Binomial Coefficients MATH Benjamin V.C. Collins, James A. Swenson MATH 2730
|
|
- Gladys Cunningham
- 5 years ago
- Views:
Transcription
1 MATH 2730 Benjamin V.C. Collins James A. Swenson
2 Binomial coefficients count subsets Definition Suppose A = n. The number of k-element subsets in A is a binomial coefficient, denoted by ( n k or n C k or C(n, k, and pronounced n choose k. Example There are six 2-element subsets in any 4-element set, so ( 4 2 = 6. In {a, b, c, d}, they are {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}.
3 First examples Example If n N: ( n 0 = ( n n = ( n = n (n ( n n = n (n
4 Triangular numbers Proposition Let n Z. If n 2, then ( n n = t. 2 t=0 Proof. We ask: How many 2-element subsets are there in {, 2,..., n}? By definition, the answer is ( n 2. On the other hand, there are: 0 subsets whose greatest element is, subset whose greatest element is 2,. t subsets whose greatest element is t +,. n subsets whose greatest element is n. n In total, this makes (n = t subsets. t=0
5 Formula for binomial coefficients Theorem If n, k N and k n, then ( n k = n! k!(n k!.
6 Formula for binomial coefficients Theorem If n, k N and k n, then ( n k = n! k!(n k!. Combinatorial proof. Let A be a set of n elements. We ask: How many k-element subsets are there in A? By definition, the answer is ( n k. On the other hand, let P denote the set of orderings of A; thus P = n!. An element of P can be used to define a k-element subset: just pick the first k elements. We say two orderings are equivalent provided they have the same first k elements, not necessarily in the same order. This is an equivalence relation, and each equivalence class represents a unique k-element subset. Each equivalence class contains k!(n k! permutations, by the n! Multiplication Principle, so there are k!(n k! k-element subsets in A.
7 Exam strategies Example Drew must answer five of the eight questions on a certain exam. In how many ways can Drew choose which questions to answer? What if Drew is required to answer the first two questions?
8 Copying strategies Example The Duplicating Center has eight high-speed copiers, and seven employees who can operate them. There are four identical jobs to be done simultaneously [4000 booklets must be made, in four boxes of 000]. How many ways are there to assign these jobs to operators and machines? Solution. There are ( ( 7 4 ways to choose the employees to do the job, and 8 4 ways to choose machines. Once these choices are made, there are 4! ways to assign operators to machines. Thus there are 4! ( 7 8 4( 4 ways to get the job done, by the Multiplication Principle.
9 Counting bitstrings Exercise A bitstring is a list of 0s and s. How many bitstrings of length n are there? How many bitstrings of length n contain exactly one zero? How many bitstrings of length n contain exactly two zeros? How many bitstrings of length n contain exactly k zeros? Prove: 2 n = n k=0 ( n. k
10 Animal committees Example Seven aardvarks, five giraffes, and three lions are on the advisory board at the Henry Virus Zoo. How many ways are there to select a committee of five animals? This means selecting a 5-element subset from the board, which is a set of = 5 animals. By definition, there are ( 5 5 ( of these. 5 5 = 5! 5!0! = ! = = ! 0! !
11 Animal committees Example Seven aardvarks, five giraffes, and three lions are on the advisory board at the Henry Virus Zoo. How many ways are there to select a committee of exactly three aardvarks and two giraffes? This means selecting 3 of the 7 aardvarks and 2 of the 5 giraffes; by the Multiplication Principle, there are ( 7 5 3( 2 choices. ( 7 ( = 7!5! 3!4!2!3! = !2! = = 350.
12 Animal committees Example Seven aardvarks, five giraffes, and three lions are on the advisory board at the Henry Virus Zoo. How many ways are there to select a 5-member committee with at least one lion? First, count the ones that don t have at least one lion! There are ( 2 5 of these. That leaves ( ( that have at least one lion; this is = 22. Another ( way: 3 ( 2 ( ( 2 ( ( = = 22. Someone might say: ( 3 4 ( 4 = 3003 by the Multiplication Principle: first choose one of the three lions, then choose any four of the remaining animals. Why doesn t this work? [The fact that ( 3 4 ( ( 4 = 5 5 is a coincidence.]
13 Animal committees Example Seven aardvarks, five giraffes, and three lions are on the advisory board at the Henry Virus Zoo. How many ways are there to select a 5-member committee with at least one lion and one giraffe? ( First, count the ones that have no lions. There are 2 5 of these. Next, count the ones that have no giraffes. There are ( 0 5 of these. Oops! Both times, we counted the ones that have neither a lion nor a giraffe! There are ( 7 5 of these. In all, there are ( ( ( 5 0 ( committees with at least one lion and one giraffe.
14 Animal committees Example Seven aardvarks, five giraffes, and three lions are on the advisory board at the Henry Virus Zoo.
15 How many are there? Theorem (principle of inclusion and exclusion If A and B are finite sets, then A + B = A B + A B. Corollary If A is the set of committees with no lions and B is the set of committees with no giraffes, then the number of committees with no lions or no giraffes is A B = A + B A B.
16 Animal committees Example Seven aardvarks, five giraffes, and three lions are on the advisory board at the Henry Virus Zoo. How many ways are there to select a 5-member committee with ( at ( least one aardvark, one giraffe, and one lion? ( 5 0 ( 5 8 ( ( ( ( 5 0 5
17 Have you seen this before? Pascal s triangle
18 Pascal s triangle ( 6 0 ( 0 ( 0 ( ( 0 2 ( 2 ( 2 ( ( 3 ( 3 ( 3 ( ( 4 ( 4 ( 4 ( 4 ( ( 5 ( 5 ( 5 ( 5 ( 5 0 ( ( 6 ( 6 ( 6 ( 6 ( Pascal s triangle
19 Pascal s triangle ( 6 0 ( 0 ( 0 ( ( 0 2 ( 2 ( 2 ( ( 3 ( 3 ( 3 ( ( 4 ( 4 ( 4 ( 4 ( ( 5 ( 5 ( 5 ( 5 ( 5 0 ( ( 6 ( 6 ( 6 ( 6 ( Pascal s triangle
20 Pascal s triangle ( ( 6 ( 6 ( ( 6 Using Pascal s identity
21 Symmetry Proposition If n, k N, then ( ( n k = n Example n k. ( 6 2 ( 6 4 = = 6! 2!(6 2! = = 5; 6! 4!(6 4! = = 5.
22 Symmetry Proposition If n, k N, then ( ( n k = n Algebraic proof. By our formula, ( n = n k n k. n! (n k!(n (n k! = n! (n k!k! = n! k!(n k! = ( n. k
23 Symmetry Proposition If n, k N, then ( ( n k = n n k. 2-elt. subsets 4-elt. subsets 2-elt. subsets 4-elt. subsets {, 2} {3, 4, 5, 6} {2, 6} {, 3, 4, 5} {, 3} {2, 4, 5, 6} {3, 4} {, 2, 5, 6} {, 4} {2, 3, 5, 6} {3, 5} {, 2, 4, 6} {, 5} {2, 3, 4, 6} {3, 6} {, 2, 4, 5} {, 6} {2, 3, 4, 5} {4, 5} {, 2, 3, 6} {2, 3} {, 4, 5, 6} {4, 6} {, 2, 3, 5} {2, 4} {, 3, 5, 6} {5, 6} {, 2, 3, 4} {2, 5} {, 3, 4, 6}
24 Symmetry Proposition If n, k N, then ( ( n k = n n k. Combinatorial proof. Let 0 k n, and let A be a universal set of n elements. For every k-element subset B A, the complement B is an (n k-element subset. Since every k-element subset has a unique complement, the number of k-element subsets in A equals the number of (n k-element subsets in A, which is what we wanted to prove.
25 Pascal s identity Theorem If n, k N and n, then ( ( n k = n ( k + n k. Blaise Pascal (
26 Pascal s identity Theorem If n, k N and n, then ( ( n k = n ( k + n k. Algebraic proof. By our formula, ( n ( n + k k = = = = (n! (k!((n (k! + (n! (k!((n k! k(n! (n k(n! + (k!(n k! (k!(n k! (n!(k + (n k (k!(n k! n! ( n (k!(n k! =. k
27 Pascal s identity Theorem If n, k N and n, then ( ( n k = n ( k + n k. Combinatorial proof. Let 0 k n, let A be a set of n elements, and let w A. We ask: How many k-element subsets are there in A? By definition, the answer is ( n k. On the other hand, ( n k is the number of k-element subsets in A that contain w, because this is the number of (k -element subsets in A \ {w}. Likewise, ( n k is the number of k-element subsets in A that do not contain w. Thus ( ( n k + n k is the answer to our question.
28 Example: combinatorial proof Exercise Prove: ( ( 2n+2 n+ = 2n ( n n n + ( 2n n. Combinatorial proof. We ask: From a group of 2n aardvarks and 2 lions, in how many ways can we form a committee of n + animals? By the definition of binomial coefficient, the answer is ( 2n+2 n+. On the other hand, let s call the lions Leo and Lola. There are ( 2n n committees that include Leo but not Lola, and ( 2n n others that include Lola but not Leo. There are also ( 2n n committees that include both lions, and ( 2n n+ committees that include neither. All told, there are ( ( 2n n n ( n + 2n n possible committees.
29 Why are they called binomial coefficients? (x + y 0 = (x + y = x + y (x + y 2 = x 2 + 2xy + y 2 (x + y 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x + y 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 Theorem (binomial theorem n If n N, then (x + y n =. k=0 ( n k x n k y k.
30 Why are they called binomial coefficients? Theorem (binomial theorem n If n N, then (x + y n = k=0 ( n k x n k y k. Exercise Find the coefficient of x 0 y 3 in (x y 3. Solution. By the binomial theorem, 3 ( 3 (x y 3 = x 3 k ( y k = k k=0 3 k=0 ( 3 ( k x 3 k y k. k So the coefficient of x 0 y 3 is ( 3( 3 3 = 3! 3!0! = = 286.
Sets II MATH Sets II. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Sets II Benjamin V.C. Collins James A. Swenson New sets from old Suppose A and B are the sets of multiples of 2 and multiples of 5: A = {n Z : 2 n} = {..., 8, 6, 4, 2, 0, 2, 4, 6, 8,... } B =
More informationCounting. Math 301. November 24, Dr. Nahid Sultana
Basic Principles Dr. Nahid Sultana November 24, 2012 Basic Principles Basic Principles The Sum Rule The Product Rule Distinguishable Pascal s Triangle Binomial Theorem Basic Principles Combinatorics: The
More informationBinomial Coefficient Identities/Complements
Binomial Coefficient Identities/Complements CSE21 Fall 2017, Day 4 Oct 6, 2017 https://sites.google.com/a/eng.ucsd.edu/cse21-fall-2017-miles-jones/ permutation P(n,r) = n(n-1) (n-2) (n-r+1) = Terminology
More informationMath.3336: Discrete Mathematics. Combinatorics: Basics of Counting
Math.3336: Discrete Mathematics Combinatorics: Basics of Counting Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu
More informationCombinatorics. But there are some standard techniques. That s what we ll be studying.
Combinatorics Problem: How to count without counting. How do you figure out how many things there are with a certain property without actually enumerating all of them. Sometimes this requires a lot of
More informationDiscrete Mathematics & Mathematical Reasoning Chapter 6: Counting
Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 39 Chapter Summary The Basics
More informationCounting Methods. CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo
Counting Methods CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 48 Need for Counting The problem of counting
More informationLecture 4: Counting, Pigeonhole Principle, Permutations, Combinations Lecturer: Lale Özkahya
BBM 205 Discrete Mathematics Hacettepe University http://web.cs.hacettepe.edu.tr/ bbm205 Lecture 4: Counting, Pigeonhole Principle, Permutations, Combinations Lecturer: Lale Özkahya Resources: Kenneth
More informationChapter 7. Inclusion-Exclusion a.k.a. The Sieve Formula
Chapter 7. Inclusion-Exclusion a.k.a. The Sieve Formula Prof. Tesler Math 184A Fall 2019 Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Fall 2019 1 / 25 Venn diagram and set sizes A = {1, 2, 3, 4,
More informationAnnouncements. CSE 321 Discrete Structures. Counting. Counting Rules. Important cases of the Product Rule. Counting examples.
Announcements CSE 321 Discrete Structures Winter 2008 Lecture 16 Counting Readings Friday, Wednesday: Counting 6 th edition: 5.1, 5.2, 5.3, 5 th edition: 4.1, 4.2. 4.3 Lecture 16 video will be posted on
More informationBasic Combinatorics. Math 40210, Section 01 Fall Homework 8 Solutions
Basic Combinatorics Math 4010, Section 01 Fall 01 Homework 8 Solutions 1.8.1 1: K n has ( n edges, each one of which can be given one of two colors; so Kn has (n -edge-colorings. 1.8.1 3: Let χ : E(K k
More informationRelations MATH Relations. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Benjamin V.C. Collins James A. Swenson among integers equals a = b is true for some pairs (a, b) Z Z, but not for all pairs. is less than a < b is true for some pairs (a, b) Z Z, but not for
More informationCounting Strategies: Inclusion-Exclusion, Categories
Counting Strategies: Inclusion-Exclusion, Categories Russell Impagliazzo and Miles Jones Thanks to Janine Tiefenbruck http://cseweb.ucsd.edu/classes/sp16/cse21-bd/ May 4, 2016 A scheduling problem In one
More informationNotes. Combinatorics. Combinatorics II. Notes. Notes. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006
Combinatorics Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 4.1-4.6 & 6.5-6.6 of Rosen cse235@cse.unl.edu
More informationCounting with Categories and Binomial Coefficients
Counting with Categories and Binomial Coefficients CSE21 Winter 2017, Day 17 (B00), Day 12 (A00) February 22, 2017 http://vlsicad.ucsd.edu/courses/cse21-w17 When sum rule fails Rosen p. 392-394 Let A =
More informationName (please print) Mathematics Final Examination December 14, 2005 I. (4)
Mathematics 513-00 Final Examination December 14, 005 I Use a direct argument to prove the following implication: The product of two odd integers is odd Let m and n be two odd integers Since they are odd,
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 information1 The Basic Counting Principles
1 The Basic Counting Principles The Multiplication Rule If an operation consists of k steps and the first step can be performed in n 1 ways, the second step can be performed in n ways [regardless of how
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 information5 Group theory. 5.1 Binary operations
5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1
More informationUNCORRECTED SAMPLE PAGES. Extension 1. The binomial expansion and. Digital resources for this chapter
15 Pascal s In Chapter 10 we discussed the factoring of a polynomial into irreducible factors, so that it could be written in a form such as P(x) = (x 4) 2 (x + 1) 3 (x 2 + x + 1). In this chapter we will
More informationSERIES
SERIES.... This chapter revisits sequences arithmetic then geometric to see how these ideas can be extended, and how they occur in other contexts. A sequence is a list of ordered numbers, whereas a series
More informationIntroduction to Decision Sciences Lecture 11
Introduction to Decision Sciences Lecture 11 Andrew Nobel October 24, 2017 Basics of Counting Product Rule Product Rule: Suppose that the elements of a collection S can be specified by a sequence of k
More information1 Counting Collections of Functions and of Subsets.
1 Counting Collections of Functions and of Subsets See p144 All page references are to PJEccles book unless otherwise stated Let X and Y be sets Definition 11 F un (X, Y will be the set of all functions
More informationThe Inclusion Exclusion Principle
The Inclusion Exclusion Principle 1 / 29 Outline Basic Instances of The Inclusion Exclusion Principle The General Inclusion Exclusion Principle Counting Derangements Counting Functions Stirling Numbers
More informationFoundations of Computer Science Lecture 14 Advanced Counting
Foundations of Computer Science Lecture 14 Advanced Counting Sequences with Repetition Union of Overlapping Sets: Inclusion-Exclusion Pigeonhole Principle Last Time To count complex objects, construct
More informationConnection MATH Connection. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Benjamin V.C. Collins James A. Swenson Traveling Salesman Problem Image: Padberg-Rinaldi, 1987: 532 cities http://www.tsp.gatech.edu/data/usa/tours.html Walks in a graph Let G = (V, E) be a graph.
More information1 Basic Combinatorics
1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set
More informationMGF 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
More informationGreatest Common Divisor MATH Greatest Common Divisor. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Greatest Common Divisor Benjamin V.C. Collins James A. Swenson The world s least necessary definition Definition Let a, b Z, not both zero. The largest integer d such that d a and d b is called
More informationApplications. More Counting Problems. Complexity of Algorithms
Recurrences Applications More Counting Problems Complexity of Algorithms Part I Recurrences and Binomial Coefficients Paths in a Triangle P(0, 0) P(1, 0) P(1,1) P(2, 0) P(2,1) P(2, 2) P(3, 0) P(3,1) P(3,
More informationMassachusetts Institute of Technology Handout J/18.062J: Mathematics for Computer Science May 3, 2000 Professors David Karger and Nancy Lynch
Massachusetts Institute of Technology Handout 48 6.042J/18.062J: Mathematics for Computer Science May 3, 2000 Professors David Karger and Nancy Lynch Quiz 2 Solutions Problem 1 [10 points] Consider n >
More informationSet theory background for probability
Set theory background for probability Defining sets (a very naïve approach) A set is a collection of distinct objects. The objects within a set may be arbitrary, with the order of objects within them having
More informationCISC-102 Fall 2018 Week 11
page! 1 of! 26 CISC-102 Fall 2018 Pascal s Triangle ( ) ( ) An easy ( ) ( way ) to calculate ( ) a table of binomial coefficients was recognized centuries ago by mathematicians in India, ) ( ) China, Iran
More informationExpectation MATH Expectation. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Expectation Benjamin V.C. Collins James A. Swenson Average value Expectation Definition If (S, P) is a sample space, then any function with domain S is called a random variable. Idea Pick a real-valued
More informationMATH 10B METHODS OF MATHEMATICS: CALCULUS, STATISTICS AND COMBINATORICS
MATH 10B METHODS OF MATHEMATICS: CALCULUS, STATISTICS AND COMBINATORICS Lior Pachter and Lawrence C. Evans Department of Mathematics University of California Berkeley, CA 94720 January 21, 2013 Lior Pachter
More informationUnit 8: Statistics. SOL Review & SOL Test * Test: Unit 8 Statistics
Name: Block: Unit 8: Statistics Day 1 Sequences Day 2 Series Day 3 Permutations & Combinations Day 4 Normal Distribution & Empirical Formula Day 5 Normal Distribution * Day 6 Standard Normal Distribution
More informationWith Question/Answer Animations. Chapter 7
With Question/Answer Animations Chapter 7 Chapter Summary Introduction to Discrete Probability Probability Theory Bayes Theorem Section 7.1 Section Summary Finite Probability Probabilities of Complements
More informationMATH PRIZE FOR GIRLS. Test Version A
Advantage Testing Foundation Ath The Eighth rize For Annual irls MATH PRIZE FOR GIRLS Saturday, September 10, 2016 TEST BOOKLET Test Version A DIRECTIONS 1. Do not open this test until your proctor instructs
More informationMathematical Structures Combinations and Permutations
Definitions: Suppose S is a (finite) set and n, k 0 are integers The set C(S, k) of k - combinations consists of all subsets of S that have exactly k elements The set P (S, k) of k - permutations consists
More informationSolution: There are 30 choices for the first person to leave, 29 for the second, etc. Thus this exodus can occur in. = P (30, 8) ways.
Math-2320 Assignment 7 Solutions Problem 1: (Section 7.1 Exercise 4) There are 30 people in a class learning about permutations. One after another, eight people gradually slip out the back door. In how
More informationMATH 433 Applied Algebra Lecture 22: Review for Exam 2.
MATH 433 Applied Algebra Lecture 22: Review for Exam 2. Topics for Exam 2 Permutations Cycles, transpositions Cycle decomposition of a permutation Order of a permutation Sign of a permutation Symmetric
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationW3203 Discrete Mathema1cs. Coun1ng. Spring 2015 Instructor: Ilia Vovsha.
W3203 Discrete Mathema1cs Coun1ng Spring 2015 Instructor: Ilia Vovsha h@p://www.cs.columbia.edu/~vovsha/w3203 Outline Bijec1on rule Sum, product, division rules Permuta1ons and combina1ons Sequences with
More information( ) is called the dependent variable because its
page 1 of 16 CLASS NOTES: 3 8 thru 4 3 and 11 7 Functions, Exponents and Polynomials 3 8: Function Notation A function is a correspondence between two sets, the domain (x) and the range (y). An example
More informationDiscrete Probability
Discrete Probability Counting Permutations Combinations r- Combinations r- Combinations with repetition Allowed Pascal s Formula Binomial Theorem Conditional Probability Baye s Formula Independent Events
More informationthen the hard copy will not be correct whenever your instructor modifies the assignments.
Assignments for Math 2030 then the hard copy will not be correct whenever your instructor modifies the assignments. exams, but working through the problems is a good way to prepare for the exams. It is
More informationCDM Combinatorial Principles
CDM Combinatorial Principles 1 Counting Klaus Sutner Carnegie Mellon University Pigeon Hole 22-in-exclusion 2017/12/15 23:16 Inclusion/Exclusion Counting 3 Aside: Ranking and Unranking 4 Counting is arguably
More informationSESSION CLASS-XI SUBJECT : MATHEMATICS FIRST TERM
TERMWISE SYLLABUS SESSION-2018-19 CLASS-XI SUBJECT : MATHEMATICS MONTH July, 2018 to September 2018 CONTENTS FIRST TERM Unit-1: Sets and Functions 1. Sets Sets and their representations. Empty set. Finite
More informationProbability 1 (MATH 11300) lecture slides
Probability 1 (MATH 11300) lecture slides Márton Balázs School of Mathematics University of Bristol Autumn, 2015 December 16, 2015 To know... http://www.maths.bris.ac.uk/ mb13434/prob1/ m.balazs@bristol.ac.uk
More informationA is a subset of (contained in) B A B iff x A = x B Socrates is a man. All men are mortal. A = B iff A B and B A. A B means A is a proper subset of B
Subsets C-N Math 207 - Massey, 71 / 125 Sets A is a subset of (contained in) B A B iff x A = x B Socrates is a man. All men are mortal. A = B iff A B and B A x A x B A B means A is a proper subset of B
More informationSets. Subsets. for any set A, A and A A vacuously true: if x then x A transitivity: A B, B C = A C N Z Q R C. C-N Math Massey, 72 / 125
Subsets Sets A is a subset of (contained in) B A B iff x A = x B Socrates is a man. All men are mortal. A = B iff A B and B A x A x B A B means A is a proper subset of B A B but A B, so x B x / A Illustrate
More informationLecture 3: Miscellaneous Techniques
Lecture 3: Miscellaneous Techniques Rajat Mittal IIT Kanpur In this document, we will take a look at few diverse techniques used in combinatorics, exemplifying the fact that combinatorics is a collection
More informationSome Review Problems for Exam 3: Solutions
Math 3355 Spring 017 Some Review Problems for Exam 3: Solutions I thought I d start by reviewing some counting formulas. Counting the Complement: Given a set U (the universe for the problem), if you want
More information1. Foundations of Numerics from Advanced Mathematics. Mathematical Essentials and Notation
1. Foundations of Numerics from Advanced Mathematics Mathematical Essentials and Notation Mathematical Essentials and Notation, October 22, 2012 1 The main purpose of this first chapter (about 4 lectures)
More informationCombinations. April 12, 2006
Combinations April 12, 2006 Combinations, April 12, 2006 Binomial Coecients Denition. The number of distinct subsets with j elements that can be chosen from a set with n elements is denoted by ( n j).
More informationNotes Week 2 Chapter 3 Probability WEEK 2 page 1
Notes Week 2 Chapter 3 Probability WEEK 2 page 1 The sample space of an experiment, sometimes denoted S or in probability theory, is the set that consists of all possible elementary outcomes of that experiment
More informationNATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 22, Time Allowed: 150 Minutes Maximum Marks: 30
NATIONAL BOARD FOR HIGHER MATHEMATICS M A and MSc Scholarship Test September 22, 2018 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions that follow INSTRUCTIONS TO CANDIDATES
More informationMULTIPLYING POLYNOMIALS. The student is expected to multiply polynomials of degree one and degree two.
MULTIPLYING POLYNOMIALS A.10B The student is expected to multiply polynomials of degree one and degree two. TELL ME MORE A polynomial is an expression that is a sum of several terms. Polynomials may contain
More informationDo not open this exam until you are told to begin. You will have 75 minutes for the exam.
Math 2603 Midterm 1 Spring 2018 Your Name Student ID # Section Do not open this exam until you are told to begin. You will have 75 minutes for the exam. Check that you have a complete exam. There are 5
More informationLecture 4/12: Polar Form and Euler s Formula. 25 Jan 2007
Lecture 4/12: Polar Form and Euler s Formula MA154: Algebra for 1st Year IT Niall Madden Niall.Madden@NUIGalway.ie 25 Jan 2007 CS457 Lecture 4/12: Polar Form and Euler s Formula 1/17 Outline 1 Recall...
More informationSets. A set is a collection of objects without repeats. The size or cardinality of a set S is denoted S and is the number of elements in the set.
Sets A set is a collection of objects without repeats. The size or cardinality of a set S is denoted S and is the number of elements in the set. If A and B are sets, then the set of ordered pairs each
More information6 CARDINALITY OF SETS
6 CARDINALITY OF SETS MATH10111 - Foundations of Pure Mathematics We all have an idea of what it means to count a finite collection of objects, but we must be careful to define rigorously what it means
More informationExecutive Assessment. Executive Assessment Math Review. Section 1.0, Arithmetic, includes the following topics:
Executive Assessment Math Review Although the following provides a review of some of the mathematical concepts of arithmetic and algebra, it is not intended to be a textbook. You should use this chapter
More informationENGG 2440B Discrete Mathematics for Engineers Tutorial 8
ENGG 440B Discrete Mathematics for Engineers Tutorial 8 Jiajin Li Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong jjli@se.cuhk.edu.hk November, 018 Jiajin
More informationCombinatorial Analysis
Chapter 1 Combinatorial Analysis STAT 302, Department of Statistics, UBC 1 A starting example: coin tossing Consider the following random experiment: tossing a fair coin twice There are four possible outcomes,
More informationNamed numbres. Ngày 25 tháng 11 năm () Named numbres Ngày 25 tháng 11 năm / 7
Named numbres Ngày 25 tháng 11 năm 2011 () Named numbres Ngày 25 tháng 11 năm 2011 1 / 7 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. () Named
More information{ 0! = 1 n! = n(n 1)!, n 1. n! =
Summations Question? What is the sum of the first 100 positive integers? Counting Question? In how many ways may the first three horses in a 10 horse race finish? Multiplication Principle: If an event
More informationCombinations and Probabilities
Combinations and Probabilities Tutor: Zhang Qi Systems Engineering and Engineering Management qzhang@se.cuhk.edu.hk November 2014 Tutor: Zhang Qi (SEEM) Tutorial 7 November 2014 1 / 16 Combination Review
More informationµ (X) := inf l(i k ) where X k=1 I k, I k an open interval Notice that is a map from subsets of R to non-negative number together with infinity
A crash course in Lebesgue measure theory, Math 317, Intro to Analysis II These lecture notes are inspired by the third edition of Royden s Real analysis. The Jordan content is an attempt to extend the
More informationNEW YORK ALGEBRA TABLE OF CONTENTS
NEW YORK ALGEBRA TABLE OF CONTENTS CHAPTER 1 NUMBER SENSE & OPERATIONS TOPIC A: Number Theory: Properties of Real Numbers {A.N.1} PART 1: Closure...1 PART 2: Commutative Property...2 PART 3: Associative
More informationKnow the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.
The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring
More information2030 LECTURES. R. Craigen. Inclusion/Exclusion and Relations
2030 LECTURES R. Craigen Inclusion/Exclusion and Relations The Principle of Inclusion-Exclusion 7 ROS enumerates the union of disjoint sets. What if sets overlap? Some 17 out of 30 students in a class
More information(1) Which of the following are propositions? If it is a proposition, determine its truth value: A propositional function, but not a proposition.
Math 231 Exam Practice Problem Solutions WARNING: This is not a sample test. Problems on the exams may or may not be similar to these problems. These problems are just intended to focus your study of the
More informationManipulating Equations
Manipulating Equations Now that you know how to set up an equation, the next thing you need to do is solve for the value that the question asks for. Above all, the most important thing to remember when
More informationCounting. Mukulika Ghosh. Fall Based on slides by Dr. Hyunyoung Lee
Counting Mukulika Ghosh Fall 2018 Based on slides by Dr. Hyunyoung Lee Counting Counting The art of counting is known as enumerative combinatorics. One tries to count the number of elements in a set (or,
More informationOn Certain Sums of Stirling Numbers with Binomial Coefficients
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 18 (2015, Article 15.9.6 On Certain Sums of Stirling Numbers with Binomial Coefficients H. W. Gould Department of Mathematics West Virginia University
More information1. How many labeled trees are there on n vertices such that all odd numbered vertices are leaves?
1. How many labeled trees are there on n vertices such that all odd numbered vertices are leaves? This is most easily done by Prüfer codes. The number of times a vertex appears is the degree of the vertex.
More informationMath 564 Homework 1. Solutions.
Math 564 Homework 1. Solutions. Problem 1. Prove Proposition 0.2.2. A guide to this problem: start with the open set S = (a, b), for example. First assume that a >, and show that the number a has the properties
More informationModular Arithmetic Instructor: Marizza Bailey Name:
Modular Arithmetic Instructor: Marizza Bailey Name: 1. Introduction to Modular Arithmetic If someone asks you what day it is 145 days from now, what would you answer? Would you count 145 days, or find
More informationEquivalence of Propositions
Equivalence of Propositions 1. Truth tables: two same columns 2. Sequence of logical equivalences: from one to the other using equivalence laws 1 Equivalence laws Table 6 & 7 in 1.2, some often used: Associative:
More informationNotes slides from before lecture. CSE 21, Winter 2017, Section A00. Lecture 16 Notes. Class URL:
Notes slides from before lecture CSE 21, Winter 2017, Section A00 Lecture 16 Notes Class URL: http://vlsicad.ucsd.edu/courses/cse21-w17/ Notes slides from before lecture Notes March 8 (1) This week: Days
More informationand Other Combinatorial Reciprocity Instances
and Other Combinatorial Reciprocity Instances Matthias Beck San Francisco State University math.sfsu.edu/beck [Courtney Gibbons] Act 1: Binomial Coefficients Not everything that can be counted counts,
More informationProblems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman.
Math 224 Fall 2017 Homework 1 Drew Armstrong Problems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman. Section 1.1, Exercises 4,5,6,7,9,12. Solutions to Book Problems.
More informationChapter 8 Sequences, Series, and Probability
Chapter 8 Sequences, Series, and Probability Overview 8.1 Sequences and Series 8.2 Arithmetic Sequences and Partial Sums 8.3 Geometric Sequences and Partial Sums 8.5 The Binomial Theorem 8.6 Counting Principles
More informationMATH 556: PROBABILITY PRIMER
MATH 6: PROBABILITY PRIMER 1 DEFINITIONS, TERMINOLOGY, NOTATION 1.1 EVENTS AND THE SAMPLE SPACE Definition 1.1 An experiment is a one-off or repeatable process or procedure for which (a there is a well-defined
More informationMath 550 Notes. Chapter 2. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 2 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 2 Fall 2010 1 / 20 Linear algebra deals with finite dimensional
More informationMATH MW Elementary Probability Course Notes Part I: Models and Counting
MATH 2030 3.00MW Elementary Probability Course Notes Part I: Models and Counting Tom Salisbury salt@yorku.ca York University Winter 2010 Introduction [Jan 5] Probability: the mathematics used for Statistics
More informationHow do we analyze, evaluate, solve, and graph quadratic functions?
Topic: 4. Quadratic Functions and Factoring Days: 18 Key Learning: Students will be able to analyze, evaluate, solve and graph quadratic functions. Unit Essential Question(s): How do we analyze, evaluate,
More informationDiscrete Mathematics, Spring 2004 Homework 4 Sample Solutions
Discrete Mathematics, Spring 2004 Homework 4 Sample Solutions 4.2 #77. Let s n,k denote the number of ways to seat n persons at k round tables, with at least one person at each table. (The numbers s n,k
More informationMath 121. Practice Problems from Chapters 9, 10, 11 Fall 2016
Math 121. Practice Problems from Chapters 9, 10, 11 Fall 2016 Topic 1. Systems of Linear Equations in Two Variables 1. Solve systems of equations using elimination. For practice see Exercises 1, 2. 2.
More informationAnimals and 2-Motzkin Paths
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 8 (2005), Article 0556 Animals and 2-Motzkin Paths Wen-jin Woan 1 Department of Mathematics Howard University Washington, DC 20059 USA wwoan@howardedu
More informationWe are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero
Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.
More informationSolutions 2017 AB Exam
1. Solve for x : x 2 = 4 x. Solutions 2017 AB Exam Texas A&M High School Math Contest October 21, 2017 ANSWER: x = 3 Solution: x 2 = 4 x x 2 = 16 8x + x 2 x 2 9x + 18 = 0 (x 6)(x 3) = 0 x = 6, 3 but x
More informationCombinatorial Proofs and Algebraic Proofs I
Combinatorial Proofs and Algebraic Proofs I Shailesh A Shirali Shailesh A Shirali is Director of Sahyadri School (KFI), Pune, and also Head of the Community Mathematics Centre in Rishi Valley School (AP).
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationRegular Languages and Finite Automata
Regular Languages and Finite Automata Theorem: Every regular language is accepted by some finite automaton. Proof: We proceed by induction on the (length of/structure of) the description of the regular
More informationIowa State University. Instructor: Alex Roitershtein Summer Exam #1. Solutions. x u = 2 x v
Math 501 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 015 Exam #1 Solutions This is a take-home examination. The exam includes 8 questions.
More informationCSE 21 Practice Exam for Midterm 2 Fall 2017
CSE 1 Practice Exam for Midterm Fall 017 These practice problems should help prepare you for the second midterm, which is on monday, November 11 This is longer than the actual exam will be, but good practice
More informationProbability (Devore Chapter Two)
Probability (Devore Chapter Two) 1016-345-01: Probability and Statistics for Engineers Fall 2012 Contents 0 Administrata 2 0.1 Outline....................................... 3 1 Axiomatic Probability 3
More information