Lecture 3: the classification of equivalence relations and the definition of a topological space
|
|
- Cornelius Sherman
- 5 years ago
- Views:
Transcription
1 Lecture 3: the classification of equivalence relations and the definition of a topological space Saul Glasman September 12, 2016 If there s a bijection f : X Y, we ll often say that X and Y are in bijection, and we ll act kind of as if X and Y are the same, since the elements of Y are in one-to-one correspondence with the elements of X; they re just the elements of X under different names. Now on the first day I promised a result on equivalence relations; today I ll be able to give the statement and the proof. After that, we ll get started on topological spaces. Let X be a set. We can form the set of all equivalence relations on X - equivalence relations are things, we can put them in a set. Call this set Eq(X). Definition 1. A partition of X is a collection (X i ) i I of subsets of X such that if i j, then X i X j =, and the union X i = X. i I In other words, each x X is in exactly one of the X i. Don t be scared of the notation (X i ) i I ; the point is that we have a big list of subsets of X, and we have to give them names, so we index them with tags from some convenient index set I. It doesn t matter exactly what I is; it just has to be the right size of set to index our collection. Now we can form the set of all partitions of X - call it Part(X). Then we have the following theorem: Theorem 2. The sets Eq(X) and Part(X) are in bijection. Remark 3. I m showing you this theorem partly because it s interesting, and partly to demonstrate that set theory can get meta : questions about the nature of sets can themselves be treated using set theory. This theorem and its proof will serve as a warm-up for the kind of level of abstraction we ll be encountering in this course. 1
2 Proof. In view of the theorem I proved last time, if I claim two sets are in bijection, there are two ways I can prove that: I can give a function one way and show that it s injective and surjective, or I can give functions both ways and show they re inverse to one another. Here we ll do the latter. First I ll prove some auxiliary results. If R is an equivalence relation on X and x X, then the equivalence class of x, EC R (x) X is defined as the set of x X such that xrx. Lemma 4. If xrx, then EC R (x ) = EC R (x). Proof. Suppose z EC R (x). Then xrx and x Rz. By transitivity, which was one of our axioms for equivalence relations, xrz. We deduce that EC R (x ) EC R (x). But by symmetry, which was another of our axioms, we have x Rx. So the same reasoning shows that EC R (x) EC R (x ). Each subset is included in the other, so the two must be equal. by Now let s define φ : Eq(X) Part(X) φ(r) = {EC R (x) x X}. Notice we ve overcounted a lot here: each equivalence class is counted separately for each element of that equivalence class. But that doesn t matter. We first have to prove that φ(r) is really a partition; otherwise the codomain of φ isn t what I claimed it is. That is, we have to show that each element x X is in exactly one equivalence class. But if an equivalence class C contains x, then by the lemma, C must be the equivalence class of x, so x is in at most one equivalence class. Since xrx, x is in its own equivalence class. We deduce that x is in exactly one equivalence class, and φ(r) is really a partition. Now define ψ : Part(X) Eq(X) by setting, for a partition ψ(p) = R P, where P = {(X i ) i I } xr P x if x X i, x X i for some i. Now we have to show that R P is an equivalence relation. But I actually gave this proof last week when I proved that landmasses give an equivalence relation! So I ll omit this in lecture, and for those of you who weren t here on the first day I ll leave it as an exercise, but for completeness, I ll give it here. 2
3 1. Each x is in some partition X i. So xr P x. 2. If x X i, y X i, and y X j, z X j, then i = j, because P is a partition. But then z X i, so xr P z. 3. If x X i, y X i, then y X i, x X i. So now we have our well-defined functions and φ : Eq(X) Part(X) ψ : Part(X) Eq(X). We ll now show that φ and ψ are mutually inverse. This is actually really clear, once you unpack the notation, but I ll go through it for completeness. First, let s show that ψ φ = id. (I ll stop writing the subscript on id when the context is clear.) This is the same as saying that by taking the equivalence relation associated to the equivalences classes, you get the same equivalence relation - that is, xrx if and only if they re in the same equivalence class for R. That, we already know. Now let s show that φ ψ = id. This is the same as saying that the equivalence classes for the equivalence relation associated to a partition are exactly the sets of that partition - that is, x and x are in the same equivalence class for ψ(p) if and only if there is some X i for which both x X i, x X i. This, too, we know; it s just the definition of ψ. So we re done! Finally. I apologize for the tedium and the length of the proof, but hopefully each step was clear individually. If you didn t follow the overall structure of the proof, I highly recommend you look over it in the notes; there will be plenty of proofs in the course which are at least this complicated. Now let s really begin the course! This subject truly starts with the definition of a topological space, and that s what we re about to cover. First a little bit of background. Topology grew out of the study of continuous functions. You ve probably encountered the definition of a continuous function from R to R using δs and ɛs (if not, don t worry, you will soon.) This definition is fine, but it does rely on the real numbers having a well-defined notion of distance - that is, the absolute difference between two real numbers. As mathematics grew in abstraction, the need for an idea of continuity that would generalize to contexts where notions of distance were unavailable or irrelevant became clear. The crucial insight was that the definition of continuity could be reformulated in terms of open sets - we ll see how this works within the next few classes. This led to an abstract, set-theoretic definition of topological space making reference only to a class of open sets. Before giving this definition, let me informally set up some intuition about what an open set should be in the context of R. I m not going to give you the precise definition of an open set just yet. Loosely speaking, an open set is a set 3
4 without any sharp edges - a set you can t cut yourself on. The archetypal example of an open set is the open interval (a, b) = {x R a < x < b}. Another example of an open set is the entire set R. Sets which are not open include the half-open interval (a, b] = {x R a < x b}, which has one sharp edge, and the closed interval [a, b] = {x R a x b}, which has two sharp edges. With this in mind, let s get to the definition of a topology: Definition 5. Let X be a set. A topology on X is a collection T of subsets of X, known as open sets satisfying the following conditions: 1. T. 2. X T. 3. Any union of elements of T is in T. 4. A finite intersection of elements of T is in T. Why do we allow arbitrary unions but only finite intersections? Consider the following example: If n is a positive integer, let I n R be the open interval (0, n ). Now let s take the intersection of all of these open sets: I = I n. n=1 What are the elements of I? Well, they re the real numbers which are greater than 0 but which are less than 1 + 1/n for all n. So certainly, Also, 1 I. But if r > 1 - say (0, 1) I. r = then there s some n such that r > 1 + 1/n, right? So I = (0, 1]. Oops - that s not open any more. If we d only taken a finite intersection, though, it would have been fine. 4
5 So that s the definition of a topology. The pair (X, T ) is called a topological space. Let s give a few easy examples. Example 6. Let X be a set. Then the discrete topology on X is the set of all subsets of X: T = P(X) (P(X) is the notation for the set of all subsets of X; it s called the power set, in case you didn t know.) It s clear that the discrete topology satisfies the axioms for a topology. Example 7. Let X be a set. Then the indiscrete topology on X is the topology for which only and X are open: T = {, X}. Again, it s clear that the indiscrete topology satisfies the axioms for a topology: the only nontrivial collection of elements of T is {, X}, and X = X, X =. Let s give the definition of an open subset of R. Definition 8. Let U R. We say U is open if for every r U, r is contained in an open interval which is contained in U; that is, for every r in U, there are positive numbers ɛ 0, ɛ 1 > 0 such that (r ɛ 0, r + ɛ 1 ) U. Proposition 9. With this definition, the open subsets of R form a topology on R. We ll deduce this result as a special case of a more general theorem next time. 5
Lecture 5: closed sets, and an introduction to continuous functions
Lecture 5: closed sets, and an introduction to continuous functions Saul Glasman September 16, 2016 Clarification on URL. To warm up today, let s talk about one more example of a topology. Definition 1.
More informationSeminaar 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 informationCosets and Lagrange s theorem
Cosets and Lagrange s theorem These are notes on cosets and Lagrange s theorem some of which may already have been lecturer. There are some questions for you included in the text. You should write the
More informationCommutative Algebra MAS439 Lecture 3: Subrings
Commutative Algebra MAS439 Lecture 3: Subrings Paul Johnson paul.johnson@sheffield.ac.uk Hicks J06b October 4th Plan: slow down a little Last week - Didn t finish Course policies + philosophy Sections
More informationLecture 4: Constructing the Integers, Rationals and Reals
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 4: Constructing the Integers, Rationals and Reals Week 5 UCSB 204 The Integers Normally, using the natural numbers, you can easily define
More informationMA554 Assessment 1 Cosets and Lagrange s theorem
MA554 Assessment 1 Cosets and Lagrange s theorem These are notes on cosets and Lagrange s theorem; they go over some material from the lectures again, and they have some new material it is all examinable,
More informationMetric spaces and metrizability
1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively
More informationLecture 3: Sizes of Infinity
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 3: Sizes of Infinity Week 2 UCSB 204 Sizes of Infinity On one hand, we know that the real numbers contain more elements than the rational
More informationSequence convergence, the weak T-axioms, and first countability
Sequence convergence, the weak T-axioms, and first countability 1 Motivation Up to now we have been mentioning the notion of sequence convergence without actually defining it. So in this section we will
More informationReading 11 : Relations and Functions
CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Reading 11 : Relations and Functions Instructor: Beck Hasti and Gautam Prakriya In reading 3, we described a correspondence between predicates
More information6 Cosets & Factor Groups
6 Cosets & Factor Groups The course becomes markedly more abstract at this point. Our primary goal is to break apart a group into subsets such that the set of subsets inherits a natural group structure.
More informationUMASS 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 informationCONSTRUCTION OF THE REAL NUMBERS.
CONSTRUCTION OF THE REAL NUMBERS. IAN KIMING 1. Motivation. It will not come as a big surprise to anyone when I say that we need the real numbers in mathematics. More to the point, we need to be able to
More informationGuide to Proofs on Discrete Structures
CS103 Handout 17 Spring 2018 Guide to Proofs on Discrete Structures In Problem Set One, you got practice with the art of proofwriting in general (as applied to numbers, sets, puzzles, etc.) Problem Set
More information1 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
More information2 Metric Spaces Definitions Exotic Examples... 3
Contents 1 Vector Spaces and Norms 1 2 Metric Spaces 2 2.1 Definitions.......................................... 2 2.2 Exotic Examples...................................... 3 3 Topologies 4 3.1 Open Sets..........................................
More informationABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS.
ABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS. ANDREW SALCH 1. Subgroups, conjugacy, normality. I think you already know what a subgroup is: Definition
More information2.23 Theorem. Let A and B be sets in a metric space. If A B, then L(A) L(B).
2.23 Theorem. Let A and B be sets in a metric space. If A B, then L(A) L(B). 2.24 Theorem. Let A and B be sets in a metric space. Then L(A B) = L(A) L(B). It is worth noting that you can t replace union
More informationEQUIVALENCE RELATIONS (NOTES FOR STUDENTS) 1. RELATIONS
EQUIVALENCE RELATIONS (NOTES FOR STUDENTS) LIOR SILBERMAN Version 1.0 compiled September 9, 2015. 1.1. List of examples. 1. RELATIONS Equality of real numbers: for some x,y R we have x = y. For other pairs
More informationWe have been going places in the car of calculus for years, but this analysis course is about how the car actually works.
Analysis I We have been going places in the car of calculus for years, but this analysis course is about how the car actually works. Copier s Message These notes may contain errors. In fact, they almost
More informationConnectedness. 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 informationWriting proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases
Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases September 22, 2018 Recall from last week that the purpose of a proof
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday
More informationFunctions and cardinality (solutions) sections A and F TA: Clive Newstead 6 th May 2014
Functions and cardinality (solutions) 21-127 sections A and F TA: Clive Newstead 6 th May 2014 What follows is a somewhat hastily written collection of solutions for my review sheet. I have omitted some
More informationCounting Colorings Cleverly
Counting Colorings Cleverly by Zev Chonoles How many ways are there to color a shape? Of course, the answer depends on the number of colors we re allowed to use. More fundamentally, the answer depends
More informationLecture 3: Latin Squares and Groups
Latin Squares Instructor: Padraic Bartlett Lecture 3: Latin Squares and Groups Week 2 Mathcamp 2012 In our last lecture, we came up with some fairly surprising connections between finite fields and Latin
More informationHandout 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 informationLecture 6: Finite Fields
CCS Discrete Math I Professor: Padraic Bartlett Lecture 6: Finite Fields Week 6 UCSB 2014 It ain t what they call you, it s what you answer to. W. C. Fields 1 Fields In the next two weeks, we re going
More information2. 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
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 informationChapter 2. Mathematical Reasoning. 2.1 Mathematical Models
Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................
More informationSemidirect products are split short exact sequences
CHAPTER 16 Semidirect products are split short exact sequences Chit-chat 16.1. Last time we talked about short exact sequences G H K. To make things easier to read, from now on we ll write L H R. The L
More informationCountability. 1 Motivation. 2 Counting
Countability 1 Motivation In topology as well as other areas of mathematics, we deal with a lot of infinite sets. However, as we will gradually discover, some infinite sets are bigger than others. Countably
More informationMath Lecture 23 Notes
Math 1010 - Lecture 23 Notes Dylan Zwick Fall 2009 In today s lecture we ll expand upon the concept of radicals and radical expressions, and discuss how we can deal with equations involving these radical
More informationDiscrete Structures Proofwriting Checklist
CS103 Winter 2019 Discrete Structures Proofwriting Checklist Cynthia Lee Keith Schwarz Now that we re transitioning to writing proofs about discrete structures like binary relations, functions, and graphs,
More informationmeans is a subset of. So we say A B for sets A and B if x A we have x B holds. BY CONTRAST, a S means that a is a member of S.
1 Notation For those unfamiliar, we have := means equal by definition, N := {0, 1,... } or {1, 2,... } depending on context. (i.e. N is the set or collection of counting numbers.) In addition, means for
More informationMATH 13 FINAL EXAM SOLUTIONS
MATH 13 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. T F
More informationCM10196 Topic 2: Sets, Predicates, Boolean algebras
CM10196 Topic 2: Sets, Predicates, oolean algebras Guy McCusker 1W2.1 Sets Most of the things mathematicians talk about are built out of sets. The idea of a set is a simple one: a set is just a collection
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 informationSpanning, linear dependence, dimension
Spanning, linear dependence, dimension In the crudest possible measure of these things, the real line R and the plane R have the same size (and so does 3-space, R 3 ) That is, there is a function between
More informationABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH
ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH 1. Homomorphisms and isomorphisms between groups. Definition 1.1. Let G, H be groups.
More informationBasic Probability. Introduction
Basic Probability Introduction The world is an uncertain place. Making predictions about something as seemingly mundane as tomorrow s weather, for example, is actually quite a difficult task. Even with
More informationNondeterministic finite automata
Lecture 3 Nondeterministic finite automata This lecture is focused on the nondeterministic finite automata (NFA) model and its relationship to the DFA model. Nondeterminism is an important concept in the
More informationBuilding Infinite Processes from Finite-Dimensional Distributions
Chapter 2 Building Infinite Processes from Finite-Dimensional Distributions Section 2.1 introduces the finite-dimensional distributions of a stochastic process, and shows how they determine its infinite-dimensional
More informationSets, 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 informationLecture 10: Everything Else
Math 94 Professor: Padraic Bartlett Lecture 10: Everything Else Week 10 UCSB 2015 This is the tenth week of the Mathematics Subject Test GRE prep course; here, we quickly review a handful of useful concepts
More information1. Continuous Functions between Euclidean spaces
Math 441 Topology Fall 2012 Metric Spaces by John M. Lee This handout should be read between Chapters 1 and 2 of the text. It incorporates material from notes originally prepared by Steve Mitchell and
More informationABOUT THE CLASS AND NOTES ON SET THEORY
ABOUT THE CLASS AND NOTES ON SET THEORY About the Class Evaluation. Final grade will be based 25%, 25%, 25%, 25%, on homework, midterm 1, midterm 2, final exam. Exam dates. Midterm 1: Oct 4. Midterm 2:
More information2. Introduction to commutative rings (continued)
2. Introduction to commutative rings (continued) 2.1. New examples of commutative rings. Recall that in the first lecture we defined the notions of commutative rings and field and gave some examples of
More informationALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.
ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add
More informationWhat if the characteristic equation has a double root?
MA 360 Lecture 17 - Summary of Recurrence Relations Friday, November 30, 018. Objectives: Prove basic facts about basic recurrence relations. Last time, we looked at the relational formula for a sequence
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationMath 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
More informationDiscrete 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
More informationCMPSCI 250: Introduction to Computation. Lecture #11: Equivalence Relations David Mix Barrington 27 September 2013
CMPSCI 250: Introduction to Computation Lecture #11: Equivalence Relations David Mix Barrington 27 September 2013 Equivalence Relations Definition of Equivalence Relations Two More Examples: Universal
More informationMATH ABSTRACT ALGEBRA DISCUSSIONS - WEEK 8
MAT 410 - ABSTRACT ALEBRA DISCUSSIONS - WEEK 8 CAN OZAN OUZ 1. Isomorphism Theorems In group theory, there are three main isomorphism theorems. They all follow from the first isomorphism theorem. Let s
More informationADVANCED 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 informationIsomorphisms and Well-definedness
Isomorphisms and Well-definedness Jonathan Love October 30, 2016 Suppose you want to show that two groups G and H are isomorphic. There are a couple of ways to go about doing this depending on the situation,
More informationLogic and Mathematics:
Logic and Mathematics: Mathematicians in Schools Program Lashi Bandara Mathematical Sciences Institute, Australian National University April 21, 2011 Contents 1 Russell s Paradox 1 2 Propositional Logic
More informationCITS2211 Discrete Structures (2017) Cardinality and Countability
CITS2211 Discrete Structures (2017) Cardinality and Countability Highlights What is cardinality? Is it the same as size? Types of cardinality and infinite sets Reading Sections 45 and 81 84 of Mathematics
More informationTheorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1)
Week 1: Logic Lecture 1, 8/1 (Sections 1.1 and 1.3) Examples of theorems and proofs Theorem (Pythagoras). Let ABC be a right triangle, with legs of lengths a and b, and hypotenuse of length c. Then a +
More informationLecture 8: Equivalence Relations
Lecture 8: Equivalence Relations 1 Equivalence Relations Next interesting relation we will study is equivalence relation. Definition 1.1 (Equivalence Relation). Let A be a set and let be a relation on
More informationABSTRACT ALGEBRA 1, LECTURE NOTES 4: DEFINITIONS AND EXAMPLES OF MONOIDS AND GROUPS.
ABSTRACT ALGEBRA 1, LECTURE NOTES 4: DEFINITIONS AND EXAMPLES OF MONOIDS AND GROUPS. ANDREW SALCH 1. Monoids. Definition 1.1. A monoid is a set M together with a function µ : M M M satisfying the following
More informationWhat is proof? Lesson 1
What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might
More informationINFINITY: 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
More informationA 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 informationThe integers. Chapter 3
Chapter 3 The integers Recall that an abelian group is a set A with a special element 0, and operation + such that x +0=x x + y = y + x x +y + z) =x + y)+z every element x has an inverse x + y =0 We also
More informationMeta-logic derivation rules
Meta-logic derivation rules Hans Halvorson February 19, 2013 Recall that the goal of this course is to learn how to prove things about (as opposed to by means of ) classical first-order logic. So, we will
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationMATH 433 Applied Algebra Lecture 14: Functions. Relations.
MATH 433 Applied Algebra Lecture 14: Functions. Relations. Cartesian product Definition. The Cartesian product X Y of two sets X and Y is the set of all ordered pairs (x,y) such that x X and y Y. The Cartesian
More informationNets and filters (are better than sequences)
Nets and filters (are better than sequences) Contents 1 Motivation 2 2 More implications we wish would reverse 2 3 Nets 4 4 Subnets 6 5 Filters 9 6 The connection between nets and filters 12 7 The payoff
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 informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset
More informationWriting proofs. Tim Hsu, San José State University. May 31, Definitions and theorems 3. 2 What is a proof? 3. 3 A word about definitions 4
Writing proofs Tim Hsu, San José State University May 31, 2006 Contents I Fundamentals 3 1 Definitions and theorems 3 2 What is a proof? 3 3 A word about definitions 4 II The structure of proofs 6 4 Assumptions
More informationDefinitions: A binary relation R on a set X is (a) reflexive if x X : xrx; (f) asymmetric if x, x X : [x Rx xr c x ]
Binary Relations Definition: A binary relation between two sets X and Y (or between the elements of X and Y ) is a subset of X Y i.e., is a set of ordered pairs (x, y) X Y. If R is a relation between X
More informationFoundations of algebra
Foundations of algebra Equivalence relations - suggested problems - solutions P1: There are several relations that you are familiar with: Relations on R (or any of its subsets): Equality. Symbol: x = y.
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More information1 k x k. d(x, y) =sup k. y k = max
1 Lecture 13: October 8 Urysohn s metrization theorem. Today, I want to explain some applications of Urysohn s lemma. The first one has to do with the problem of characterizing metric spaces among all
More informationTopology Math Conrad Plaut
Topology Math 467 2010 Conrad Plaut Contents Chapter 1. Background 1 1. Set Theory 1 2. Finite and Infinite Sets 3 3. Indexed Collections of Sets 4 Chapter 2. Topology of R and Beyond 7 1. The Topology
More informationarxiv: v1 [math.ac] 25 Jul 2017
Primary Decomposition in Boolean Rings David C. Vella, Skidmore College arxiv:1707.07783v1 [math.ac] 25 Jul 2017 1. Introduction Let R be a commutative ring with identity. The Lasker- Noether theorem on
More informationSpectral Graph Theory Lecture 2. The Laplacian. Daniel A. Spielman September 4, x T M x. ψ i = arg min
Spectral Graph Theory Lecture 2 The Laplacian Daniel A. Spielman September 4, 2015 Disclaimer These notes are not necessarily an accurate representation of what happened in class. The notes written before
More informationSymmetries and Polynomials
Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we ll learn how to solve a cubic. We ll also sketch how to solve a quartic. We ll explore the connections
More informationCSCI3390-Lecture 6: An Undecidable Problem
CSCI3390-Lecture 6: An Undecidable Problem September 21, 2018 1 Summary The language L T M recognized by the universal Turing machine is not decidable. Thus there is no algorithm that determines, yes or
More informationEconomics 204 Summer/Fall 2017 Lecture 1 Monday July 17, 2017
Economics 04 Summer/Fall 07 Lecture Monday July 7, 07 Section.. Methods of Proof We begin by looking at the notion of proof. What is a proof? Proof has a formal definition in mathematical logic, and a
More informationMath 31 Lesson Plan. Day 2: Sets; Binary Operations. Elizabeth Gillaspy. September 23, 2011
Math 31 Lesson Plan Day 2: Sets; Binary Operations Elizabeth Gillaspy September 23, 2011 Supplies needed: 30 worksheets. Scratch paper? Sign in sheet Goals for myself: Tell them what you re going to tell
More information0 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 informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationHOW 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
More informationMATH 22 FUNCTIONS: COMPOSITION & INVERSES. Lecture N: 10/16/2003. Mad world! mad kings! mad composition! Shakespeare, King John, II:1
MATH 22 Lecture N: 10/16/2003 FUNCTIONS: COMPOSITION & INVERSES Mad world! mad kings! mad composition! Shakespeare, King John, II:1 Copyright 2003 Larry Denenberg Administrivia http://denenberg.com/lecturen.pdf
More informationShort Introduction to Admissible Recursion Theory
Short Introduction to Admissible Recursion Theory Rachael Alvir November 2016 1 Axioms of KP and Admissible Sets An admissible set is a transitive set A satisfying the axioms of Kripke-Platek Set Theory
More informationJOURNAL OF INQUIRY-BASED LEARNING IN MATHEMATICS
Edited: 4:18pm, July 17, 2012 JOURNAL OF INQUIRY-BASED LEARNING IN MATHEMATICS Analysis 1 W. Ted Mahavier < W. S. Mahavier Lamar University 1 If each of X and Y is a person, then the relation X < Y indicates
More informationScott 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 informationACCESS 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 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 informationSupplementary Material for MTH 299 Online Edition
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
More informationa (b + c) = a b + a c
Chapter 1 Vector spaces In the Linear Algebra I module, we encountered two kinds of vector space, namely real and complex. The real numbers and the complex numbers are both examples of an algebraic structure
More informationLECTURE 5, FRIDAY
LECTURE 5, FRIDAY 20.02.04 FRANZ LEMMERMEYER Before we start with the arithmetic of elliptic curves, let us talk a little bit about multiplicities, tangents, and singular points. 1. Tangents How do we
More information5 Set Operations, Functions, and Counting
5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,
More information