Chapter 4. Basic Set Theory. 4.1 The Language of Set Theory


 Philomena Hutchinson
 7 months ago
 Views:
Transcription
1 Chapter 4 Basic Set Theory There are two good reasons for studying set theory. First, it s a indispensable tool for both logic and mathematics, and even for other fields including computer science, linguistics, and so on. But there is also a second reason it s philosophically interesting. For many, the philosophy of mathematics is really just the philosophy of set theory. This is probably a mistaken view, but it is not wholly implausible, since there seems to be counterparts in set theory for all mathematical objects (i.e., numbers are sets, functions are sets, integrals and derivatives are sets, and so on). In any case, familiarity with set theory is essential. 4.1 The Language of Set Theory There are two basic notions, set and member. Strictly speaking these must remain undefined, since otherwise we would end up giving definitions that are either circular or lead to an infinite regress. Nevertheless, we can step outside of the theory and give what Frege called an explication of the concepts. Things such as a flock of birds or a pack of wolves are sets. The individual birds and wolves are members of the sets. Examples such as these help us to get a grip on set and member. But beware. A pack of wolves has a location, whereas a set does not. Sets are abstract entities that exist outside of spacetime. They have no mass, or charge, or shape, or any other physical property. Sometimes it s useful to distinguish a set of bricks that make up a wall from the physical collection of those bricks that make up the wall. The former is located no where and won t harm you, while the latter can give you 27
2 CHAPTER 4. BASIC SET THEORY 28 a nasty bump if you run into it. We will use the symbol for the membership relation, and use A, B, C,..., {},a,b,c,... for both sets and their members. Curley brackets are a common way to specify sets. Thus, {a, b} is the set with the two members a and b. We will define many more important settheoretical concepts as we move along. Though it is convenient to use upper case letters for sets and lower case for members, sets can be members of other sets; so A B,B a, and a b are perfectly legitimate. We will also use the standard symbols from logic, such as,,,, and so on. We adopt the same formation rules as before, with the following additions. When A is a set and a is a member of it, then the following are wffs: a A, {a} = A, and so on. Exercise 11 Which of the following are wffs? (Don t worry about which are true.) 1. a b 2. A a 3. a {b, c} 4. {b, c} a 5. x X(x X) 6. a/ A (this means (a A)) 7. A 8. a B b A 9. {a, b} = x(x {b, a}) HINTS & ANSWERS 1. yes 2. yes (don t be misled by the letters, A could be a member and a aset) 3. yes
3 CHAPTER 4. BASIC SET THEORY yes 5. yes (we ll let quantifiers range over set variables such as X) 6. yes 7. no (here A is a set, not a sentence) 8. yes 9. no (the left side is a set, the right side is a sentence) 4.2 Axioms Axiom 1 (Extensionality): Two sets are identical, if and only if they have the same members. Symbolically: A B x((x A x B) A = B)) This axiom implies the following: a {a}, {a, b} = {b, a} (order is irrelevant), {a} = {a, a, a, a, a} (there is just one yhing is in the set; repeating it is merely redundant), b 6= a b/ {a}, anda 6= {a} 6= {{a}} 6= {{{a}}} (each of these is a different entity). Axiom 2 (Empty set): There is a set which has no members. A x x/ A (It is usually called the empty set, or the null set, and is denoted φ.) Each of the following is equal to the empty set: φ = the set of humans over 10 feet tall = the set of unicorns = the set of even prime numbers that are greater than 2. You shouldn t think of these as different sets that happen to be empty. There is really only one empty set (and these are different ways of describing it). Theorem 1 The empty set is unique. Proof. Suppose that φ 1 and φ 2 are both empty sets. Because an empty set has no members, it follows that x φ 1 x φ 2. Given the axiom of extensionality, it follows that φ 1 = φ 2. Axiom 3 (Pairing): For any a and b, there is a set which has both of them and nothing else as members. a b A x(x A x = a x = b)
4 CHAPTER 4. BASIC SET THEORY 30 Given that Bob exists and the proposition P exists, then there exists a set containing both of them, {Bob, P}. We can use the axiom again. given that Bob exists and the set {Bob, P} exists, then there exists a set {Bob, {Bob, P}}. Obviously, we can build up all sorts of new sets in this way. Axiom 4 (Union): For any sets A and B, there is a set C whose members are exactly the members of A and B. A B C x (x C x A x B) For example, if A = {a, b, c} and B = {1, 2}, thenthereisasetc, such that C = {a, b, c, 1, 2}. Notation 1 {x : P (x)} means the set of all x such that the condition P(x) holds. (For example, {x : x is red} is the set of all red things; {y : y Z y>27} is the set of all integers that are greater than 27.) Axiom 5 (Comprehension (also know as specification, separation, and aussonderung)): Let U be any set. Then the members of U that satisfy a condition P determine a set. A(A = {x : x U P (x)} If we start with the class members as the set U, andletf mean is a woman and P mean is a philosophy student, then {x : x U Fx Px} is the set of women philosophy students in the class. Exercise 12 Let A = {1, 2, 3,...},B = {x : x is a prime number},c = {a, b, c,..., z},d = x : x is an even number},e = x : x is an odd number}. 1. What is the result of applying the Union Axion to E and D? 2. What is the result of applying the Pairing Axiom to E and D? 3. What is {x : x A x<8}? 4. What is {x : x B x C}? 5. What is {x : x D x E}? ANSWERS: 1. E D = A
5 CHAPTER 4. BASIC SET THEORY {E,D} 3. {1, 2, 3, 4, 5, 6, 7} 4. {a, b, c,..., z, 2, 3, 5, 7, 11, 13,...} 5. A 4.3 Russell s Paradox Let s take a moment to look at Russell s paradox. The axiom of comprehension has the slightly odd form that it does have in order to get around this problem. The current axiom is due to Zermelo, who proposed it to block a number of paradoxes that had arisen in set theory in its early days (in the late 19th and early 20th century). A very natural and intuitive principle of reasoning that was widely used is this: Every condition determines a set. If the condition (or property) is being red then there is a set of red things, and if the condition is being a prime number, then there is a set of prime numbers, and so on. The principle seems completely selfevident. Unfortunately, it leads to a contradiction. (Set theory with this principle is often called naive set theory. ) Notice that some sets seem to be members of themselves and others not. The set of apples, for instance, is not an apple, so it is not a member of itself. On the other hand, the set of abstract entities is an abstract entity, so it would be a member of itself. So far, so good. There is nothing problematic about the conditions being an apple or being an abstract entity. But now consider the condition of being not a member of itself. The set of apples seems to satisfy this condition while the set of abstract entities does not. Let s form the set which corresponds to the condition (that is, the set of all things that are not members of themselves), and call it R. R = {x : x/ x} It looks like a legitimate set. The set of apples will be in R, butthesetof abstract entities will not. Now let s ask the question: Is it true or false that R R? The answer must be either Yes or No. Let s assume Yes: But if R R, thenr {x : x/ x}, and so must satisfy the condition x/ x, so R/ R. Now let s assume No. But if R/ R, thenr does satisfy the condition x/ x, so it must be a
6 CHAPTER 4. BASIC SET THEORY 32 member after all; hence R R. Either assumption leads to its opposite, so we have an outright contraction, R R R/ R. There have been a number of reactions to Russell s paradox. Zermelo s modification of the axiom of comprehension is the most popular. The key idea is that the defining condition is limited to the background set; it is not allowed to apply to everything. After almost 100 years, no one has been able to derive a contradiction. Exercise 13 Try to derive Russell s paradox using the axioms given so far as a way to convincing yourself that Zermelo s modification is plausible. Prior to Zermelo s axiomatization (in 1908), there was thought to be a universal set, that is, a set that contained everything. (Naturally, it would have to contain itself, as well.) This is no longer believed to be the case. The reasoning behind Russell s paradox is used to prove there is no universal set. Theorem 2 There is no universe. That is, U x x U. Proof. Suppose there is a universal set U that contains all sets. Define asetr as follows: R = {x : x U x/ x}. Since U contains everything, R U. By the same reasoning as that involved in Russell s paradox, we have R R R/ R. Since this is a contradiction, we blame the premiss that lead to the absurdity, namely, the assumption that U exists. Thus, there is no such set U. 4.4 Some Key Concepts Using the concepts, axioms, and notation developed so far, we can now define a number of very central concepts. Definition 9 (Union): A B = {x : x A x B} For example, {a, b} {1, 2, 3} = {a, b, 1, 2, 3} Definition 10 (Intersection): A B = {x : x A x B} For example, {a, b} {b, c, 2} = {b} Definition 11 (Subset): A B ( x x A x B) For example, {a, b} {a, b, c}; notice two special cases, A A, φ A
7 CHAPTER 4. BASIC SET THEORY 33 Definition 12 (Proper subset)a B (A B A 6= B) Definition 13 (Difference): A B = {x : x A x/ B} For example, {a, b} {b, c} = {a} Definition 14 (Complement): A 0 = {x : x U x/ A} For example, if U is the set of natural numbers and A is the set of even numbers, then A 0 is the set of odd numbers. Exercise 14 Let U = {1, 2, 3, 4, 5} be the background set and let A = {1, 2}, B = {2, 3}, and C = {3, 4, 5}. What are each of the following? 1. A 0 2. B 0 3. A B 4. A 0 B 0 5. C 0 A 6. U A 7. A U 8. A φ 9. B φ 10. U φ A (A B) B φ Answers: (1) A 0 = {3, 4, 5} = C, (4)A 0 B 0 = {1, 3, 4, 5}, (11)φ 0 = U, (14) B φ = B
8 CHAPTER 4. BASIC SET THEORY More Axioms Axiom 6 (Power Set): For any set A, there is a set whose members are exactly the subsets of A. (The power set of A is usually denoted A.). A B x (x B x A) (Here B is the powerset A) For example, if A = {a, b} then A = {A, φ, {a}, {b}}; and if B = {1, 2, 3}, then B = {B,φ, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}}. Note the special case, φ = {φ}. Notice the pattern. If a set has n members, then its power set has 2 n members. This is why it s called the power set. Exercise 15 (1) What is the powerst of {Bob,Alice}? (2) How many members in the powerset of the alphabet {a, b, c,..., z}? Axiom 7 (Infinity): There is a set with infinitely many members. Sincewehaven tyet defined infinite, the axiom does not really make sense yet. Later we will define it, but for now an intuitive understanding will do. So far I have stated nine axioms. There are more, but we won t be needing them. So I ll only mention some of them: There is the axiom of choice, the axiom of replacement, the axiom of regularity. These are pretty much standard and are in common use by logicians and mathematicians. There are still others, for example, socalled large cardinal axioms, that are quite controversial. These are not part of regular mathematics (at least not yet), but rather are the subject of ongoing research. Now we shall develop some of the key concepts. 4.6 The Algebra of Sets We will now develop the notions of intersection, subset, and so on. This basic part of set theory is known as the algebra of sets. Theorem 3 (Commutative law): A B = B A Proof. x A B x A x B Def x B x A tautology x B A Def
9 CHAPTER 4. BASIC SET THEORY 35 A B = B A Ax. of Extensionality This is a typical proof in the algebra of sets. Notice how it works. We need to show that two sets are identical, which we do by showing that they have exactly the same members. The crucial step in the proof is to use a fact from elementary logic, namely that P Q is equivalent to Q P. A proof does not have to be laid out exactly as above. You may be a bit more or less formal. The crucial point is to include the relevant information in a way that is as clear to the reader as possible. You might find something like this better: Proof. Let us assume that an arbitrary x is a member of the set A B. By the definition of union, this means that x is a member of A or is a member of B. Clearly, this is the same as saying that x is a member of B or x is a member of A. Again, by the definition of union, this means that x is a member of B A. The equality of these two sets follows from the Axiom of Extensionality. Pick the style of writing up proof that you like best. Proofs must be correct. After that make it as easy on your reader as you can. Theorem 4 (Associative laws): Theorem 5 (Distributive laws) Theorem 6 (DeMorgan s laws) A (B C) = (A B) C A (B C) = (A B) C A (B C) = (A B) (A C) A (B C) = (A B) (C B) (A B) 0 = A 0 B 0 (A B) 0 = A 0 B 0
10 CHAPTER 4. BASIC SET THEORY 36 Proof. (of the first) x (A B) 0 x/ A B (x A B) Def / (x A x B) Def x/ A x/ B tautology, Def / x A 0 x B 0 Def 0 x A 0 B 0 Def (A B) 0 = A 0 B 0 Axiom of Extensionality Noticeonceagainthatthekeystepisatransformationinlogic. Inthis proof it is the step from (P Q) to P Q. It is no accident that this equivalence is called De Morgan s in logic as well as in set theory. Theorem 7 A φ = A and A φ = φ Proof. (of the first) x A φ x A x φ Def x A (since x/ φ) A φ = A Axiom of Extensionality Theorem 8 A 00 = A Hint: P P Theorem 9 A B A C B C Proof. A B (x A x B) Def (x A x C x B x C) tautology (x A C x B C) Def A C B C Def Theorem 10 A B A C B C
11 CHAPTER 4. BASIC SET THEORY 37 Theorem 11 A B B 0 A 0 Theorem 12 φ A Hint: P Q is true if P is false. Theorem 13 A A = φ and A φ = A Proof. (of the first) x A A x A x/ A Def x φ (equivalent to a contradiction) A A = φ Axiom of Extensionality Exercise 16 Prove each of the theorems above that were not proven. 4.7 CounterExamples Theorems, of course, are true for any set whatsoever. A sentence which is not a theorem will have counterexamples, that is, example sets which make the sentence false. Of course, there might be other sets which make it true, in which case the sentence is satisfiable. When faced with a sentence that might be a theorem, try to prove it. If you can t, then try to find a counterexample. For example, we might wonder if A B = A B. To show that it is not atheoremweleta = {a},b = {b}. Then A B = {a} {b} = {a, b}. But A B = {a} {b} = φ 6= A B. So, A and B defined this way provide a counterexample to the alleged theorem. Exercise 17 Prove or give a counterexample to each of the following. 1. A B = B A 2. A 0 B 0 = B 0 A 0 3. A B = B 0 A 4. A B 6= φ A 6= φ 5. A B 6= φ A 6= φ
12 CHAPTER 4. BASIC SET THEORY A C B C A B C 7. (B A) A = A 8. (A B) A = φ 9. (A (B C) =(A B) (A B) 10. (A B B A) A = B Hints and Answers: (4) Counterexample: Let B = {a} and A = φ, then A B = {a} 6= φ, (5) Proof. Suppose A B 6= φ, then x x A B; thus, x A x B. Therefore, A is not empty, ie, A 6= φ. 4.8 Further Reading Enderton, Elements of Set Theory Potter, Set Theory and Its Philosophy Halmos, Naive Set Theory (Thisisagoodshortbook; butitis not naive in the technical sense.)
Russell s logicism. Jeff Speaks. September 26, 2007
Russell s logicism Jeff Speaks September 26, 2007 1 Russell s definition of number............................ 2 2 The idea of reducing one theory to another.................... 4 2.1 Axioms and theories.............................
More informationThis section will take the very naive point of view that a set is a collection of objects, the collection being regarded as a single object.
1.10. BASICS CONCEPTS OF SET THEORY 193 1.10 Basics Concepts of Set Theory Having learned some fundamental notions of logic, it is now a good place before proceeding to more interesting things, such as
More informationTutorial on Axiomatic Set Theory. Javier R. Movellan
Tutorial on Axiomatic Set Theory Javier R. Movellan Intuitively we think of sets as collections of elements. The crucial part of this intuitive concept is that we are willing to treat sets as entities
More informationChapter 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
More informationWe introduce one more operation on sets, perhaps the most important
11. The power set Please accept my resignation. I don t want to belong to any club that will accept me as a member. Groucho Marx We introduce one more operation on sets, perhaps the most important one:
More informationMath 144 Summer 2012 (UCR) ProNotes June 24, / 15
Before we start, I want to point out that these notes are not checked for typos. There are prbally many typeos in them and if you find any, please let me know as it s extremely difficult to find them all
More informationCS 173: Discrete Structures. Eric Shaffer Office Hour: Wed. 12, 2215 SC
CS 173: Discrete Structures Eric Shaffer Office Hour: Wed. 12, 2215 SC shaffer1@illinois.edu Agenda Sets (sections 2.1, 2.2) 2 Set Theory Sets you should know: Notation you should know: 3 Set Theory 
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 informationDirect Proof and Counterexample I:Introduction. Copyright Cengage Learning. All rights reserved.
Direct Proof and Counterexample I:Introduction Copyright Cengage Learning. All rights reserved. Goal Importance of proof Building up logic thinking and reasoning reading/using definition interpreting statement:
More informationSet 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
More information(2) Generalize De Morgan s laws for n sets and prove the laws by induction. 1
ARS DIGITA UNIVERSITY MONTH 2: DISCRETE MATHEMATICS PROFESSOR SHAI SIMONSON PROBLEM SET 2 SOLUTIONS SET, FUNCTIONS, BIGO, RATES OF GROWTH (1) Prove by formal logic: (a) The complement of the union of
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 information3 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;
More informationMathematical Preliminaries. Sipser pages 128
Mathematical Preliminaries Sipser pages 128 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation
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 informationBoolean 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 settheoretic operations on P(X) are the binary operations of union
More informationMathmatics 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)
More information12. INFINITE ABELIAN GROUPS
12. INFINITE ABELIAN GROUPS 12.1. Examples of Infinite Abelian Groups Many of the groups which arise in various parts of mathematics are abelian. That is, they satisfy the commutative law: xy = yx. If
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 informationCHAPTER 10. Gentzen Style Proof Systems for Classical Logic
CHAPTER 10 Gentzen Style Proof Systems for Classical Logic Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning
More informationReal Analysis: Part I. William G. Faris
Real Analysis: Part I William G. Faris February 2, 2004 ii Contents 1 Mathematical proof 1 1.1 Logical language........................... 1 1.2 Free and bound variables...................... 3 1.3 Proofs
More informationCS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:
x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which
More informationMath 13, Spring 2013, Lecture B: Midterm
Math 13, Spring 2013, Lecture B: Midterm Name Signature UCI ID # Email address Each numbered problem is worth 12 points, for a total of 84 points. Present your work, especially proofs, as clearly as possible.
More informationSec$on Summary. Definition of sets Describing Sets
Section 2.1 Sec$on Summary Definition of sets Describing Sets Roster Method SetBuilder Notation Some Important Sets in Mathematics Empty Set and Universal Set Subsets and Set Equality Cardinality of Sets
More informationCHAPTER 11. Introduction to Intuitionistic Logic
CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated
More informationBasic counting techniques. Periklis A. Papakonstantinou Rutgers Business School
Basic counting techniques Periklis A. Papakonstantinou Rutgers Business School i LECTURE NOTES IN Elementary counting methods Periklis A. Papakonstantinou MSIS, Rutgers Business School ALL RIGHTS RESERVED
More informationPEANO 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
More information1 Completeness Theorem for Classical Predicate
1 Completeness Theorem for Classical Predicate Logic The relationship between the first order models defined in terms of structures M = [M, I] and valuations s : V AR M and propositional models defined
More informationSets. Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing Spring, Outline Sets An Algebra on Sets Summary
An Algebra on Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing Spring, 2018 Alice E. Fischer... 1/37 An Algebra on 1 Definitions and Notation Venn Diagrams 2 An Algebra on 3 Alice E. Fischer...
More informationAnalysis I. Classroom Notes. H.D. Alber
Analysis I Classroom Notes HD 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
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 informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationContradiction MATH Contradiction. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Contradiction Benjamin V.C. Collins James A. Swenson Contrapositive The contrapositive of the statement If A, then B is the statement If not B, then not A. A statement and its contrapositive
More informationMATH10040: Chapter 0 Mathematics, Logic and Reasoning
MATH10040: Chapter 0 Mathematics, Logic and Reasoning 1. What is Mathematics? There is no definitive answer to this question. 1 Indeed, the answer given by a 21stcentury mathematician would differ greatly
More information8. Reductio ad absurdum
8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, 10 Galileo Galilea (15641642) gives several arguments meant to demonstrate that there can be no such thing as actual
More informationCHAPTER 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?
More information2. Ordered sets. A theorem of Hausdorff.
ORDERED SETS. 2. Ordered sets. A theorem of Hausdorff. One obtains a more complete idea of Cantor's work by studying his theory of ordered sets. As to the notion "ordered set" this is nowadays mostly defined
More informationLogic of Sentences (Propositional Logic) is interested only in true or false statements; does not go inside.
You are a mathematician if 1.1 Overview you say to a car dealer, I ll take the red car or the blue one, but then you feel the need to add, but not both.  1. Logic and Mathematical Notation (not in the
More informationSets. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Fall 2007
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 1 / 42 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.1, 2.2 of Rosen Introduction I Introduction
More informationSection 3.1: Direct Proof and Counterexample 1
Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion
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 informationAnalysis 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
More informationThe 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
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 informationGödel Numbering. Substitute {x: x is not an element of itself} for y, and we get a contradiction:
Gödel Numbering {x: x is a horse} is a collection that has all the worlds horses as elements, and nothing else. Thus we have For any y, y 0 {x: x is a horse} if and only if y is a horse. Traveler, for
More informationDirect 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
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 informationPropositional Logic Review
Propositional Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane The task of describing a logical system comes in three parts: Grammar Describing what counts as a formula Semantics Defining
More informationAn 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
More informationOn an Unsound Proof of the Existence of Possible Worlds
598 Notre Dame Journal of Formal Logic Volume 30, Number 4, Fall 1989 On an Unsound Proof of the Existence of Possible Worlds CHRISTOPHER MENZEL* Abstract In this paper, an argument of Alvin Plantinga's
More informationChapter 2 Sets, Relations and Functions
Chapter 2 Sets, Relations and Functions Key Topics Sets Set Operations Russell s Paradox Relations Composition of Relations Reflexive, Symmetric and Transitive Relations Functions Partial and Total Functions
More informationSection 20: Arrow Diagrams on the Integers
Section 0: Arrow Diagrams on the Integers Most of the material we have discussed so far concerns the idea and representations of functions. A function is a relationship between a set of inputs (the leave
More informationSequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.
Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence
More informationSolving Equations by Adding and Subtracting
SECTION 2.1 Solving Equations by Adding and Subtracting 2.1 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the addition property to solve equations 3. Determine whether
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 informationHANDOUT 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
More informationMath 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
More informationAfter taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.
Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric
More informationIntroduction to Karnaugh Maps
Introduction to Karnaugh Maps Review So far, you (the students) have been introduced to truth tables, and how to derive a Boolean circuit from them. We will do an example. Consider the truth table for
More informationAn 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
More informationDiscrete Mathematical Structures: Theory and Applications
Chapter 1: Foundations: Sets, Logic, and Algorithms Discrete Mathematical Structures: Theory and Applications Learning Objectives Learn about sets Explore various operations on sets Become familiar with
More informationAxioms of Kleene Algebra
Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 3
EECS 70 Discrete Mathematics and Probability Theory Spring 014 Anant Sahai Note 3 Induction Induction is an extremely powerful tool in mathematics. It is a way of proving propositions that hold for all
More informationLogic. Propositional Logic: Syntax
Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about
More informationChoosing Logical Connectives
Choosing Logical Connectives 1. Too Few Connectives?: We have chosen to use only 5 logical connectives in our constructed language of logic, L1 (they are:,,,, and ). But, we might ask, are these enough?
More informationINTRODUCTION TO LOGIC 8 Identity and Definite Descriptions
8.1 Qualitative and Numerical Identity INTRODUCTION TO LOGIC 8 Identity and Definite Descriptions Volker Halbach Keith and Volker have the same car. Keith and Volker have identical cars. Keith and Volker
More informationINTRODUCTION TO LOGIC. Propositional Logic. Examples of syntactic claims
Introduction INTRODUCTION TO LOGIC 2 Syntax and Semantics of Propositional Logic Volker Halbach In what follows I look at some formal languages that are much simpler than English and define validity of
More informationChapter 2: Introduction to Propositional Logic
Chapter 2: Introduction to Propositional Logic PART ONE: History and Motivation Origins: Stoic school of philosophy (3rd century B.C.), with the most eminent representative was Chryssipus. Modern Origins:
More informationMATH 201 Solutions: TEST 3A (in class)
MATH 201 Solutions: TEST 3A (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
More informationInfinite constructions in set theory
VI : Infinite constructions in set theory In elementary accounts of set theory, examples of finite collections of objects receive a great deal of attention for several reasons. For example, they provide
More informationUC Berkeley, Philosophy 142, Spring 2016 John MacFarlane Philosophy 142
Plural Quantifiers UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane Philosophy 142 1 Expressive limitations of firstorder logic Firstorder logic uses only quantifiers that bind variables in name
More informationLecture 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,
More informationRoberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7. Inverse matrices
Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7 Inverse matrices What you need to know already: How to add and multiply matrices. What elementary matrices are. What you can learn
More information(D) Introduction to order types and ordinals
(D) Introduction to order types and ordinals Linear orders are one of the mathematical tools that are used all over the place. Wellordered sets are a special kind of linear order. At first sight wellorders
More informationDominoes and Counting
Giuseppe Peano (Public Domain) Dominoes and Counting All of us have an intuitive feeling or innate sense for the counting or natural numbers, including a sense for infinity: ={1,, 3, }. The ability to
More information8. Reductio ad absurdum
8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, Galileo Galilea (15641642) gives several arguments meant to demonstrate that there can be no such thing as actual infinities
More informationPL Proofs Introduced. Chapter A1. A1.1 Choices, choices
Chapter A1 PL Proofs Introduced Outside the logic classroom, when we want to convince ourselves that an inference is valid, we don t often use techniques like the truthtable test or tree test. Instead
More informationMA103 STATEMENTS, PROOF, LOGIC
MA103 STATEMENTS, PROOF, LOGIC Abstract Mathematics is about making precise mathematical statements and establishing, by proof or disproof, whether these statements are true or false. We start by looking
More informationMathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras. Lecture  15 Propositional Calculus (PC)
Mathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras Lecture  15 Propositional Calculus (PC) So, now if you look back, you can see that there are three
More informationUnary negation: T F F T
Unary negation: ϕ 1 ϕ 1 T F F T Binary (inclusive) or: ϕ 1 ϕ 2 (ϕ 1 ϕ 2 ) T T T T F T F T T F F F Binary (exclusive) or: ϕ 1 ϕ 2 (ϕ 1 ϕ 2 ) T T F T F T F T T F F F Classical (material) conditional: ϕ 1
More informationKaplan s Paradox and Epistemically Possible Worlds
Kaplan s Paradox and Epistemically Possible Worlds 1. Epistemically possible worlds David Chalmers Metaphysically possible worlds: S is metaphysically possible iff S is true in some metaphysically possible
More informationCHAPTER 1: THE PEANO AXIOMS
CHAPTER 1: THE PEANO AXIOMS MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction We begin our exploration of number systems with the most basic number system: the natural numbers N. Informally, natural
More informationCHAPTER 1. MATHEMATICAL LOGIC 1.1 Fundamentals of Mathematical Logic
CHAPER 1 MAHEMAICAL LOGIC 1.1 undamentals of Mathematical Logic Logic is commonly known as the science of reasoning. Some of the reasons to study logic are the following: At the hardware level the design
More informationMATH 2200 Final Review
MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 810 proofs It will take place on Tuesday, December 11, from 10:30 AM  1:30 PM, in the usual room Topics
More informationcse303 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
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 informationNatural deduction for truthfunctional logic
Natural deduction for truthfunctional logic Phil 160  Boston University Why natural deduction? After all, we just found this nice method of truthtables, which can be used to determine the validity or
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 informationLecture 2: Syntax. January 24, 2018
Lecture 2: Syntax January 24, 2018 We now review the basic definitions of firstorder logic in more detail. Recall that a language consists of a collection of symbols {P i }, each of which has some specified
More informationLebesgue measure on R is just one of many important measures in mathematics. In these notes we introduce the general framework for measures.
Measures In General Lebesgue measure on R is just one of many important measures in mathematics. In these notes we introduce the general framework for measures. Definition: σalgebra Let X be a set. A
More informationZeno s Paradox #1. The Achilles
Zeno s Paradox #1. The Achilles Achilles, who is the fastest runner of antiquity, is racing to catch the tortoise that is slowly crawling away from him. Both are moving along a linear path at constant
More information1 Proof techniques. CS 224W Linear Algebra, Probability, and Proof Techniques
1 Proof techniques Here we will learn to prove universal mathematical statements, like the square of any odd number is odd. It s easy enough to show that this is true in specific cases for example, 3 2
More informationLogical Reasoning. Chapter Statements and Logical Operators
Chapter 2 Logical Reasoning 2.1 Statements and Logical Operators Preview Activity 1 (Compound Statements) Mathematicians often develop ways to construct new mathematical objects from existing mathematical
More informationProofs. 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,
More informationFollow links for Class Use and other Permissions. For more information send to:
COPYRIGHT NOTICE: John J. Watkins: Topics in Commutative Ring Theory is published by Princeton University Press and copyrighted, 2007, by Princeton University Press. All rights reserved. No part of this
More informationChapter 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
More informationSolutions to Problem Set 1
Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 21 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 21, 2005, 1076 minutes Problem
More informationIsomorphisms and Welldefinedness
Isomorphisms and Welldefinedness 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 informationSemantics and Generative Grammar. The Semantics of Adjectival Modification 1. (1) Our Current Assumptions Regarding Adjectives and Common Ns
The Semantics of Adjectival Modification 1 (1) Our Current Assumptions Regarding Adjectives and Common Ns a. Both adjectives and common nouns denote functions of type (i) [[ male ]] = [ λx : x D
More informationCS 360, Winter Morphology of Proof: An introduction to rigorous proof techniques
CS 30, Winter 2011 Morphology of Proof: An introduction to rigorous proof techniques 1 Methodology of Proof An example Deep down, all theorems are of the form If A then B, though they may be expressed
More information