

MATH 13 SAMPLE FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers.

### Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007)

Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007) Problem 1: Specify two different predicates P (x) and

### Chapter 1. Logic and Proof

Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

### 2 so Q[ 2] is closed under both additive and multiplicative inverses. a 2 2b 2 + b

. FINITE-DIMENSIONAL VECTOR SPACES.. Fields By now you ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices,

### Contribution of Problems

Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions

### CHAPTER 3: THE INTEGERS Z

CHAPTER 3: THE INTEGERS Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction The natural numbers are designed for measuring the size of finite sets, but what if you want to compare the sizes of two sets?

### 0.2 Vector spaces. J.A.Beachy 1

J.A.Beachy 1 0.2 Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a

### Math 110, Spring 2015: Midterm Solutions

Math 11, Spring 215: Midterm Solutions These are not intended as model answers ; in many cases far more explanation is provided than would be necessary to receive full credit. The goal here is to make

### Abstract Vector Spaces and Concrete Examples

LECTURE 18 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.

### APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF

ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done

### Sanjay Mishra. Topology. Dr. Sanjay Mishra. A Profound Subtitle

Topology A Profound Subtitle Dr. Copyright c 2017 Contents I General Topology 1 Topological Spaces............................................ 7 1.1 Introduction 7 1.2 Topological Space 7 1.2.1 Topological

### Proofs. Chapter 2 P P Q Q

Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,

### Set Theory. CSE 215, Foundations of Computer Science Stony Brook University

Set Theory CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Set theory Abstract set theory is one of the foundations of mathematical thought Most mathematical

### 1 Basics of vector space

Linear Algebra- Review And Beyond Lecture 1 In this lecture, we will talk about the most basic and important concept of linear algebra vector space. After the basics of vector space, I will introduce dual

### 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

### Exercises for Unit VI (Infinite constructions in set theory)

Exercises for Unit VI (Infinite constructions in set theory) VI.1 : Indexed families and set theoretic operations (Halmos, 4, 8 9; Lipschutz, 5.3 5.4) Lipschutz : 5.3 5.6, 5.29 5.32, 9.14 1. Generalize

### EXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)

EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily

### Matrices. Chapter What is a Matrix? We review the basic matrix operations. An array of numbers a a 1n A = a m1...

Chapter Matrices We review the basic matrix operations What is a Matrix? An array of numbers a a n A = a m a mn with m rows and n columns is a m n matrix Element a ij in located in position (i, j The elements

### Sets 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

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Analysis 1. Lecture Notes 2013/2014. The original version of these Notes was written by. Vitali Liskevich

Analysis 1 Lecture Notes 2013/2014 The original version of these Notes was written by Vitali Liskevich followed by minor adjustments by many Successors, and presently taught by Misha Rudnev University

### Analysis I. Classroom Notes. H.-D. Alber

Analysis I Classroom Notes H-D Alber Contents 1 Fundamental notions 1 11 Sets 1 12 Product sets, relations 5 13 Composition of statements 7 14 Quantifiers, negation of statements 9 2 Real numbers 11 21

### 1. General Vector Spaces

1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

### Boolean Algebra CHAPTER 15

CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an

### Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

### Propositional 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

### Logic and Propositional Calculus

CHAPTER 4 Logic and Propositional Calculus 4.1 INTRODUCTION Many algorithms and proofs use logical expressions such as: IF p THEN q or If p 1 AND p 2, THEN q 1 OR q 2 Therefore it is necessary to know

### Direct Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24

Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using

### Contribution of Problems

Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions

### A Semester Course in Basic Abstract Algebra

A Semester Course in Basic Abstract Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved December 29, 2011 1 PREFACE This book is an introduction to abstract algebra course for undergraduates

### PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION. The Peano axioms

PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION The Peano axioms The following are the axioms for the natural numbers N. You might think of N as the set of integers {0, 1, 2,...}, but it turns

### Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant

### Gaussian elimination

Gaussian elimination October 14, 2013 Contents 1 Introduction 1 2 Some definitions and examples 2 3 Elementary row operations 7 4 Gaussian elimination 11 5 Rank and row reduction 16 6 Some computational

### Boolean Algebras. Chapter 2

Chapter 2 Boolean Algebras Let X be an arbitrary set and let P(X) be the class of all subsets of X (the power set of X). Three natural set-theoretic operations on P(X) are the binary operations of union

### Linear Algebra Notes. Lecture Notes, University of Toronto, Fall 2016

Linear Algebra Notes Lecture Notes, University of Toronto, Fall 2016 (Ctd ) 11 Isomorphisms 1 Linear maps Definition 11 An invertible linear map T : V W is called a linear isomorphism from V to W Etymology:

### MATH 369 Linear Algebra

Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine

### MATH 201 Solutions: TEST 3-A (in class)

MATH 201 Solutions: TEST 3-A (in class) (revised) God created infinity, and man, unable to understand infinity, had to invent finite sets. - Gian Carlo Rota Part I [5 pts each] 1. Let X be a set. Define

### WUCT121. Discrete Mathematics. Logic. Tutorial Exercises

WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the

### MAT 243 Test 1 SOLUTIONS, FORM A

t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,

### INFINITY: CARDINAL NUMBERS

INFINITY: CARDINAL NUMBERS BJORN POONEN 1 Some terminology of set theory N := {0, 1, 2, 3, } Z := {, 2, 1, 0, 1, 2, } Q := the set of rational numbers R := the set of real numbers C := the set of complex

### Handout #6 INTRODUCTION TO ALGEBRAIC STRUCTURES: Prof. Moseley AN ALGEBRAIC FIELD

Handout #6 INTRODUCTION TO ALGEBRAIC STRUCTURES: Prof. Moseley Chap. 2 AN ALGEBRAIC FIELD To introduce the notion of an abstract algebraic structure we consider (algebraic) fields. (These should not to

### Integer-Valued Polynomials

Integer-Valued Polynomials LA Math Circle High School II Dillon Zhi October 11, 2015 1 Introduction Some polynomials take integer values p(x) for all integers x. The obvious examples are the ones where

### cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 1 Course Web Page www3.cs.stonybrook.edu/ cse303 The webpage contains: lectures notes slides; very detailed solutions to

### 17. C M 2 (C), the set of all 2 2 matrices with complex entries. 19. Is C 3 a real vector space? Explain.

250 CHAPTER 4 Vector Spaces 14. On R 2, define the operation of addition by (x 1,y 1 ) + (x 2,y 2 ) = (x 1 x 2,y 1 y 2 ). Do axioms A5 and A6 in the definition of a vector space hold? Justify your answer.

### Foundations Revision Notes

oundations Revision Notes hese notes are designed as an aid not a substitute for revision. A lot of proofs have not been included because you should have them in your notes, should you need them. Also,

### Criteria for Determining If A Subset is a Subspace

These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

### Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)

Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational

### Math 13, Spring 2013, Lecture B: Midterm

Math 13, Spring 2013, Lecture B: Midterm Name Signature UCI ID # E-mail address Each numbered problem is worth 12 points, for a total of 84 points. Present your work, especially proofs, as clearly as possible.

### Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.

Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary

### HANDOUT AND SET THEORY. Ariyadi Wijaya

HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics

### a. Define a function called an inner product on pairs of points x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ) in R n by

Real Analysis Homework 1 Solutions 1. Show that R n with the usual euclidean distance is a metric space. Items a-c will guide you through the proof. a. Define a function called an inner product on pairs

### 2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a.

Chapter 2 Groups Groups are the central objects of algebra. In later chapters we will define rings and modules and see that they are special cases of groups. Also ring homomorphisms and module homomorphisms

Mathematical Olympiad Training Polynomials Definition A polynomial over a ring R(Z, Q, R, C) in x is an expression of the form p(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0, a i R, for 0 i n. If a n 0,

### Math 24 Spring 2012 Questions (mostly) from the Textbook

Math 24 Spring 2012 Questions (mostly) from the Textbook 1. TRUE OR FALSE? (a) The zero vector space has no basis. (F) (b) Every vector space that is generated by a finite set has a basis. (c) Every vector

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### 8. Limit Laws. lim(f g)(x) = lim f(x) lim g(x), (x) = lim x a f(x) g lim x a g(x)

8. Limit Laws 8.1. Basic Limit Laws. If f and g are two functions and we know the it of each of them at a given point a, then we can easily compute the it at a of their sum, difference, product, constant

### Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.

The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or

### The Integers. Math 3040: Spring Contents 1. The Basic Construction 1 2. Adding integers 4 3. Ordering integers Multiplying integers 12

Math 3040: Spring 2011 The Integers Contents 1. The Basic Construction 1 2. Adding integers 4 3. Ordering integers 11 4. Multiplying integers 12 Before we begin the mathematics of this section, it is worth

### HOW TO CREATE A PROOF. Writing proofs is typically not a straightforward, algorithmic process such as calculating

HOW TO CREATE A PROOF ALLAN YASHINSKI Abstract We discuss how to structure a proof based on the statement being proved Writing proofs is typically not a straightforward, algorithmic process such as calculating

### Vector Spaces - Definition

Vector Spaces - Definition Definition Let V be a set of vectors equipped with two operations: vector addition and scalar multiplication. Then V is called a vector space if for all vectors u,v V, the following

### Mathmatics 239 solutions to Homework for Chapter 2

Mathmatics 239 solutions to Homework for Chapter 2 Old version of 8.5 My compact disc player has space for 5 CDs; there are five trays numbered 1 through 5 into which I load the CDs. I own 100 CDs. a)

### Some Background Material

Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

### Why study probability? Set theory. ECE 6010 Lecture 1 Introduction; Review of Random Variables

ECE 6010 Lecture 1 Introduction; Review of Random Variables Readings from G&S: Chapter 1. Section 2.1, Section 2.3, Section 2.4, Section 3.1, Section 3.2, Section 3.5, Section 4.1, Section 4.2, Section

### Calculating determinants for larger matrices

Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

### Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009

Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009 Our main goal is here is to do counting using functions. For that, we

### CS Discrete Mathematics Dr. D. Manivannan (Mani)

CS 275 - Discrete Mathematics Dr. D. Manivannan (Mani) Department of Computer Science University of Kentucky Lexington, KY 40506 Course Website: www.cs.uky.edu/~manivann/cs275 Notes based on Discrete Mathematics

### Notes on Linear Algebra I. # 1

Notes on Linear Algebra I. # 1 Oussama Moutaoikil Contents 1 Introduction 1 2 On Vector Spaces 5 2.1 Vectors................................... 5 2.2 Vector Spaces................................ 7 2.3

### Packet #1: Logic & Proofs. Applied Discrete Mathematics

Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should

### UMA Putnam Talk LINEAR ALGEBRA TRICKS FOR THE PUTNAM

UMA Putnam Talk LINEAR ALGEBRA TRICKS FOR THE PUTNAM YUFEI ZHAO In this talk, I want give some examples to show you some linear algebra tricks for the Putnam. Many of you probably did math contests in

### Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )

Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O

### NONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction

NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques

### Mathematics Department Stanford University Math 61CM/DM Vector spaces and linear maps

Mathematics Department Stanford University Math 61CM/DM Vector spaces and linear maps We start with the definition of a vector space; you can find this in Section A.8 of the text (over R, but it works

### CS2742 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

### 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

### An Introduction to Mathematical Reasoning

An Introduction to Mathematical Reasoning Matthew M. Conroy and Jennifer L. Taggart University of Washington 2 Version: December 28, 2016 Contents 1 Preliminaries 7 1.1 Axioms and elementary properties

### Chapter 1 Vector Spaces

Chapter 1 Vector Spaces Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field

### NOTES (1) FOR MATH 375, FALL 2012

NOTES 1) FOR MATH 375, FALL 2012 1 Vector Spaces 11 Axioms Linear algebra grows out of the problem of solving simultaneous systems of linear equations such as 3x + 2y = 5, 111) x 3y = 9, or 2x + 3y z =

### Rings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.

Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over

### Real Analysis - Notes and After Notes Fall 2008

Real Analysis - Notes and After Notes Fall 2008 October 29, 2008 1 Introduction into proof August 20, 2008 First we will go through some simple proofs to learn how one writes a rigorous proof. Let start

### ABSTRACT VECTOR SPACES AND THE CONCEPT OF ISOMORPHISM. Executive summary

ABSTRACT VECTOR SPACES AND THE CONCEPT OF ISOMORPHISM MATH 196, SECTION 57 (VIPUL NAIK) Corresponding material in the book: Sections 4.1 and 4.2. General stuff... Executive summary (1) There is an abstract

### Unit 1: Equations & Inequalities in One Variable

Date Period Unit 1: Equations & Inequalities in One Variable Day Topic 1 Properties of Real Numbers Algebraic Expressions Solving Equations 3 Solving Inequalities 4 QUIZ 5 Absolute Value Equations 6 Double

### 3 The language of proof

3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;

### MATH 2200 Final Review

MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics

### Sanjay Mishra. Topology. Dr. Sanjay Mishra. A Profound Subtitle

Topology A Profound Subtitle Dr. Copyright c 2017 Contents I General Topology 1 Compactness of Topological Space............................ 7 1.1 Introduction 7 1.2 Compact Space 7 1.2.1 Compact Space.................................................

### Ch 7 Summary - POLYNOMIAL FUNCTIONS

Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)

### An Introduction to University Level Mathematics

An Introduction to University Level Mathematics Alan Lauder May 22, 2017 First, a word about sets. These are the most primitive objects in mathematics, so primitive in fact that it is not possible to give

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. College Algebra for STEM

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics College Algebra for STEM Marcel B. Finan c All Rights Reserved 2015 Edition To my children Amin & Nadia Preface From