29. Army Housing (a) (b) (c) (d) (e) (f ) Totals Totals (a) (b) (c) (d) (e) (f) Basketball Positions 32. Guard Forward Center

Size: px
Start display at page:

Download "29. Army Housing (a) (b) (c) (d) (e) (f ) Totals Totals (a) (b) (c) (d) (e) (f) Basketball Positions 32. Guard Forward Center"

Transcription

1 Infinite Sets and Their Cardinalities As mentioned at the beginning of this chapter, most of the early work in set theory was done by Georg Cantor He devoted much of his life to a study of the cardinal numbers of sets Recall that the cardinal number, or cardinality, of a finite set is the number of elements that it contains For example, the set, 9, contains elements and has a cardinal number of The cardinal number of 0 is 0 Cantor proved many results about the cardinal numbers of infinite sets The proofs of Cantor are quite different from the type of proofs you may have seen in an algebra or geometry course Because of the novelty of Cantor s methods, they were not quickly accepted by the mathematicians of his day (In fact, some other aspects

2 8 CHAPTER The Basic Concepts of Set Theory of Cantor s theory lead to paradoxes) The results we will discuss here, however, are commonly accepted today The idea of the cardinal number of an infinite set depends on the idea of one-toone correspondence For example, each of the sets,,, and 9, 0,, has four elements Corresponding elements of the two sets could be paired off in the following manner (among many other ways): Paradox The word in Greek originally meant wrong opinion as opposed to orthodox, which meant right opinion Over the years, the word came to mean selfcontradiction An example is the statement This sentence is false By assuming it is true, we get a contradiction; likewise, by assuming it is false, we get a contradiction Thus, it s a paradox Before the twentieth century it was considered a paradox that any set could be placed into one-toone correspondence with a proper subset of itself This paradox, called Galileo s paradox after the sixteenth-century mathematician and scientist Galileo (see the picture), is now explained by saying that the ability to make such a correspondence is how we distinguish infinite sets from finite sets What is true for finite sets is not necessarily true for infinite sets,,, j j j j 9, 0,, Such a pairing is a one-to-one correspondence between the two sets The one-toone refers to the fact that each element of the first set is paired with exactly one element of the second set and each element of the second set is paired with exactly one element of the first set Two sets A and B which may be put in a one-to-one correspondence are said to be equivalent Symbolically, this is written A B Do you see that the two sets shown above are equivalent but not equal? The following correspondence between sets, 8, and,,, 8, j hg,, is not one-to-one since the elements 8 and from the first set are both paired with the element from the second set These sets are not equivalent It seems reasonable to say that if two non-empty sets have the same cardinal number, then a one-to-one correspondence can be established between the two sets Also, if a one-to-one correspondence can be established between two sets, then the two sets must have the same cardinal number These two facts are fundamental in discussing the cardinal numbers of infinite sets The basic set used in discussing infinite sets is the set of counting numbers,,,,,, The set of counting numbers is said to have the infinite cardinal number (the first Hebrew letter, aleph, with a zero subscript, read aleph-null ) Think of as being the smallest infinite cardinal number To the question How many counting numbers are there? answer There are of them From the discussion above, any set that can be placed in a one-to-one correspondence with the counting numbers will have the same cardinal number as the set of counting numbers, or It turns out that many sets of numbers have cardinal number EXAMPLE Show that the set of whole numbers 0,,,, has cardinal number This problem is easily stated, but not quite so easily solved All we really know about is that it is the cardinal number of the set of counting numbers (by definition) To show that another set, such as the whole numbers, also has as its cardinal number, we must apparently show that set to be equivalent to the set of counting numbers And equivalence is established by a one-to-one correspondence between the two sets This sequence of thoughts, involving just a few basic ideas, leads us to a plan: exhibit a one-to-one correspondence between the counting numbers and the whole

3 Infinite Sets and Their Cardinalities 8 numbers Our strategy will be to sketch such a correspondence, showing exactly how each counting number is paired with a unique whole number In the correspondence,,,,,,, n, j j j j j j j 0,,,,,,, n, Counting numbers Whole numbers Number Lore of the Aleph-bet Aleph and other letters of the Hebrew alphabet are shown on a Kabbalistic diagram representing one of the ten emanations of God during Creation Kabbalah, the ultramystical tradition within Judaism, arose in the fifth century and peaked in the sixteenth century in both Palestine and Poland Kabbalists believed that the Bible held mysteries that could be discovered in permutations, combinations, and anagrams of its very letters They also read the numerical value of letters in a word by the technique called Gematria (from geometry?) This was possible since each letter in the aleph-bet has a numerical value (aleph ), and thus a numeration system exists The letter Y stands for 0, so should be YH (0 ) However, YH is a form of the Holy Name, so instead TW (9 ) is the symbol the pairing of the counting number n with the whole number n continues indefinitely, with neither set containing any element not used up in the pairing process So, even though the set of whole numbers has one more element (the number 0) than the set of counting numbers, and thus should have cardinal number, the above correspondence proves that both sets have the same cardinal number That is, This result shows that intuition is a poor guide for dealing with infinite sets Intuitively, it is obvious that there are more whole numbers than counting numbers However, since the sets can be placed in a one-to-one correspondence, the two sets have the same cardinal number The set,, 7 is a proper subset of the set,, 7, 8, and there is no way to place these two sets in a one-to-one correspondence On the other hand, the set of counting numbers is a proper subset of the set of whole numbers, and Example showed that these two sets can be placed in a one-to-one correspondence The only way a proper subset of a set can possibly be placed in a one-to-one correspondence with the set itself is if both sets are infinite In fact, this important property is used as the definition of an infinite set Infinite Set A set is infinite if it can be placed in a one-to-one correspondence with a proper subset of itself EXAMPLE Show that the set of integers,,,, 0,,,, has cardinal number A one-to-one correspondence can be set up between the set of integers and the set of counting numbers, as follows:,,,,,, 7,, n, n, j j j j j j j j j 0,,,,,,,, n, n, Because of this one-to-one correspondence, the cardinal number of the set of integers is the same as the cardinal number of the set of counting numbers, If the elements of a set can be placed in a specific order, such as the set of integers in Example, that set has cardinal number

4 8 CHAPTER The Basic Concepts of Set Theory Notice that the one-to-one correspondence of Example also proves that the set of integers is infinite The set of integers was placed in a one-to-one correspondence with a proper subset of itself As shown by Example, there are just as many integers as there are counting numbers This result is not at all intuitive However, the next result is even less intuitive We know that there is an infinite number of fractions between any two counting numbers For example, there is an infinite set of fractions,, 7 8,,, between the counting numbers 0 and This should imply that there are more fractions than counting numbers It turns out, however, that there are just as many fractions as counting numbers, as shown by the next example EXAMPLE Show that the cardinal number of the set of rational numbers is To show that the cardinal number of the set of rational numbers is, first show that a one-to-one correspondence may be set up between the set of nonnegative rational numbers and the counting numbers This is done by the following ingenious scheme, devised by Georg Cantor Look at Figure The nonnegative rational numbers whose denominators are are written in the first row; those whose denominators are are written in the second row, and so on Every nonnegative rational number appears in this list sooner or later For example, 7 89 is in row 89 and column 7 0 _ FIGURE To set up a one-to-one correspondence between the set of nonnegative rationals and the set of counting numbers, follow the path drawn in Figure Let 0 correspond to, let correspond to, to, to (skip, since ), to, to, and so on The numbers under the colored disks are omitted, since they can be reduced to lower terms, and were thus included earlier in the listing This procedure sets up a one-to-one correspondence between the set of nonnegative rationals and the counting numbers, showing that both of these sets have the same cardinal number, Now by using the method of Example, that is, letting each negative number follow its corresponding positive number, we can extend this correspondence to include negative rational numbers as well Thus, the set of all rational numbers has cardinal number

5 Infinite Sets and Their Cardinalities 87 A set is called countable if it is finite, or if it has cardinal number All the infinite sets of numbers discussed so far the counting numbers, the whole numbers, the integers, and the rational numbers are countable It seems now that every set is countable, but this is not true The next example shows that the set of real numbers is not countable EXAMPLE Show that the set of all real numbers does not have cardinal number There are two possibilities: The Barber Paradox This is a version of a paradox of set theory that Bertrand Russell proposed in the early twentieth century The men in a village are of two types: men who do not shave themselves and men who do The village barber shaves all men who do not shave themselves and he shaves only those men But who shaves the barber? The barber cannot shave himself If he did, he would fall into the category of men who shave themselves However, () above states that the barber does not shave such men So the barber does not shave himself But then he falls into the category of men who do not shave themselves According to (), the barber shaves all of these men; hence, the barber shaves himself, too We find that the barber cannot shave himself, yet the barber does shave himself a paradox The set of real numbers has cardinal number The set of real numbers does not have cardinal number Assume for the time being that the first statement is true If the first statement is true, then a one-to-one correspondence can be set up between the set of real numbers and the set of counting numbers We do not know what sort of correspondence this might be, but assume it can be done In a later chapter we show that every real number can be written as a decimal number (or simply decimal ) Thus, in the one-to-one correspondence we are assuming, some decimal corresponds to the counting number, some decimal corresponds to, and so on Suppose the correspondence is as follows: i i i 798 i 97 and so on Assuming the existence of a one-to-one correspondence between the counting numbers and the real numbers means that every decimal is in the list above Let s construct a new decimal K as follows The first decimal in the above list has as its first digit; let K start as K We picked since ; we could have used any other digit except Since the second digit of the second decimal in the list is, we let K (since ) The third digit of the third decimal is, so let K 7 (since 7 ) The fourth digit of the fourth decimal is, so let K 7 (since ) Continue K in this way Is K in the list that we assumed to contain all decimals? The first decimal in the list differs from K in at least the first position (K starts with, and the first decimal in the list starts with ) The second decimal in the list differs from K in at least the second position, and the nth decimal in the list differs from K in at least the nth position Every decimal in the list differs from K in at least one position, so that K cannot possibly be in the list In summary: We assume every decimal is in the list above The decimal K is not in the list Since these statements cannot both be true, the original assumption has led to a contradiction This forces the acceptance of the only possible alternative to the original assumption: it is not possible to set up a one-to-one correspondence between the set of reals and the set of counting numbers; the cardinal number of the set of reals is not equal to

6 88 CHAPTER The Basic Concepts of Set Theory The set of counting numbers is a proper subset of the set of real numbers Because of this, it would seem reasonable to say that the cardinal number of the set of reals, commonly written c, is greater than (The letter c here represents continuum) Other, even larger, infinite cardinal numbers can be constructed For example, the set of all subsets of the set of real numbers has a cardinal number larger than c Continuing this process of finding cardinal numbers of sets of subsets, more and more, larger and larger infinite cardinal numbers are produced The six most important infinite sets of numbers were listed in an earlier section All of them have been dealt with in this section except the irrational numbers The irrationals have decimal representations, so they are all included among the real numbers Since the irrationals are a subset of the reals, you might guess that the irrationals have cardinal number, just like the rationals However, since the union of the rationals and the irrationals is all the reals, that would imply that c But Example showed that this is not the case Hence, a better guess is that the cardinal number of the irrationals is c (the same as that of the reals) This is, in fact, true The major infinite sets of numbers, with their cardinal numbers, are now summarized Cardinal Numbers of Infinite Number Sets Infinite Set Natural or Counting numbers Whole numbers Integers Rational numbers Irrational numbers Real numbers Cardinal Number c c

Cardinality. Hebrew alphabet). We write S = ℵ 0 and say that S has cardinality aleph null.

Cardinality. Hebrew alphabet). We write S = ℵ 0 and say that S has cardinality aleph null. Section 2.5 1 Cardinality Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a one-to-one correspondence (i.e., a bijection) from A to

More information

DVA325 Formal Languages, Automata and Models of Computation (FABER)

DVA325 Formal Languages, Automata and Models of Computation (FABER) DVA325 Formal Languages, Automata and Models of Computation (FABER) Lecture 1 - Introduction School of Innovation, Design and Engineering Mälardalen University 11 November 2014 Abu Naser Masud FABER November

More information

Cardinality revisited

Cardinality revisited Cardinality revisited A set is finite (has finite cardinality) if its cardinality is some (finite) integer n. Two sets A,B have the same cardinality iff there is a one-to-one correspondence from A to B

More information

Sets. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, Outline Sets Equality Subset Empty Set Cardinality Power Set

Sets. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, Outline Sets Equality Subset Empty Set Cardinality Power Set Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, 2012 Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) Gazihan Alankuş (Based on original slides by Brahim Hnich

More information

1. The empty set is a proper subset of every set. Not true because the empty set is not a proper subset of itself! is the power set of A.

1. The empty set is a proper subset of every set. Not true because the empty set is not a proper subset of itself! is the power set of A. MAT 101 Solutions to Sample Questions for Exam 1 True or False Questions Answers: 1F, 2F, 3F, 4T, 5T, 6T, 7T 1. The empty set is a proper subset of every set. Not true because the empty set is not a proper

More information

CSE 20 DISCRETE MATH. Fall

CSE 20 DISCRETE MATH. Fall CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Define and compute the cardinality of a set. Use functions to compare the sizes of sets. Classify sets

More information

Finite and Infinite Sets

Finite and Infinite Sets Finite and Infinite Sets MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Basic Definitions Definition The empty set has 0 elements. If n N, a set S is said to have

More information

Lecture 18 - Counting

Lecture 18 - Counting Lecture 18 - Counting 6.0 - April, 003 One of the most common mathematical problems in computer science is counting the number of elements in a set. This is often the core difficulty in determining a program

More information

Cardinality and Bijections

Cardinality and Bijections Countable and Cardinality and Bijections Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 13, 2012 Countable and Countable and Countable and How to count elements in a set? How

More information

ECS 20 (Spring 2013) Phillip Rogaway Lecture 1

ECS 20 (Spring 2013) Phillip Rogaway Lecture 1 ECS 20 (Spring 2013) Phillip Rogaway Lecture 1 Today: Introductory comments Some example problems Announcements course information sheet online (from my personal homepage: Rogaway ) first HW due Wednesday

More information

Counting. Chapter 6. With Question/Answer Animations

Counting. Chapter 6. With Question/Answer Animations . All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Counting Chapter

More information

An ordered collection of counters in rows or columns, showing multiplication facts.

An ordered collection of counters in rows or columns, showing multiplication facts. Addend A number which is added to another number. Addition When a set of numbers are added together. E.g. 5 + 3 or 6 + 2 + 4 The answer is called the sum or the total and is shown by the equals sign (=)

More information

Sets. Definition A set is an unordered collection of objects called elements or members of the set.

Sets. Definition A set is an unordered collection of objects called elements or members of the set. Sets Definition A set is an unordered collection of objects called elements or members of the set. Sets Definition A set is an unordered collection of objects called elements or members of the set. Examples:

More information

Grade 7/8 Math Circles February 21 st /22 nd, Sets

Grade 7/8 Math Circles February 21 st /22 nd, Sets Faculty of Mathematics Waterloo, Ontario N2L 3G1 Sets Grade 7/8 Math Circles February 21 st /22 nd, 2017 Sets Centre for Education in Mathematics and Computing A set is a collection of unique objects i.e.

More information

1. Functions and set sizes 2. Infinite set sizes. ! Let X,Y be finite sets, f:x!y a function. ! Theorem: If f is injective then X Y.

1. Functions and set sizes 2. Infinite set sizes. ! Let X,Y be finite sets, f:x!y a function. ! Theorem: If f is injective then X Y. 2 Today s Topics: CSE 20: Discrete Mathematics for Computer Science Prof. Miles Jones 1. Functions and set sizes 2. 3 4 1. Functions and set sizes! Theorem: If f is injective then Y.! Try and prove yourself

More information

Countability. Jason Filippou UMCP. Jason Filippou UMCP) Countability / 12

Countability. Jason Filippou UMCP. Jason Filippou UMCP) Countability / 12 Countability Jason Filippou CMSC250 @ UMCP 06-23-2016 Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 1 / 12 Outline 1 Infinity 2 Countability of integers and rationals 3 Uncountability of R Jason

More information

Solution: This is sampling without repetition and order matters. Therefore

Solution: This is sampling without repetition and order matters. Therefore June 27, 2001 Your name It is important that you show your work. The total value of this test is 220 points. 1. (10 points) Use the Euclidean algorithm to solve the decanting problem for decanters of sizes

More information

The next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:

The next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following: CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such

More information

EXPLAINING THE SHAPE OF RSK

EXPLAINING THE SHAPE OF RSK EXPLAINING THE SHAPE OF RSK SIMON RUBINSTEIN-SALZEDO 1. Introduction There is an algorithm, due to Robinson, Schensted, and Knuth (henceforth RSK), that gives a bijection between permutations σ S n and

More information

MULTIPLES, FACTORS AND POWERS

MULTIPLES, FACTORS AND POWERS The Improving Mathematics Education in Schools (TIMES) Project MULTIPLES, FACTORS AND POWERS NUMBER AND ALGEBRA Module 19 A guide for teachers - Years 7 8 June 2011 7YEARS 8 Multiples, Factors and Powers

More information

CITS2211 Discrete Structures Turing Machines

CITS2211 Discrete Structures Turing Machines CITS2211 Discrete Structures Turing Machines October 23, 2017 Highlights We have seen that FSMs and PDAs are surprisingly powerful But there are some languages they can not recognise We will study a new

More information

Class 8 - Sets (Lecture Notes)

Class 8 - Sets (Lecture Notes) Class 8 - Sets (Lecture Notes) What is a Set? A set is a well-defined collection of distinct objects. Example: A = {1, 2, 3, 4, 5} What is an element of a Set? The objects in a set are called its elements.

More information

With Question/Answer Animations. Chapter 6

With Question/Answer Animations. Chapter 6 With Question/Answer Animations Chapter 6 Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations Binomial Coefficients and Identities Generalized Permutations and

More information

Norman Do. Department of Mathematics and Statistics, The University of Melbourne, VIC

Norman Do. Department of Mathematics and Statistics, The University of Melbourne, VIC Norman Do Welcome to the Australian Mathematical Society Gazette s Puzzle Corner. Each issue will include a handful of entertaining puzzles for adventurous readers to try. The puzzles cover a range of

More information

Olympiad Combinatorics. Pranav A. Sriram

Olympiad Combinatorics. Pranav A. Sriram Olympiad Combinatorics Pranav A. Sriram August 2014 Chapter 2: Algorithms - Part II 1 Copyright notices All USAMO and USA Team Selection Test problems in this chapter are copyrighted by the Mathematical

More information

1. The chance of getting a flush in a 5-card poker hand is about 2 in 1000.

1. The chance of getting a flush in a 5-card poker hand is about 2 in 1000. CS 70 Discrete Mathematics for CS Spring 2008 David Wagner Note 15 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette wheels. Today

More information

The topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:

The topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following: CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of

More information

17. Symmetries. Thus, the example above corresponds to the matrix: We shall now look at how permutations relate to trees.

17. Symmetries. Thus, the example above corresponds to the matrix: We shall now look at how permutations relate to trees. 7 Symmetries 7 Permutations A permutation of a set is a reordering of its elements Another way to look at it is as a function Φ that takes as its argument a set of natural numbers of the form {, 2,, n}

More information

A theorem on the cores of partitions

A theorem on the cores of partitions A theorem on the cores of partitions Jørn B. Olsson Department of Mathematical Sciences, University of Copenhagen Universitetsparken 5,DK-2100 Copenhagen Ø, Denmark August 9, 2008 Abstract: If s and t

More information

12. 6 jokes are minimal.

12. 6 jokes are minimal. Pigeonhole Principle Pigeonhole Principle: When you organize n things into k categories, one of the categories has at least n/k things in it. Proof: If each category had fewer than n/k things in it then

More information

n! = n(n 1)(n 2) 3 2 1

n! = n(n 1)(n 2) 3 2 1 A Counting A.1 First principles If the sample space Ω is finite and the outomes are equally likely, then the probability measure is given by P(E) = E / Ω where E denotes the number of outcomes in the event

More information

SETS OBJECTIVES EXPECTED BACKGROUND KNOWLEDGE 1.1 SOME STANDARD NOTATIONS. Sets. MODULE - I Sets, Relations and Functions

SETS OBJECTIVES EXPECTED BACKGROUND KNOWLEDGE 1.1 SOME STANDARD NOTATIONS. Sets. MODULE - I Sets, Relations and Functions 1 SETS Let us consider the following situation : One day Mrs. and Mr. Mehta went to the market. Mr. Mehta purchased the following objects/items. "a toy, one kg sweets and a magazine". Where as Mrs. Mehta

More information

Lecture 2: Sum rule, partition method, difference method, bijection method, product rules

Lecture 2: Sum rule, partition method, difference method, bijection method, product rules Lecture 2: Sum rule, partition method, difference method, bijection method, product rules References: Relevant parts of chapter 15 of the Math for CS book. Discrete Structures II (Summer 2018) Rutgers

More information

CIS 2033 Lecture 6, Spring 2017

CIS 2033 Lecture 6, Spring 2017 CIS 2033 Lecture 6, Spring 2017 Instructor: David Dobor February 2, 2017 In this lecture, we introduce the basic principle of counting, use it to count subsets, permutations, combinations, and partitions,

More information

The Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n

The Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n Chapter 5 Chapter Summary 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.5 Generalized Permutations and Combinations Section 5.1 The Product Rule The Product

More information

In this paper, we discuss strings of 3 s and 7 s, hereby dubbed dreibens. As a first step

In this paper, we discuss strings of 3 s and 7 s, hereby dubbed dreibens. As a first step Dreibens modulo A New Formula for Primality Testing Arthur Diep-Nguyen In this paper, we discuss strings of s and s, hereby dubbed dreibens. As a first step towards determining whether the set of prime

More information

Ivan Guo. Broken bridges There are thirteen bridges connecting the banks of River Pluvia and its six piers, as shown in the diagram below:

Ivan Guo. Broken bridges There are thirteen bridges connecting the banks of River Pluvia and its six piers, as shown in the diagram below: Ivan Guo Welcome to the Australian Mathematical Society Gazette s Puzzle Corner No. 20. Each issue will include a handful of fun, yet intriguing, puzzles for adventurous readers to try. The puzzles cover

More information

COUNTING AND PROBABILITY

COUNTING AND PROBABILITY CHAPTER 9 COUNTING AND PROBABILITY Copyright Cengage Learning. All rights reserved. SECTION 9.2 Possibility Trees and the Multiplication Rule Copyright Cengage Learning. All rights reserved. Possibility

More information

Cardinality of Accumulation Points of Infinite Sets

Cardinality of Accumulation Points of Infinite Sets International Mathematical Forum, Vol. 11, 2016, no. 11, 539-546 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6224 Cardinality of Accumulation Points of Infinite Sets A. Kalapodi CTI

More information

10 GRAPHING LINEAR EQUATIONS

10 GRAPHING LINEAR EQUATIONS 0 GRAPHING LINEAR EQUATIONS We now expand our discussion of the single-variable equation to the linear equation in two variables, x and y. Some examples of linear equations are x+ y = 0, y = 3 x, x= 4,

More information

MATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability)

MATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability) MATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability) Last modified: November 10, 2004 This follows very closely Apostol, Chapter 13, the course pack. Attachments

More information

CALCULATING SQUARE ROOTS BY HAND By James D. Nickel

CALCULATING SQUARE ROOTS BY HAND By James D. Nickel By James D. Nickel Before the invention of electronic calculators, students followed two algorithms to approximate the square root of any given number. First, we are going to investigate the ancient Babylonian

More information

Final exam. Question Points Score. Total: 150

Final exam. Question Points Score. Total: 150 MATH 11200/20 Final exam DECEMBER 9, 2016 ALAN CHANG Please present your solutions clearly and in an organized way Answer the questions in the space provided on the question sheets If you run out of room

More information

6.2 Modular Arithmetic

6.2 Modular Arithmetic 6.2 Modular Arithmetic Every reader is familiar with arithmetic from the time they are three or four years old. It is the study of numbers and various ways in which we can combine them, such as through

More information

Order and Compare Rational and Irrational numbers and Locate on the number line

Order and Compare Rational and Irrational numbers and Locate on the number line 806.2.1 Order and Compare Rational and Irrational numbers and Locate on the number line Rational Number ~ any number that can be made by dividing one integer by another. The word comes from the word "ratio".

More information

Reading 14 : Counting

Reading 14 : Counting CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 14 : Counting In this reading we discuss counting. Often, we are interested in the cardinality

More information

It is important that you show your work. The total value of this test is 220 points.

It is important that you show your work. The total value of this test is 220 points. June 27, 2001 Your name It is important that you show your work. The total value of this test is 220 points. 1. (10 points) Use the Euclidean algorithm to solve the decanting problem for decanters of sizes

More information

Beyond Infinity? Joel Feinstein. School of Mathematical Sciences University of Nottingham

Beyond Infinity? Joel Feinstein. School of Mathematical Sciences University of Nottingham Beyond Infinity? Joel Feinstein School of Mathematical Sciences University of Nottingham 2006-2007 The serious mathematics behind this talk is due to the great mathematicians David Hilbert (1862 1943)

More information

Notes for Recitation 3

Notes for Recitation 3 6.042/18.062J Mathematics for Computer Science September 17, 2010 Tom Leighton, Marten van Dijk Notes for Recitation 3 1 State Machines Recall from Lecture 3 (9/16) that an invariant is a property of a

More information

Pattern Avoidance in Unimodal and V-unimodal Permutations

Pattern Avoidance in Unimodal and V-unimodal Permutations Pattern Avoidance in Unimodal and V-unimodal Permutations Dido Salazar-Torres May 16, 2009 Abstract A characterization of unimodal, [321]-avoiding permutations and an enumeration shall be given.there is

More information

MATH 2420 Discrete Mathematics Lecture notes

MATH 2420 Discrete Mathematics Lecture notes MATH 2420 Discrete Mathematics Lecture notes Series and Sequences Objectives: Introduction. Find the explicit formula for a sequence. 2. Be able to do calculations involving factorial, summation and product

More information

Combinatorial Proofs

Combinatorial Proofs Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A

More information

Combinatorics. Chapter Permutations. Counting Problems

Combinatorics. Chapter Permutations. Counting Problems Chapter 3 Combinatorics 3.1 Permutations Many problems in probability theory require that we count the number of ways that a particular event can occur. For this, we study the topics of permutations and

More information

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13 CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13 Introduction to Discrete Probability In the last note we considered the probabilistic experiment where we flipped a

More information

Chapter 1. Set Theory

Chapter 1. Set Theory Chapter 1 Set Theory 1 Section 1.1: Types of Sets and Set Notation Set: A collection or group of distinguishable objects. Ex. set of books, the letters of the alphabet, the set of whole numbers. You can

More information

Tiling Problems. This document supersedes the earlier notes posted about the tiling problem. 1 An Undecidable Problem about Tilings of the Plane

Tiling Problems. This document supersedes the earlier notes posted about the tiling problem. 1 An Undecidable Problem about Tilings of the Plane Tiling Problems This document supersedes the earlier notes posted about the tiling problem. 1 An Undecidable Problem about Tilings of the Plane The undecidable problems we saw at the start of our unit

More information

Tilings with T and Skew Tetrominoes

Tilings with T and Skew Tetrominoes Quercus: Linfield Journal of Undergraduate Research Volume 1 Article 3 10-8-2012 Tilings with T and Skew Tetrominoes Cynthia Lester Linfield College Follow this and additional works at: http://digitalcommons.linfield.edu/quercus

More information

Math Fundamentals for Statistics (Math 52) Unit 2:Number Line and Ordering. By Scott Fallstrom and Brent Pickett The How and Whys Guys.

Math Fundamentals for Statistics (Math 52) Unit 2:Number Line and Ordering. By Scott Fallstrom and Brent Pickett The How and Whys Guys. Math Fundamentals for Statistics (Math 52) Unit 2:Number Line and Ordering By Scott Fallstrom and Brent Pickett The How and Whys Guys Unit 2 Page 1 2.1: Place Values We just looked at graphing ordered

More information

Topics to be covered

Topics to be covered Basic Counting 1 Topics to be covered Sum rule, product rule, generalized product rule Permutations, combinations Binomial coefficients, combinatorial proof Inclusion-exclusion principle Pigeon Hole Principle

More information

Pre-Test Unit 7: Real Numbers KEY

Pre-Test Unit 7: Real Numbers KEY Pre-Test Unit 7: Real Numbers KEY No calculator necessary. Please do not use a calculator. Convert the following fraction to a decimal or decimal to a fraction. (5 pts; 3 pts for correct set-up/work, 2

More information

Real Numbers and the Number Line. Unit 1 Lesson 3

Real Numbers and the Number Line. Unit 1 Lesson 3 Real Numbers and the Number Line Unit 1 Lesson 3 Students will be able to: graph and compare real numbers using the number line. Key Vocabulary: Real Number Rational Number Irrational number Non-Integers

More information

Compound Probability. Set Theory. Basic Definitions

Compound Probability. Set Theory. Basic Definitions Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic

More information

Sample Spaces, Events, Probability

Sample Spaces, Events, Probability Sample Spaces, Events, Probability CS 3130/ECE 3530: Probability and Statistics for Engineers August 28, 2014 Sets A set is a collection of unique objects. Sets A set is a collection of unique objects.

More information

Game Theory and Algorithms Lecture 19: Nim & Impartial Combinatorial Games

Game Theory and Algorithms Lecture 19: Nim & Impartial Combinatorial Games Game Theory and Algorithms Lecture 19: Nim & Impartial Combinatorial Games May 17, 2011 Summary: We give a winning strategy for the counter-taking game called Nim; surprisingly, it involves computations

More information

Math236 Discrete Maths with Applications

Math236 Discrete Maths with Applications Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 43 The Multiplication Principle Theorem Let S be a set of k-tuples (s 1,

More information

CLASS NOTES. A mathematical proof is an argument which convinces other people that something is true.

CLASS NOTES. A mathematical proof is an argument which convinces other people that something is true. Propositional Statements A mathematical proof is an argument which convinces other people that something is true. The implication If p then q written as p q means that if p is true, then q must also be

More information

Proof that Mersenne Prime Numbers are Infinite and that Even Perfect Numbers are Infinite

Proof that Mersenne Prime Numbers are Infinite and that Even Perfect Numbers are Infinite Proof that Mersenne Prime Numbers are Infinite and that Even Perfect Numbers are Infinite Stephen Marshall 7 November 208 Abstract Mersenne prime is a prime number that is one less than a power of two.

More information

Pre-Algebra Unit 1: Number Sense Unit 1 Review Packet

Pre-Algebra Unit 1: Number Sense Unit 1 Review Packet Pre-Algebra Unit 1: Number Sense Unit 1 Review Packet Target 1: Writing Repeating Decimals in Rational Form Remember the goal is to get rid of the repeating decimal so we can write the number in rational

More information

Discrete Mathematics and Probability Theory Spring 2018 Ayazifar and Rao Midterm 2 Solutions

Discrete Mathematics and Probability Theory Spring 2018 Ayazifar and Rao Midterm 2 Solutions CS 70 Discrete Mathematics and Probability Theory Spring 2018 Ayazifar and Rao Midterm 2 Solutions PRINT Your Name: Oski Bear SIGN Your Name: OS K I PRINT Your Student ID: CIRCLE your exam room: Pimentel

More information

Some Fine Combinatorics

Some Fine Combinatorics Some Fine Combinatorics David P. Little Department of Mathematics Penn State University University Park, PA 16802 Email: dlittle@math.psu.edu August 3, 2009 Dedicated to George Andrews on the occasion

More information

Chapter 1. Probability

Chapter 1. Probability Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

More information

The study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability

The study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch

More information

Pythagorean Theorem Unit

Pythagorean Theorem Unit Pythagorean Theorem Unit TEKS covered: ~ Square roots and modeling square roots, 8.1(C); 7.1(C) ~ Real number system, 8.1(A), 8.1(C); 7.1(A) ~ Pythagorean Theorem and Pythagorean Theorem Applications,

More information

POKER (AN INTRODUCTION TO COUNTING)

POKER (AN INTRODUCTION TO COUNTING) POKER (AN INTRODUCTION TO COUNTING) LAMC INTERMEDIATE GROUP - 10/27/13 If you want to be a succesful poker player the first thing you need to do is learn combinatorics! Today we are going to count poker

More information

Figurate Numbers. by George Jelliss June 2008 with additions November 2008

Figurate Numbers. by George Jelliss June 2008 with additions November 2008 Figurate Numbers by George Jelliss June 2008 with additions November 2008 Visualisation of Numbers The visual representation of the number of elements in a set by an array of small counters or other standard

More information

Optimal Results in Staged Self-Assembly of Wang Tiles

Optimal Results in Staged Self-Assembly of Wang Tiles Optimal Results in Staged Self-Assembly of Wang Tiles Rohil Prasad Jonathan Tidor January 22, 2013 Abstract The subject of self-assembly deals with the spontaneous creation of ordered systems from simple

More information

Discrete Structures Lecture Permutations and Combinations

Discrete Structures Lecture Permutations and Combinations Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these

More information

Week 1: Probability models and counting

Week 1: Probability models and counting Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model

More information

A Group-theoretic Approach to Human Solving Strategies in Sudoku

A Group-theoretic Approach to Human Solving Strategies in Sudoku Colonial Academic Alliance Undergraduate Research Journal Volume 3 Article 3 11-5-2012 A Group-theoretic Approach to Human Solving Strategies in Sudoku Harrison Chapman University of Georgia, hchaps@gmail.com

More information

Dyck paths, standard Young tableaux, and pattern avoiding permutations

Dyck paths, standard Young tableaux, and pattern avoiding permutations PU. M. A. Vol. 21 (2010), No.2, pp. 265 284 Dyck paths, standard Young tableaux, and pattern avoiding permutations Hilmar Haukur Gudmundsson The Mathematics Institute Reykjavik University Iceland e-mail:

More information

Honors Precalculus Chapter 9 Summary Basic Combinatorics

Honors Precalculus Chapter 9 Summary Basic Combinatorics Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each

More information

TOPOLOGY, LIMITS OF COMPLEX NUMBERS. Contents 1. Topology and limits of complex numbers 1

TOPOLOGY, LIMITS OF COMPLEX NUMBERS. Contents 1. Topology and limits of complex numbers 1 TOPOLOGY, LIMITS OF COMPLEX NUMBERS Contents 1. Topology and limits of complex numbers 1 1. Topology and limits of complex numbers Since we will be doing calculus on complex numbers, not only do we need

More information

Discrete Mathematics with Applications MATH236

Discrete Mathematics with Applications MATH236 Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet

More information

Teaching the TERNARY BASE

Teaching the TERNARY BASE Features Teaching the TERNARY BASE Using a Card Trick SUHAS SAHA Any sufficiently advanced technology is indistinguishable from magic. Arthur C. Clarke, Profiles of the Future: An Inquiry Into the Limits

More information

Focus on Mathematics

Focus on Mathematics Focus on Mathematics Year 4 Pre-Learning Tasks Number Pre-learning tasks are used at the start of each new topic in Maths. The children are grouped after the pre-learning task is marked to ensure the work

More information

From a Ball Game to Incompleteness

From a Ball Game to Incompleteness From a Ball Game to Incompleteness Arindama Singh We present a ball game that can be continued as long as we wish. It looks as though the game would never end. But by applying a result on trees, we show

More information

Section 8.1. Sequences and Series

Section 8.1. Sequences and Series Section 8.1 Sequences and Series Sequences Definition A sequence is a list of numbers. Definition A sequence is a list of numbers. A sequence could be finite, such as: 1, 2, 3, 4 Definition A sequence

More information

Halting Problem. Implement HALT? Today. Halt does not exist. Halt and Turing. Another view of proof: diagonalization. P - program I - input.

Halting Problem. Implement HALT? Today. Halt does not exist. Halt and Turing. Another view of proof: diagonalization. P - program I - input. Today. Halting Problem. Implement HALT? Finish undecidability. Start counting. HALT (P,I) P - program I - input. Determines if P(I) (P run on I) halts or loops forever. Notice: Need a computer with the

More information

I.M.O. Winter Training Camp 2008: Invariants and Monovariants

I.M.O. Winter Training Camp 2008: Invariants and Monovariants I.M.. Winter Training Camp 2008: Invariants and Monovariants n math contests, you will often find yourself trying to analyze a process of some sort. For example, consider the following two problems. Sample

More information

By Scott Fallstrom and Brent Pickett The How and Whys Guys

By Scott Fallstrom and Brent Pickett The How and Whys Guys Math Fundamentals for Statistics I (Math 52) Unit 2:Number Line and Ordering By Scott Fallstrom and Brent Pickett The How and Whys Guys This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike

More information

Mathematics Competition Practice Session 6. Hagerstown Community College: STEM Club November 20, :00 pm - 1:00 pm STC-170

Mathematics Competition Practice Session 6. Hagerstown Community College: STEM Club November 20, :00 pm - 1:00 pm STC-170 2015-2016 Mathematics Competition Practice Session 6 Hagerstown Community College: STEM Club November 20, 2015 12:00 pm - 1:00 pm STC-170 1 Warm-Up (2006 AMC 10B No. 17): Bob and Alice each have a bag

More information

Sec 5.1 The Basics of Counting

Sec 5.1 The Basics of Counting 1 Sec 5.1 The Basics of Counting Combinatorics, the study of arrangements of objects, is an important part of discrete mathematics. In this chapter, we will learn basic techniques of counting which has

More information

MUMS seminar 24 October 2008

MUMS seminar 24 October 2008 MUMS seminar 24 October 2008 Tiles have been used in art and architecture since the dawn of civilisation. Toddlers grapple with tiling problems when they pack away their wooden blocks and home renovators

More information

Team Round University of South Carolina Math Contest, 2018

Team Round University of South Carolina Math Contest, 2018 Team Round University of South Carolina Math Contest, 2018 1. This is a team round. You have one hour to solve these problems as a team, and you should submit one set of answers for your team as a whole.

More information

RMT 2015 Power Round Solutions February 14, 2015

RMT 2015 Power Round Solutions February 14, 2015 Introduction Fair division is the process of dividing a set of goods among several people in a way that is fair. However, as alluded to in the comic above, what exactly we mean by fairness is deceptively

More information

14.2 Limits and Continuity

14.2 Limits and Continuity 14 Partial Derivatives 14.2 Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Let s compare the behavior of the functions Tables 1 2 show values of f(x,

More information

Permutations and Combinations

Permutations and Combinations Permutations and Combinations Introduction Permutations and combinations refer to number of ways of selecting a number of distinct objects from a set of distinct objects. Permutations are ordered selections;

More information

NON-OVERLAPPING PERMUTATION PATTERNS. To Doron Zeilberger, for his Sixtieth Birthday

NON-OVERLAPPING PERMUTATION PATTERNS. To Doron Zeilberger, for his Sixtieth Birthday NON-OVERLAPPING PERMUTATION PATTERNS MIKLÓS BÓNA Abstract. We show a way to compute, to a high level of precision, the probability that a randomly selected permutation of length n is nonoverlapping. As

More information

Permutation Groups. Every permutation can be written as a product of disjoint cycles. This factorization is unique up to the order of the factors.

Permutation Groups. Every permutation can be written as a product of disjoint cycles. This factorization is unique up to the order of the factors. Permutation Groups 5-9-2013 A permutation of a set X is a bijective function σ : X X The set of permutations S X of a set X forms a group under function composition The group of permutations of {1,2,,n}

More information

Building Number Sense K-2

Building Number Sense K-2 Roll A Tower - Let's Race! 1 2 3 4 5 6 Building Number Sense K-2 Counting Activities Building Instructional Leaders Across Oregon Developing Algebraic Thinking Session 1 Winter 2009 Fifty Chart 1 2 3 4

More information