# Sec\$on Summary. Definition of sets Describing Sets

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Section 2.1

2 Sec\$on Summary Definition of sets Describing Sets Roster Method Set-Builder Notation Some Important Sets in Mathematics Empty Set and Universal Set Subsets and Set Equality Cardinality of Sets Tuples Cartesian Products 2

3 Introduc\$on 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 is an important branch of mathematics. Many different systems of axioms have been used to develop set theory. Here we are not concerned with a formal set of axioms for set theory. Instead, we will use what is called naïve set theory. 3

4 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain its elements. The notation a A denotes that a is an element of the set A. If a is not a member of A, write a A 4

5 Describing a Set: Roster Method S = {a,b,c,d} Order not important S = {a,b,c,d} = {b,c,a,d} Each distinct object is either a member or not; listing more than once does not change the set. S = {a,b,c,d} = {a,b,c,b,c,d} Elipses ( ) may be used to describe a set without listing all of the members when the pattern is clear. S = {a,b,c,d,,z } 5

6 Roster Method Set of all vowels in the English alphabet: V = {a,e,i,o,u} Set of all odd positive integers less than 10: O = {1,3,5,7,9} Set of all positive integers less than 100: S = {1,2,3,..,99} Set of all integers less than 0: S = {., -3,-2,-1} 6

7 Some Important Sets N = natural numbers = {0,1,2,3.} Z = integers = {,-3,-2,-1,0,1,2,3, } Z+ = positive integers = {1,2,3,..} Q = set of rational numbers R = set of real numbers R + = set of positive real numbers C = set of complex numbers. C R Q Z N 7

8 Set-Builder Nota\$on Specify the property or properties that all members must satisfy: S = {x x is a positive integer less than 100} O = {x x is an odd positive integer less than 10} O = {x Z+ x is odd and x < 10} A predicate may be used: S = {x P(x)} Example: S = {x Prime(x)} Positive rational numbers: Q + = {x R x = p/q, for some positive integers p,q} 8

9 Interval Nota\$on [a,b] = {x a x b} [a,b) = {x a x < b} (a,b] = {x a < x b} (a,b) = {x a < x < b} closed interval [a,b] open interval (a,b) [2, 7] [2, 7) (2, 7] (2, 7)

10 Universal Set and Empty Set The universal set U is the set containing everything currently under consideration. Sometimes implicit Sometimes explicitly stated. Contents depend on the context. The empty set is the set with no elements. Symbolized, but {} also used. Venn Diagram V a e i o u U John Venn ( ) Cambridge, UK 10

11 Russell s Paradox Let S be the set of all sets which are not members of themselves. A paradox results from trying to answer the question Is S a member of itself? Related Paradox: Henry is a barber who shaves all people who do not shave themselves. A paradox results from trying to answer the question Does Henry shave himself? Bertrand Russell ( ) Cambridge, UK Nobel Prize Winner 11

12 Some things to remember Sets can be elements of sets. {{1,2,3},a, {b,c}} {N,Z,Q,R} The empty set is different from a set containing the empty set. { } 12

13 Set Equality Definition: Two sets are equal if and only if they have the same elements. Therefore if A and B are sets, then A and B are equal if and only if. We write A = B if A and B are equal sets. {1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5} 13

14 Subsets Definition: The set A is a subset of B, if and only if every element of A is also an element of B. The notation A B is used to indicate that A is a subset of the set B. A B holds if and only if 1. Because a is always false, S, for every set S. 2. Because a S a S, S S, for every set S. is true. U A B B A 14

15 Showing a Set is or is not a Subset of Another Set A is a Subset of B: To show A B, show that if x belongs to A, then x also belongs to B. A is not a Subset of B: To show A B, find an element x A such that x B. (Such an x is a counterexample to the claim that x A implies x B.) Examples: 1. The set of all computer science majors at your school is a subset of all students at your school. 2. The set of integers with squares less than 100 is not a subset of the set of nonnegative integers. 15

16 Another look at Equality of Sets Recall that two sets A and B are equal, denoted by A = B, iff Using logical equivalences we have that A = B iff This is equivalent to A B and B A 16

17 Proper Subsets Definition: If A B, but A B, then we say A is a proper subset of B, denoted by A B. If A B, then is true. Venn Diagram B A U 17

18 Set Cardinality Definition: If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is finite. Otherwise it is infinite. Definition: The cardinality of a finite set A, denoted by A, is the number of (distinct) elements of A. Examples: 1. ø = 0 2. Let S be the letters of the English alphabet. Then S = {1,2,3} = 3 4. {ø} = 1 5. The set of integers is infinite. 18

19 Power Sets Definition: The set of all subsets of a set A, denoted P(A), is called the power set of A. Ex: If A = {a,b} then P(A) = {ø, {a},{b},{a,b}} If a set has n elements, then the cardinality of the power set is 2ⁿ. (In Ch. 5 and 6, we ll discuss different ways to show this.) 19

20 Tuples The ordered n-tuple (a 1,a 2,..,a n ) is the ordered collection that has a 1 as its first element and a 2 as its second element and so on until a n as its last element. Two n-tuples are equal if and only if their corresponding elements are equal. 2-tuples are called ordered pairs. The ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d. 20

21 René Descartes ( ) Cartesian Product Definition: The Cartesian Product of two sets A and B, denoted by A B is the set of ordered pairs (a,b) where a A and b B. Example: A = {a,b} B = {1,2,3} A B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)} B A = {(1,a),(1,b), (2,a),(2,b), (3,a),(3,b)} Definition: A subset R of the Cartesian product A B is called a relation from the set A to the set B. (More in Chapter 9. ) 21

22 Cartesian Product Definition: The cartesian products of the sets A 1,A 2,,A n, denoted by A 1 A 2 A n, is the set of ordered n-tuples (a 1,a 2,,a n ) where a i belongs to A i for i = 1, n. Ex: What is A B C where A = {0,1}, B = {1,2} and C = {0,1,2} Solution: A B C = {(0,1,0), (0,1,1), (0,1,2),(0,2,0), (0,2,1), (0,2,2),(1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)} 22

23 Set Nota\$on with Quan\$fiers x S (P(x)) is shorthand for x(x S P(x)) x S (P(x)) is shorthand for x(x S P(x)) Ex: Express the following in English 1. x R (x 2 0) The square of every real number is nonnegative. 2. x Z (x 2 = 1) There is an integer whose square is one. 23

24 Truth Sets and Quan\$fiers Given a predicate P and a domain D, we define the truth set of P to be the set of elements in D for which P(x) is true. The truth set of P(x) is denoted by Ex: The truth set of P(x) where the domain is the integers and P(x) is x = 1 is the set {-1,1} 24

### 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

### CS 173: Discrete Structures. Eric Shaffer Office Hour: Wed. 1-2, 2215 SC

CS 173: Discrete Structures Eric Shaffer Office Hour: Wed. 1-2, 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 -

### Sets. Introduction to Set Theory ( 2.1) Basic notations for sets. Basic properties of sets CMSC 302. Vojislav Kecman

Introduction to Set Theory ( 2.1) VCU, Department of Computer Science CMSC 302 Sets Vojislav Kecman A set is a new type of structure, representing an unordered collection (group, plurality) of zero or

### 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

### Discrete 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

### 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

### Chapter 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

### Sets. 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

### LIN1032 Formal Foundations for Linguistics

LIN1032 Formal Foundations for Lecture 1 Albert Gatt Practical stuff Course tutors: Albert Gatt (first half) albert.gatt@um.edu.mt Ray Fabri (second half) ray.fabri@um.edu.mt Course website: TBA Practical

### Section 2.1: Introduction to the Logic of Quantified Statements

Section 2.1: Introduction to the Logic of Quantified Statements In the previous chapter, we studied a branch of logic called propositional logic or propositional calculus. Loosely speaking, propositional

### 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

### Solutions 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

### Mathematical Preliminaries. Sipser pages 1-28

Mathematical Preliminaries Sipser pages 1-28 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation

### 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

### CISC 1100: Structures of Computer Science

CISC 1100: Structures of Computer Science Chapter 2 Sets and Sequences Fordham University Department of Computer and Information Sciences Fall, 2010 CISC 1100/Fall, 2010/Chapter 2 1 / 49 Outline Sets Basic

### Topics in Logic and Proofs

Chapter 2 Topics in Logic and Proofs Some mathematical statements carry a logical value of being true or false, while some do not. For example, the statement 4 + 5 = 9 is true, whereas the statement 2

### Logic. Facts (with proofs) CHAPTER 1. Definitions

CHAPTER 1 Logic Definitions D1. Statements (propositions), compound statements. D2. Truth values for compound statements p q, p q, p q, p q. Truth tables. D3. Converse and contrapositive. D4. Tautologies

### 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,

### Logical 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

### Math 144 Summer 2012 (UCR) Pro-Notes 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

### CSE 20. Lecture 4: Introduction to Boolean algebra. CSE 20: Lecture4

CSE 20 Lecture 4: Introduction to Boolean algebra Reminder First quiz will be on Friday (17th January) in class. It is a paper quiz. Syllabus is all that has been done till Wednesday. If you want you may

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

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,

### Section 1: Sets and Interval Notation

PART 1 From Sets to Functions Section 1: Sets and Interval Notation Introduction Set concepts offer the means for understanding many different aspects of mathematics and its applications to other branches

### CHAPTER 1. Basic Ideas

CHPTER 1 asic Ideas In the end, all mathematics can be boiled down to logic and set theory. ecause of this, any careful presentation of fundamental mathematical ideas is inevitably couched in the language

### Lecture Notes for MATH Mathematical Logic 1

Lecture Notes for MATH2040 - Mathematical Logic 1 Michael Rathjen School of Mathematics University of Leeds Autumn 2009 Chapter 0. Introduction Maybe not all areas of human endeavour, but certainly the

### This 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

### 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

### Tutorial 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

### Chapter 0: Sets and Numbers

Haberman MTH 112 Section I: Periodic Functions and Trigonometry Chapter 0: Sets and Numbers DEFINITION: A set is a collection of objects specified in a manner that enables one to determine if a given object

### 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

### Chapter 2: The Logic of Quantified Statements. January 22, 2010

Chapter 2: The Logic of Quantified Statements January 22, 2010 Outline 1 2.1- Introduction to Predicates and Quantified Statements I 2 2.2 - Introduction to Predicates and Quantified Statements II 3 2.3

### Propositions and Proofs

Propositions and Proofs Gert Smolka, Saarland University April 25, 2018 Proposition are logical statements whose truth or falsity can be established with proofs. Coq s type theory provides us with a language

### 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

### 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

### CSC173 Workshop: 13 Sept. Notes

CSC173 Workshop: 13 Sept. Notes Frank Ferraro Department of Computer Science University of Rochester September 14, 2010 1 Regular Languages and Equivalent Forms A language can be thought of a set L of

### 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

### Supplementary 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

### Propositional Logic Not Enough

Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks

### Predicate Calculus lecture 1

Predicate Calculus lecture 1 Section 1.3 Limitation of Propositional Logic Consider the following reasoning All cats have tails Gouchi is a cat Therefore, Gouchi has tail. MSU/CSE 260 Fall 2009 1 MSU/CSE

### Section 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

### MATH 114 Fall 2004 Solutions to practice problems for Final Exam

MATH 11 Fall 00 Solutions to practice problems for Final Exam Reminder: the final exam is on Monday, December 13 from 11am - 1am. Office hours: Thursday, December 9 from 1-5pm; Friday, December 10 from

### Section 1.3 Ordered Structures

Section 1.3 Ordered Structures Tuples Have order and can have repetitions. (6,7,6) is a 3-tuple () is the empty tuple A 2-tuple is called a pair and a 3-tuple is called a triple. We write (x 1,, x n )

### Proposi'onal Logic Not Enough

Section 1.4 Proposi'onal Logic Not Enough If we have: All men are mortal. Socrates is a man. Socrates is mortal Compare to: If it is snowing, then I will study discrete math. It is snowing. I will study

### { } Notation, Functions, Domain & Range

Lesson 02, page 1 of 9 Déjà Vu, It's Algebra 2! Lesson 02 Notation, Functions, Domain & Range Since Algebra grew out of arithmetic, which is operations on number sets, it is important early on to understand

### One-to-one functions and onto functions

MA 3362 Lecture 7 - One-to-one and Onto Wednesday, October 22, 2008. Objectives: Formalize definitions of one-to-one and onto One-to-one functions and onto functions At the level of set theory, there are

### Solutions to Homework Set 1

Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement

### September 7, Formal Definition of a Nondeterministic Finite Automaton

Formal Definition of a Nondeterministic Finite Automaton September 7, 2014 A comment first The formal definition of an NFA is similar to that of a DFA. Both have states, an alphabet, transition function,

### 1.A Sets, Relations, Graphs, and Functions 1.A.1 Set a collection of objects(element) Let A be a set and a be an elements in A, then we write a A.

1.A Sets, Relations, Graphs, and Functions 1.A.1 Set a collection of objects(element) Let A be a set and a be an elements in A, then we write a A. How to specify sets 1. to enumerate all of the elements

### The Nature of Mathematics 13th Edition, Smith Notes. Korey Nishimoto Math Department, Kapiolani Community College October

Mathematics 13th Edition, Smith Notes Korey Nishimoto Math Department, Kapiolani Community College October 9 2017 Expanded Introduction to Mathematical Reasoning Page 3 Contents Contents Nature of Logic....................................

### 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

### Lecture 15: Validity and Predicate Logic

Lecture 15: Validity and Predicate Logic 1 Goals Today Learn the definition of valid and invalid arguments in terms of the semantics of predicate logic, and look at several examples. Learn how to get equivalents

### The Limit of Humanly Knowable Mathematical Truth

The Limit of Humanly Knowable Mathematical Truth Gödel s Incompleteness Theorems, and Artificial Intelligence Santa Rosa Junior College December 12, 2015 Another title for this talk could be... An Argument

### CSE 1400 Applied Discrete Mathematics Proofs

CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4

### Real 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

### Sets. 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...

### 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)

### MGF 1106: Exam 1 Solutions

MGF 1106: Exam 1 Solutions 1. (15 points total) True or false? Explain your answer. a) A A B Solution: Drawn as a Venn diagram, the statement says: This is TRUE. The union of A with any set necessarily

### 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

### INTRODUCTION 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

### Chapter 6. Relations. 6.1 Relations

Chapter 6 Relations Mathematical relations are an extremely general framework for specifying relationships between pairs of objects. This chapter surveys the types of relations that can be constructed

### Introduction to Metalogic 1

Philosophy 135 Spring 2012 Tony Martin Introduction to Metalogic 1 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: (i) sentence letters p 0, p 1, p 2,... (ii) connectives,

### Chapter 3. The Logic of Quantified Statements

Chapter 3. The Logic of Quantified Statements 3.1. Predicates and Quantified Statements I Predicate in grammar Predicate refers to the part of a sentence that gives information about the subject. Example:

### 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

### - 1.2 Implication P. Danziger. Implication

Implication There is another fundamental type of connectives between statements, that of implication or more properly conditional statements. In English these are statements of the form If p then q or

### Lecture : Set Theory and Logic

Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Contrapositive and Converse 1 Contrapositive and Converse 2 3 4 5 Contrapositive and Converse

### Context-free grammars and languages

Context-free grammars and languages The next class of languages we will study in the course is the class of context-free languages. They are defined by the notion of a context-free grammar, or a CFG for

### Logic: The Big Picture

Logic: The Big Picture A typical logic is described in terms of syntax: what are the legitimate formulas semantics: under what circumstances is a formula true proof theory/ axiomatization: rules for proving

### Notes on Satisfiability-Based Problem Solving. First Order Logic. David Mitchell October 23, 2013

Notes on Satisfiability-Based Problem Solving First Order Logic David Mitchell mitchell@cs.sfu.ca October 23, 2013 Preliminary draft. Please do not distribute. Corrections and suggestions welcome. In this

### Section 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

### 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 61-02: PRACTICE PROBLEMS FOR FINAL EXAM

MATH 61-02: PRACTICE PROBLEMS FOR FINAL EXAM (FP1) The exclusive or operation, denoted by and sometimes known as XOR, is defined so that P Q is true iff P is true or Q is true, but not both. Prove (through

### ! Predicates! Variables! Quantifiers. ! Universal Quantifier! Existential Quantifier. ! Negating Quantifiers. ! De Morgan s Laws for Quantifiers

Sec\$on Summary (K. Rosen notes for Ch. 1.4, 1.5 corrected and extended by A.Borgida)! Predicates! Variables! Quantifiers! Universal Quantifier! Existential Quantifier! Negating Quantifiers! De Morgan s

### MATH 120. Test 1 Spring, 2012 DO ALL ASSIGNED PROBLEMS. Things to particularly study

MATH 120 Test 1 Spring, 2012 DO ALL ASSIGNED PROBLEMS Things to particularly study 1) Critical Thinking Basic strategies Be able to solve using the basic strategies, such as finding patterns, questioning,

### 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.

### 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

### Preface. [ Uj \m Vk NVp U\k Y rj U! - kndh kùa«bg. Rªjudh. (iii)

015 Preface [ Uj \m Vk NVp U\k Y rj U! - kndh kùa«bg. Rªjudh The Government of Tamil Nadu has decided to evolve a uniform system of school education in the state to ensure social justice and provide quality

### Section-A. Short Questions

Section-A Short Questions Question1: Define Problem? : A Problem is defined as a cultural artifact, which is especially visible in a society s economic and industrial decision making process. Those managers

### Section Summary. Predicate logic Quantifiers. Negating Quantifiers. Translating English to Logic. Universal Quantifier Existential Quantifier

Section 1.4 Section Summary Predicate logic Quantifiers Universal Quantifier Existential Quantifier Negating Quantifiers De Morgan s Laws for Quantifiers Translating English to Logic Propositional Logic

### Math 205 April 28, 2009 Final Exam Review

1 Definitions 1. Some question will ask you to statea a definition and/or give an example of a defined term. The terms are: subset proper subset intersection union power set cartesian product relation

### Notes for Recitation 1

6.042/18.062J Mathematics for Computer Science September 10, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 1 1 Logic How can one discuss mathematics with logical precision, when the English

### Lecture Notes for MATH 2040 Mathematical Logic I Semester 1, 2009/10. Chapter 0. Introduction

Lecture Notes for MATH 2040 Mathematical Logic I Semester 1, 2009/10 Michael Rathjen Chapter 0. Introduction Maybe not all areas of human endeavour, but certainly the sciences presuppose an underlying

### Exercises. Exercise Sheet 1: Propositional Logic

B Exercises Exercise Sheet 1: Propositional Logic 1. Let p stand for the proposition I bought a lottery ticket and q for I won the jackpot. Express the following as natural English sentences: (a) p (b)

### 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

### The λ-calculus and Curry s Paradox Drew McDermott , revised

The λ-calculus and Curry s Paradox Drew McDermott drew.mcdermott@yale.edu 2015-09-23, revised 2015-10-24 The λ-calculus was invented by Alonzo Church, building on earlier work by Gottlob Frege and Moses

### Monadic Predicate Logic is Decidable. Boolos et al, Computability and Logic (textbook, 4 th Ed.)

Monadic Predicate Logic is Decidable Boolos et al, Computability and Logic (textbook, 4 th Ed.) These slides use A instead of E instead of & instead of - instead of Nota>on Equality statements are atomic

### 2.2 Lowenheim-Skolem-Tarski theorems

Logic SEP: Day 1 July 15, 2013 1 Some references Syllabus: http://www.math.wisc.edu/graduate/guide-qe Previous years qualifying exams: http://www.math.wisc.edu/ miller/old/qual/index.html Miller s Moore

### 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

### Choosing 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?

### Lecture 4: Proposition, Connectives and Truth Tables

Discrete Mathematics (II) Spring 2017 Lecture 4: Proposition, Connectives and Truth Tables Lecturer: Yi Li 1 Overview In last lecture, we give a brief introduction to mathematical logic and then redefine

### 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

### Introduction to Metalogic

Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)

### LECTURE 1. Logic and Proofs

LECTURE 1 Logic and Proofs The primary purpose of this course is to introduce you, most of whom are mathematics majors, to the most fundamental skills of a mathematician; the ability to read, write, and

### Sets and Games. Logic Tea, ILLC Amsterdam. 14. September Stefan Bold. Institute for Logic, Language and Computation. Department of Mathematics

Sets and Games 14. September 2004 Logic Tea, ILLC Amsterdam Stefan Bold Mathematical Logic Group Department of Mathematics University of Bonn Institute for Logic, Language and Computation University of

### Sec\$on Summary. Sequences. Recurrence Relations. Summations. Ex: Geometric Progression, Arithmetic Progression. Ex: Fibonacci Sequence

Section 2.4 Sec\$on Summary Sequences Ex: Geometric Progression, Arithmetic Progression Recurrence Relations Ex: Fibonacci Sequence Summations 2 Introduc\$on Sequences are ordered lists of elements. 1, 2,

### Solving 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

### 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.............................

### A Transition to Advanced Mathematics

A Transition to Advanced Mathematics Darrin Doud and Pace P. Nielsen Darrin Doud Department of Mathematics Brigham Young University Provo, UT 84602 doud@math.byu.edu Pace P. Nielsen Department of Mathematics